3.406 \(\int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\)

Optimal. Leaf size=1519 \[ -\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^2}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right ) a^4}{b^3 \left (a^2+b^2\right ) d^4}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {(e+f x)^3 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) a^3}{b^2 \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right ) a^3}{b^2 \left (a^2+b^2\right ) d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right ) a^3}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) a^3}{b^2 \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right ) a^3}{4 b^2 \left (a^2+b^2\right ) d^4}+\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right ) a^2}{b^3 d}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right ) a^2}{b^3 d^2}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right ) a^2}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right ) a^2}{b^3 d^3}-\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right ) a^2}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right ) a^2}{b^3 d^4}+\frac {(e+f x)^4 a}{4 b^2 f}-\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right ) a}{b^2 d}-\frac {3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right ) a}{2 b^2 d^2}+\frac {3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right ) a}{2 b^2 d^3}-\frac {3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right ) a}{4 b^2 d^4}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3} \]

[Out]

-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^4-6*a^3*f^3*polylog(4,-b*exp(d*x+c)/(a
+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^4+a^3*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/b^2/(a^2+b^2)/d-a^3*(f*x+e)^3*ln(1+b*e
xp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d-a^3*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b
^2)/d+2*a^2*(f*x+e)^3*arctan(exp(d*x+c))/b^3/d-3/4*a*f^3*polylog(4,-exp(2*d*x+2*c))/b^2/d^4-6*I*f^3*polylog(4,
I*exp(d*x+c))/b/d^4+3/2*a*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b^2/d^3+3/4*a^3*f^3*polylog(4,-exp(2*d*x+2*c)
)/b^2/(a^2+b^2)/d^4-3*I*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/b/d^2-6*I*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b/d
^3-6*I*a^2*f^3*polylog(4,-I*exp(d*x+c))/b^3/d^4-2*a^4*(f*x+e)^3*arctan(exp(d*x+c))/b^3/(a^2+b^2)/d-3/2*a*f*(f*
x+e)^2*polylog(2,-exp(2*d*x+2*c))/b^2/d^2+3*I*a^2*f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/b^3/d^2+6*I*a^2*f^2*(f*x
+e)*polylog(3,-I*exp(d*x+c))/b^3/d^3+6*I*a^4*f^3*polylog(4,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^4-3*I*a^4*f*(f*x+e)^
2*polylog(2,I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-6*I*a^4*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^3+3*I
*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b/d^2+3/2*a^3*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^2+6
*I*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b/d^3-3/2*a^3*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b^2/(a^2+b^2)/d^3+
6*I*a^2*f^3*polylog(4,I*exp(d*x+c))/b^3/d^4-3*I*a^2*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^3/d^2-6*I*a^2*f^2*(
f*x+e)*polylog(3,I*exp(d*x+c))/b^3/d^3-6*I*a^4*f^3*polylog(4,I*exp(d*x+c))/b^3/(a^2+b^2)/d^4+(f*x+e)^3*sinh(d*
x+c)/b/d-2*(f*x+e)^3*arctan(exp(d*x+c))/b/d-6*f^3*cosh(d*x+c)/b/d^4-3*f*(f*x+e)^2*cosh(d*x+c)/b/d^2+6*f^2*(f*x
+e)*sinh(d*x+c)/b/d^3+6*I*f^3*polylog(4,-I*exp(d*x+c))/b/d^4+1/4*a*(f*x+e)^4/b^2/f+6*I*a^4*f^2*(f*x+e)*polylog
(3,I*exp(d*x+c))/b^3/(a^2+b^2)/d^3+3*I*a^4*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^3/(a^2+b^2)/d^2-3*a^3*f*(f*x
+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2-3*a^3*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)
/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^2+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b^2/(a^
2+b^2)/d^3+6*a^3*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b^2/(a^2+b^2)/d^3-a*(f*x+e)^3*ln(1+e
xp(2*d*x+2*c))/b^2/d

________________________________________________________________________________________

Rubi [A]  time = 2.15, antiderivative size = 1519, normalized size of antiderivative = 1.00, number of steps used = 61, number of rules used = 15, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.441, Rules used = {5581, 5449, 3296, 2638, 4180, 2531, 6609, 2282, 6589, 3718, 2190, 5567, 5573, 5561, 6742} \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Int[((e + f*x)^3*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(a*(e + f*x)^4)/(4*b^2*f) + (2*a^2*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^3*d) - (2*(e + f*x)^3*ArcTan[E^(c + d*x
)])/(b*d) - (2*a^4*(e + f*x)^3*ArcTan[E^(c + d*x)])/(b^3*(a^2 + b^2)*d) - (6*f^3*Cosh[c + d*x])/(b*d^4) - (3*f
*(e + f*x)^2*Cosh[c + d*x])/(b*d^2) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])])/(b^2*(a
^2 + b^2)*d) - (a^3*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^2*(a^2 + b^2)*d) - (a*(e +
f*x)^3*Log[1 + E^(2*(c + d*x))])/(b^2*d) + (a^3*(e + f*x)^3*Log[1 + E^(2*(c + d*x))])/(b^2*(a^2 + b^2)*d) - ((
3*I)*a^2*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*d^2) + ((3*I)*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d
*x)])/(b*d^2) + ((3*I)*a^4*f*(e + f*x)^2*PolyLog[2, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) + ((3*I)*a^2*f*(e
+ f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^3*d^2) - ((3*I)*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b*d^2) - ((3
*I)*a^4*f*(e + f*x)^2*PolyLog[2, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*
E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (3*a^3*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x)
)/(a + Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^2) - (3*a*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*d^2
) + (3*a^3*f*(e + f*x)^2*PolyLog[2, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^2) + ((6*I)*a^2*f^2*(e + f*x)*Poly
Log[3, (-I)*E^(c + d*x)])/(b^3*d^3) - ((6*I)*f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b*d^3) - ((6*I)*a^4*
f^2*(e + f*x)*PolyLog[3, (-I)*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) - ((6*I)*a^2*f^2*(e + f*x)*PolyLog[3, I*E^(c
+ d*x)])/(b^3*d^3) + ((6*I)*f^2*(e + f*x)*PolyLog[3, I*E^(c + d*x)])/(b*d^3) + ((6*I)*a^4*f^2*(e + f*x)*PolyL
og[3, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2
+ b^2]))])/(b^2*(a^2 + b^2)*d^3) + (6*a^3*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/
(b^2*(a^2 + b^2)*d^3) + (3*a*f^2*(e + f*x)*PolyLog[3, -E^(2*(c + d*x))])/(2*b^2*d^3) - (3*a^3*f^2*(e + f*x)*Po
lyLog[3, -E^(2*(c + d*x))])/(2*b^2*(a^2 + b^2)*d^3) - ((6*I)*a^2*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^3*d^4) +
((6*I)*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b*d^4) + ((6*I)*a^4*f^3*PolyLog[4, (-I)*E^(c + d*x)])/(b^3*(a^2 + b
^2)*d^4) + ((6*I)*a^2*f^3*PolyLog[4, I*E^(c + d*x)])/(b^3*d^4) - ((6*I)*f^3*PolyLog[4, I*E^(c + d*x)])/(b*d^4)
- ((6*I)*a^4*f^3*PolyLog[4, I*E^(c + d*x)])/(b^3*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(
a - Sqrt[a^2 + b^2]))])/(b^2*(a^2 + b^2)*d^4) - (6*a^3*f^3*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))
])/(b^2*(a^2 + b^2)*d^4) - (3*a*f^3*PolyLog[4, -E^(2*(c + d*x))])/(4*b^2*d^4) + (3*a^3*f^3*PolyLog[4, -E^(2*(c
+ d*x))])/(4*b^2*(a^2 + b^2)*d^4) + (6*f^2*(e + f*x)*Sinh[c + d*x])/(b*d^3) + ((e + f*x)^3*Sinh[c + d*x])/(b*
d)

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]

Rule 5449

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 5561

Int[(Cosh[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Symbol] :
> -Simp[(e + f*x)^(m + 1)/(b*f*(m + 1)), x] + (Int[((e + f*x)^m*E^(c + d*x))/(a - Rt[a^2 + b^2, 2] + b*E^(c +
d*x)), x] + Int[((e + f*x)^m*E^(c + d*x))/(a + Rt[a^2 + b^2, 2] + b*E^(c + d*x)), x]) /; FreeQ[{a, b, c, d, e,
f}, x] && IGtQ[m, 0] && NeQ[a^2 + b^2, 0]

Rule 5567

Int[(((e_.) + (f_.)*(x_))^(m_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[1/b, Int[(e + f*x)^m*Sech[c + d*x]*Tanh[c + d*x]^(n - 1), x], x] - Dist[a/b, Int[((e + f*x)^m*Sec
h[c + d*x]*Tanh[c + d*x]^(n - 1))/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
&& IGtQ[n, 0]

Rule 5573

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sech[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sinh[(c_.) + (d_.)*(x_)]), x_Sym
bol] :> Dist[b^2/(a^2 + b^2), Int[((e + f*x)^m*Sech[c + d*x]^(n - 2))/(a + b*Sinh[c + d*x]), x], x] + Dist[1/(
a^2 + b^2), Int[(e + f*x)^m*Sech[c + d*x]^n*(a - b*Sinh[c + d*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && I
GtQ[m, 0] && NeQ[a^2 + b^2, 0] && IGtQ[n, 0]

Rule 5581

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sinh[(c_.) + (d_.)*(x_)]^(p_.)*Tanh[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*S
inh[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[1/b, Int[(e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n, x], x]
- Dist[a/b, Int[((e + f*x)^m*Sinh[c + d*x]^(p - 1)*Tanh[c + d*x]^n)/(a + b*Sinh[c + d*x]), x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && IGtQ[m, 0] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \sinh ^2(c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \sinh (c+d x) \tanh (c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {a \int (e+f x)^3 \tanh (c+d x) \, dx}{b^2}+\frac {a^2 \int \frac {(e+f x)^3 \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b^2}+\frac {\int (e+f x)^3 \cosh (c+d x) \, dx}{b}-\frac {\int (e+f x)^3 \text {sech}(c+d x) \, dx}{b}\\ &=\frac {a (e+f x)^4}{4 b^2 f}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {a^2 \int (e+f x)^3 \text {sech}(c+d x) \, dx}{b^3}-\frac {a^3 \int \frac {(e+f x)^3 \text {sech}(c+d x)}{a+b \sinh (c+d x)} \, dx}{b^3}-\frac {(2 a) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{b^2}+\frac {(3 i f) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b d}-\frac {(3 i f) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b d}-\frac {(3 f) \int (e+f x)^2 \sinh (c+d x) \, dx}{b d}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {a^3 \int (e+f x)^3 \text {sech}(c+d x) (a-b \sinh (c+d x)) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {(e+f x)^3 \cosh (c+d x)}{a+b \sinh (c+d x)} \, dx}{b \left (a^2+b^2\right )}-\frac {\left (3 i a^2 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 d}+\frac {\left (3 i a^2 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 d}+\frac {(3 a f) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 d}-\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b d^2}+\frac {\left (6 f^2\right ) \int (e+f x) \cosh (c+d x) \, dx}{b d^2}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {a^3 (e+f x)^4}{4 b^2 \left (a^2+b^2\right ) f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {a^3 \int \left (a (e+f x)^3 \text {sech}(c+d x)-b (e+f x)^3 \tanh (c+d x)\right ) \, dx}{b^3 \left (a^2+b^2\right )}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^3}{a-\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}-\frac {a^3 \int \frac {e^{c+d x} (e+f x)^3}{a+\sqrt {a^2+b^2}+b e^{c+d x}} \, dx}{b \left (a^2+b^2\right )}+\frac {\left (6 i a^2 f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 d^2}-\frac {\left (6 i a^2 f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 d^2}+\frac {\left (3 a f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 d^2}+\frac {\left (6 i f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{b d^3}-\frac {\left (6 i f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{b d^3}-\frac {\left (6 f^3\right ) \int \sinh (c+d x) \, dx}{b d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {a^3 (e+f x)^4}{4 b^2 \left (a^2+b^2\right ) f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {a^4 \int (e+f x)^3 \text {sech}(c+d x) \, dx}{b^3 \left (a^2+b^2\right )}+\frac {a^3 \int (e+f x)^3 \tanh (c+d x) \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^4}-\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b d^4}-\frac {\left (6 i a^2 f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{b^3 d^3}+\frac {\left (6 i a^2 f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{b^3 d^3}-\frac {\left (3 a f^3\right ) \int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 b^2 d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {\left (2 a^3\right ) \int \frac {e^{2 (c+d x)} (e+f x)^3}{1+e^{2 (c+d x)}} \, dx}{b^2 \left (a^2+b^2\right )}+\frac {\left (3 i a^4 f\right ) \int (e+f x)^2 \log \left (1-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}-\frac {\left (3 i a^4 f\right ) \int (e+f x)^2 \log \left (1+i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}+\frac {\left (6 a^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}-\frac {\left (6 i a^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4}+\frac {\left (6 i a^2 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 d^4}-\frac {\left (3 a f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 b^2 d^4}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {6 i a^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac {3 a f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {\left (3 a^3 f\right ) \int (e+f x)^2 \log \left (1+e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d}-\frac {\left (6 i a^4 f^2\right ) \int (e+f x) \text {Li}_2\left (-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}+\frac {\left (6 i a^4 f^2\right ) \int (e+f x) \text {Li}_2\left (i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^2}-\frac {\left (6 a^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^3}-\frac {\left (6 a^3 f^3\right ) \int \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {6 i a^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac {3 a f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}-\frac {\left (3 a^3 f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{2 (c+d x)}\right ) \, dx}{b^2 \left (a^2+b^2\right ) d^2}-\frac {\left (6 a^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {b x}{-a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {\left (6 a^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {b x}{a+\sqrt {a^2+b^2}}\right )}{x} \, dx,x,e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {\left (6 i a^4 f^3\right ) \int \text {Li}_3\left (-i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^3}-\frac {\left (6 i a^4 f^3\right ) \int \text {Li}_3\left (i e^{c+d x}\right ) \, dx}{b^3 \left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac {6 i a^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {3 a f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {\left (6 i a^4 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac {\left (6 i a^4 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac {\left (3 a^3 f^3\right ) \int \text {Li}_3\left (-e^{2 (c+d x)}\right ) \, dx}{2 b^2 \left (a^2+b^2\right ) d^3}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac {6 i a^4 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac {6 i a^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac {6 i a^4 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {3 a f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}+\frac {\left (3 a^3 f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{2 (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right ) d^4}\\ &=\frac {a (e+f x)^4}{4 b^2 f}+\frac {2 a^2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 d}-\frac {2 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b d}-\frac {2 a^4 (e+f x)^3 \tan ^{-1}\left (e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d}-\frac {6 f^3 \cosh (c+d x)}{b d^4}-\frac {3 f (e+f x)^2 \cosh (c+d x)}{b d^2}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a^3 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {a (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 d}+\frac {a^3 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b^2 \left (a^2+b^2\right ) d}-\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 d^2}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b d^2}+\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}+\frac {3 i a^2 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b d^2}-\frac {3 i a^4 f (e+f x)^2 \text {Li}_2\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 a f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 d^2}+\frac {3 a^3 f (e+f x)^2 \text {Li}_2\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^2}+\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 d^3}-\frac {6 i f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b d^3}-\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 d^3}+\frac {6 i f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b d^3}+\frac {6 i a^4 f^2 (e+f x) \text {Li}_3\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 a^3 f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {3 a f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 d^3}-\frac {3 a^3 f^2 (e+f x) \text {Li}_3\left (-e^{2 (c+d x)}\right )}{2 b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a^2 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 d^4}+\frac {6 i f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b d^4}+\frac {6 i a^4 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}+\frac {6 i a^2 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 d^4}-\frac {6 i f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b d^4}-\frac {6 i a^4 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^3 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 a^3 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {3 a f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 d^4}+\frac {3 a^3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b^2 \left (a^2+b^2\right ) d^4}+\frac {6 f^2 (e+f x) \sinh (c+d x)}{b d^3}+\frac {(e+f x)^3 \sinh (c+d x)}{b d}\\ \end {align*}

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Mathematica [A]  time = 23.37, size = 2861, normalized size = 1.88 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)^3*Sinh[c + d*x]^2*Tanh[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-1/4*(8*b*d^3*e^3*(1 + E^(2*c))*ArcTan[E^(c + d*x)] - 4*a*d^3*e^3*E^(2*c)*(2*d*x - Log[1 + E^(2*(c + d*x))]) +
4*a*d^3*e^3*Log[1 + E^(2*(c + d*x))] + (12*I)*b*d^2*e^2*(1 + E^(2*c))*f*(d*x*(Log[1 - I*E^(c + d*x)] - Log[1
+ I*E^(c + d*x)]) - PolyLog[2, (-I)*E^(c + d*x)] + PolyLog[2, I*E^(c + d*x)]) - 6*a*d^2*e^2*E^(2*c)*f*(2*d*x*(
d*x - Log[1 + E^(2*(c + d*x))]) - PolyLog[2, -E^(2*(c + d*x))]) + 6*a*d^2*e^2*f*(2*d*x*Log[1 + E^(2*(c + d*x))
] + PolyLog[2, -E^(2*(c + d*x))]) + (12*I)*b*d*e*(1 + E^(2*c))*f^2*(d^2*x^2*Log[1 - I*E^(c + d*x)] - d^2*x^2*L
og[1 + I*E^(c + d*x)] - 2*d*x*PolyLog[2, (-I)*E^(c + d*x)] + 2*d*x*PolyLog[2, I*E^(c + d*x)] + 2*PolyLog[3, (-
I)*E^(c + d*x)] - 2*PolyLog[3, I*E^(c + d*x)]) + 6*a*d*e*f^2*(2*d^2*x^2*Log[1 + E^(2*(c + d*x))] + 2*d*x*PolyL
og[2, -E^(2*(c + d*x))] - PolyLog[3, -E^(2*(c + d*x))]) - 2*a*d*e*E^(2*c)*f^2*(2*d^2*x^2*(2*d*x - 3*Log[1 + E^
(2*(c + d*x))]) - 6*d*x*PolyLog[2, -E^(2*(c + d*x))] + 3*PolyLog[3, -E^(2*(c + d*x))]) + (4*I)*b*(1 + E^(2*c))
*f^3*(d^3*x^3*Log[1 - I*E^(c + d*x)] - d^3*x^3*Log[1 + I*E^(c + d*x)] - 3*d^2*x^2*PolyLog[2, (-I)*E^(c + d*x)]
+ 3*d^2*x^2*PolyLog[2, I*E^(c + d*x)] + 6*d*x*PolyLog[3, (-I)*E^(c + d*x)] - 6*d*x*PolyLog[3, I*E^(c + d*x)]
- 6*PolyLog[4, (-I)*E^(c + d*x)] + 6*PolyLog[4, I*E^(c + d*x)]) - a*E^(2*c)*f^3*(2*d^4*x^4 - 4*d^3*x^3*Log[1 +
E^(2*(c + d*x))] - 6*d^2*x^2*PolyLog[2, -E^(2*(c + d*x))] + 6*d*x*PolyLog[3, -E^(2*(c + d*x))] - 3*PolyLog[4,
-E^(2*(c + d*x))]) + a*f^3*(4*d^3*x^3*Log[1 + E^(2*(c + d*x))] + 6*d^2*x^2*PolyLog[2, -E^(2*(c + d*x))] - 6*d
*x*PolyLog[3, -E^(2*(c + d*x))] + 3*PolyLog[4, -E^(2*(c + d*x))]))/((a^2 + b^2)*d^4*(1 + E^(2*c))) + (a^3*(4*e
^3*E^(2*c)*x + 6*e^2*E^(2*c)*f*x^2 + 4*e*E^(2*c)*f^2*x^3 + E^(2*c)*f^3*x^4 + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(
2*c)*ArcTan[(a + b*E^(c + d*x))/Sqrt[-a^2 - b^2]])/((a^2 + b^2)^(3/2)*d) + (4*a*Sqrt[-(a^2 + b^2)^2]*e^3*E^(2*
c)*ArcTanh[(a + b*E^(c + d*x))/Sqrt[a^2 + b^2]])/((-a^2 - b^2)^(3/2)*d) + (2*e^3*Log[b - 2*a*E^(c + d*x) - b*E
^(2*(c + d*x))])/d - (2*e^3*E^(2*c)*Log[2*a*E^(c + d*x) + b*(-1 + E^(2*(c + d*x)))])/d + (6*e^2*f*x*Log[1 + (b
*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c
- Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])]
)/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[
1 + (b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(
a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e^2*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c
)])])/d - (6*e^2*E^(2*c)*f*x*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (6*e*f^2*x^2*
Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*e*E^(2*c)*f^2*x^2*Log[1 + (b*E^(2*c + d
*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d + (2*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*
E^(2*c)])])/d - (2*E^(2*c)*f^3*x^3*Log[1 + (b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)])])/d - (6*(-1
+ E^(2*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (6*(-1 +
E^(2*c))*f*(e + f*x)^2*PolyLog[2, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^2 - (12*e*f^2*P
olyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e*E^(2*c)*f^2*PolyLog[3, -((b*E
^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - S
qrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*
E^(2*c)]))])/d^3 - (12*e*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*e
*E^(2*c)*f^2*PolyLog[3, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 - (12*f^3*x*PolyLog[3,
-((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*E^(2*c)*f^3*x*PolyLog[3, -((b*E^(2*c + d*
x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^3 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b
^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c - Sqrt[(a^2 + b^2)*E^(2*c)]))])/d
^4 + (12*f^3*PolyLog[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4 - (12*E^(2*c)*f^3*PolyL
og[4, -((b*E^(2*c + d*x))/(a*E^c + Sqrt[(a^2 + b^2)*E^(2*c)]))])/d^4))/(2*b^2*(a^2 + b^2)*(-1 + E^(2*c))) - (a
*e^3*x*(a^2 - b^2 + (a^2 + b^2)*Cosh[2*c])*Csch[c]*Sech[c])/(2*b^2*(a^2 + b^2)) - (3*a*e^2*f*x^2*(a^2 - b^2 +
(a^2 + b^2)*Cosh[2*c])*Csch[c]*Sech[c])/(4*b^2*(a^2 + b^2)) - (a*e*f^2*x^3*(a^2 - b^2 + (a^2 + b^2)*Cosh[2*c])
*Csch[c]*Sech[c])/(2*b^2*(a^2 + b^2)) - (a*f^3*x^4*(a^2 - b^2 + (a^2 + b^2)*Cosh[2*c])*Csch[c]*Sech[c])/(8*b^2
*(a^2 + b^2)) + ((6*f^3 + 6*d*f^2*(e + f*x) + 3*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(-Cosh[c + d*x] + Sinh[c
+ d*x]))/(2*b*d^4) + ((-6*f^3 + 6*d*f^2*(e + f*x) - 3*d^2*f*(e + f*x)^2 + d^3*(e + f*x)^3)*(Cosh[c + d*x] + Si
nh[c + d*x]))/(2*b*d^4)

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fricas [C]  time = 0.74, size = 4546, normalized size = 2.99 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/4*(2*(a^2*b + b^3)*d^3*f^3*x^3 + 2*(a^2*b + b^3)*d^3*e^3 + 6*(a^2*b + b^3)*d^2*e^2*f + 12*(a^2*b + b^3)*d*e
*f^2 + 12*(a^2*b + b^3)*f^3 + 6*((a^2*b + b^3)*d^3*e*f^2 + (a^2*b + b^3)*d^2*f^3)*x^2 - 2*((a^2*b + b^3)*d^3*f
^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3
*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^
2 + 2*(a^2*b + b^3)*d*f^3)*x)*cosh(d*x + c)^2 - 2*((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*
b + b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3
)*d^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x)*sinh(d*x +
c)^2 + 6*((a^2*b + b^3)*d^3*e^2*f + 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x - ((a^3 + a*b^2)*d^4
*f^3*x^4 + 4*(a^3 + a*b^2)*d^4*e*f^2*x^3 + 6*(a^3 + a*b^2)*d^4*e^2*f*x^2 + 4*(a^3 + a*b^2)*d^4*e^3*x + 8*(a^3
+ a*b^2)*c*d^3*e^3 - 12*(a^3 + a*b^2)*c^2*d^2*e^2*f + 8*(a^3 + a*b^2)*c^3*d*e*f^2 - 2*(a^3 + a*b^2)*c^4*f^3)*c
osh(d*x + c) + 12*((a^3*d^2*f^3*x^2 + 2*a^3*d^2*e*f^2*x + a^3*d^2*e^2*f)*cosh(d*x + c) + (a^3*d^2*f^3*x^2 + 2*
a^3*d^2*e*f^2*x + a^3*d^2*e^2*f)*sinh(d*x + c))*dilog((a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) +
b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + 12*((a^3*d^2*f^3*x^2 + 2*a^3*d^2*e*f^2*x + a^3*d^2*e^2*f)
*cosh(d*x + c) + (a^3*d^2*f^3*x^2 + 2*a^3*d^2*e*f^2*x + a^3*d^2*e^2*f)*sinh(d*x + c))*dilog((a*cosh(d*x + c) +
a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b + 1) + ((12*a*b^2*d^2*f^3*
x^2 + 12*I*b^3*d^2*f^3*x^2 + 24*a*b^2*d^2*e*f^2*x + 24*I*b^3*d^2*e*f^2*x + 12*a*b^2*d^2*e^2*f + 12*I*b^3*d^2*e
^2*f)*cosh(d*x + c) + (12*a*b^2*d^2*f^3*x^2 + 12*I*b^3*d^2*f^3*x^2 + 24*a*b^2*d^2*e*f^2*x + 24*I*b^3*d^2*e*f^2
*x + 12*a*b^2*d^2*e^2*f + 12*I*b^3*d^2*e^2*f)*sinh(d*x + c))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + ((12*a
*b^2*d^2*f^3*x^2 - 12*I*b^3*d^2*f^3*x^2 + 24*a*b^2*d^2*e*f^2*x - 24*I*b^3*d^2*e*f^2*x + 12*a*b^2*d^2*e^2*f - 1
2*I*b^3*d^2*e^2*f)*cosh(d*x + c) + (12*a*b^2*d^2*f^3*x^2 - 12*I*b^3*d^2*f^3*x^2 + 24*a*b^2*d^2*e*f^2*x - 24*I*
b^3*d^2*e*f^2*x + 12*a*b^2*d^2*e^2*f - 12*I*b^3*d^2*e^2*f)*sinh(d*x + c))*dilog(-I*cosh(d*x + c) - I*sinh(d*x
+ c)) + 4*((a^3*d^3*e^3 - 3*a^3*c*d^2*e^2*f + 3*a^3*c^2*d*e*f^2 - a^3*c^3*f^3)*cosh(d*x + c) + (a^3*d^3*e^3 -
3*a^3*c*d^2*e^2*f + 3*a^3*c^2*d*e*f^2 - a^3*c^3*f^3)*sinh(d*x + c))*log(2*b*cosh(d*x + c) + 2*b*sinh(d*x + c)
+ 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 4*((a^3*d^3*e^3 - 3*a^3*c*d^2*e^2*f + 3*a^3*c^2*d*e*f^2 - a^3*c^3*f^3)*co
sh(d*x + c) + (a^3*d^3*e^3 - 3*a^3*c*d^2*e^2*f + 3*a^3*c^2*d*e*f^2 - a^3*c^3*f^3)*sinh(d*x + c))*log(2*b*cosh(
d*x + c) + 2*b*sinh(d*x + c) - 2*b*sqrt((a^2 + b^2)/b^2) + 2*a) + 4*((a^3*d^3*f^3*x^3 + 3*a^3*d^3*e*f^2*x^2 +
3*a^3*d^3*e^2*f*x + 3*a^3*c*d^2*e^2*f - 3*a^3*c^2*d*e*f^2 + a^3*c^3*f^3)*cosh(d*x + c) + (a^3*d^3*f^3*x^3 + 3*
a^3*d^3*e*f^2*x^2 + 3*a^3*d^3*e^2*f*x + 3*a^3*c*d^2*e^2*f - 3*a^3*c^2*d*e*f^2 + a^3*c^3*f^3)*sinh(d*x + c))*lo
g(-(a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + 4*
((a^3*d^3*f^3*x^3 + 3*a^3*d^3*e*f^2*x^2 + 3*a^3*d^3*e^2*f*x + 3*a^3*c*d^2*e^2*f - 3*a^3*c^2*d*e*f^2 + a^3*c^3*
f^3)*cosh(d*x + c) + (a^3*d^3*f^3*x^3 + 3*a^3*d^3*e*f^2*x^2 + 3*a^3*d^3*e^2*f*x + 3*a^3*c*d^2*e^2*f - 3*a^3*c^
2*d*e*f^2 + a^3*c^3*f^3)*sinh(d*x + c))*log(-(a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*
x + c))*sqrt((a^2 + b^2)/b^2) - b)/b) + ((4*a*b^2*d^3*e^3 + 4*I*b^3*d^3*e^3 - 12*a*b^2*c*d^2*e^2*f - 12*I*b^3*
c*d^2*e^2*f + 12*a*b^2*c^2*d*e*f^2 + 12*I*b^3*c^2*d*e*f^2 - 4*a*b^2*c^3*f^3 - 4*I*b^3*c^3*f^3)*cosh(d*x + c) +
(4*a*b^2*d^3*e^3 + 4*I*b^3*d^3*e^3 - 12*a*b^2*c*d^2*e^2*f - 12*I*b^3*c*d^2*e^2*f + 12*a*b^2*c^2*d*e*f^2 + 12*
I*b^3*c^2*d*e*f^2 - 4*a*b^2*c^3*f^3 - 4*I*b^3*c^3*f^3)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + I) +
((4*a*b^2*d^3*e^3 - 4*I*b^3*d^3*e^3 - 12*a*b^2*c*d^2*e^2*f + 12*I*b^3*c*d^2*e^2*f + 12*a*b^2*c^2*d*e*f^2 - 12
*I*b^3*c^2*d*e*f^2 - 4*a*b^2*c^3*f^3 + 4*I*b^3*c^3*f^3)*cosh(d*x + c) + (4*a*b^2*d^3*e^3 - 4*I*b^3*d^3*e^3 - 1
2*a*b^2*c*d^2*e^2*f + 12*I*b^3*c*d^2*e^2*f + 12*a*b^2*c^2*d*e*f^2 - 12*I*b^3*c^2*d*e*f^2 - 4*a*b^2*c^3*f^3 + 4
*I*b^3*c^3*f^3)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - I) + ((4*a*b^2*d^3*f^3*x^3 - 4*I*b^3*d^3*f^
3*x^3 + 12*a*b^2*d^3*e*f^2*x^2 - 12*I*b^3*d^3*e*f^2*x^2 + 12*a*b^2*d^3*e^2*f*x - 12*I*b^3*d^3*e^2*f*x + 12*a*b
^2*c*d^2*e^2*f - 12*I*b^3*c*d^2*e^2*f - 12*a*b^2*c^2*d*e*f^2 + 12*I*b^3*c^2*d*e*f^2 + 4*a*b^2*c^3*f^3 - 4*I*b^
3*c^3*f^3)*cosh(d*x + c) + (4*a*b^2*d^3*f^3*x^3 - 4*I*b^3*d^3*f^3*x^3 + 12*a*b^2*d^3*e*f^2*x^2 - 12*I*b^3*d^3*
e*f^2*x^2 + 12*a*b^2*d^3*e^2*f*x - 12*I*b^3*d^3*e^2*f*x + 12*a*b^2*c*d^2*e^2*f - 12*I*b^3*c*d^2*e^2*f - 12*a*b
^2*c^2*d*e*f^2 + 12*I*b^3*c^2*d*e*f^2 + 4*a*b^2*c^3*f^3 - 4*I*b^3*c^3*f^3)*sinh(d*x + c))*log(I*cosh(d*x + c)
+ I*sinh(d*x + c) + 1) + ((4*a*b^2*d^3*f^3*x^3 + 4*I*b^3*d^3*f^3*x^3 + 12*a*b^2*d^3*e*f^2*x^2 + 12*I*b^3*d^3*e
*f^2*x^2 + 12*a*b^2*d^3*e^2*f*x + 12*I*b^3*d^3*e^2*f*x + 12*a*b^2*c*d^2*e^2*f + 12*I*b^3*c*d^2*e^2*f - 12*a*b^
2*c^2*d*e*f^2 - 12*I*b^3*c^2*d*e*f^2 + 4*a*b^2*c^3*f^3 + 4*I*b^3*c^3*f^3)*cosh(d*x + c) + (4*a*b^2*d^3*f^3*x^3
+ 4*I*b^3*d^3*f^3*x^3 + 12*a*b^2*d^3*e*f^2*x^2 + 12*I*b^3*d^3*e*f^2*x^2 + 12*a*b^2*d^3*e^2*f*x + 12*I*b^3*d^3
*e^2*f*x + 12*a*b^2*c*d^2*e^2*f + 12*I*b^3*c*d^2*e^2*f - 12*a*b^2*c^2*d*e*f^2 - 12*I*b^3*c^2*d*e*f^2 + 4*a*b^2
*c^3*f^3 + 4*I*b^3*c^3*f^3)*sinh(d*x + c))*log(-I*cosh(d*x + c) - I*sinh(d*x + c) + 1) + 24*(a^3*f^3*cosh(d*x
+ c) + a^3*f^3*sinh(d*x + c))*polylog(4, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x +
c))*sqrt((a^2 + b^2)/b^2))/b) + 24*(a^3*f^3*cosh(d*x + c) + a^3*f^3*sinh(d*x + c))*polylog(4, (a*cosh(d*x + c)
+ a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b) + ((24*a*b^2*f^3 + 24*I*b^3
*f^3)*cosh(d*x + c) + (24*a*b^2*f^3 + 24*I*b^3*f^3)*sinh(d*x + c))*polylog(4, I*cosh(d*x + c) + I*sinh(d*x + c
)) + ((24*a*b^2*f^3 - 24*I*b^3*f^3)*cosh(d*x + c) + (24*a*b^2*f^3 - 24*I*b^3*f^3)*sinh(d*x + c))*polylog(4, -I
*cosh(d*x + c) - I*sinh(d*x + c)) - 24*((a^3*d*f^3*x + a^3*d*e*f^2)*cosh(d*x + c) + (a^3*d*f^3*x + a^3*d*e*f^2
)*sinh(d*x + c))*polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) + (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2
+ b^2)/b^2))/b) - 24*((a^3*d*f^3*x + a^3*d*e*f^2)*cosh(d*x + c) + (a^3*d*f^3*x + a^3*d*e*f^2)*sinh(d*x + c))*
polylog(3, (a*cosh(d*x + c) + a*sinh(d*x + c) - (b*cosh(d*x + c) + b*sinh(d*x + c))*sqrt((a^2 + b^2)/b^2))/b)
- ((24*a*b^2*d*f^3*x + 24*I*b^3*d*f^3*x + 24*a*b^2*d*e*f^2 + 24*I*b^3*d*e*f^2)*cosh(d*x + c) + (24*a*b^2*d*f^3
*x + 24*I*b^3*d*f^3*x + 24*a*b^2*d*e*f^2 + 24*I*b^3*d*e*f^2)*sinh(d*x + c))*polylog(3, I*cosh(d*x + c) + I*sin
h(d*x + c)) - ((24*a*b^2*d*f^3*x - 24*I*b^3*d*f^3*x + 24*a*b^2*d*e*f^2 - 24*I*b^3*d*e*f^2)*cosh(d*x + c) + (24
*a*b^2*d*f^3*x - 24*I*b^3*d*f^3*x + 24*a*b^2*d*e*f^2 - 24*I*b^3*d*e*f^2)*sinh(d*x + c))*polylog(3, -I*cosh(d*x
+ c) - I*sinh(d*x + c)) - ((a^3 + a*b^2)*d^4*f^3*x^4 + 4*(a^3 + a*b^2)*d^4*e*f^2*x^3 + 6*(a^3 + a*b^2)*d^4*e^
2*f*x^2 + 4*(a^3 + a*b^2)*d^4*e^3*x + 8*(a^3 + a*b^2)*c*d^3*e^3 - 12*(a^3 + a*b^2)*c^2*d^2*e^2*f + 8*(a^3 + a*
b^2)*c^3*d*e*f^2 - 2*(a^3 + a*b^2)*c^4*f^3 + 4*((a^2*b + b^3)*d^3*f^3*x^3 + (a^2*b + b^3)*d^3*e^3 - 3*(a^2*b +
b^3)*d^2*e^2*f + 6*(a^2*b + b^3)*d*e*f^2 - 6*(a^2*b + b^3)*f^3 + 3*((a^2*b + b^3)*d^3*e*f^2 - (a^2*b + b^3)*d
^2*f^3)*x^2 + 3*((a^2*b + b^3)*d^3*e^2*f - 2*(a^2*b + b^3)*d^2*e*f^2 + 2*(a^2*b + b^3)*d*f^3)*x)*cosh(d*x + c)
)*sinh(d*x + c))/((a^2*b^2 + b^4)*d^4*cosh(d*x + c) + (a^2*b^2 + b^4)*d^4*sinh(d*x + c))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 2.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \left (\sinh ^{2}\left (d x +c \right )\right ) \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

int((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, {\left (\frac {2 \, a^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{2} b^{2} + b^{4}\right )} d} - \frac {4 \, b \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {2 \, a \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{2} + b^{2}\right )} d} + \frac {2 \, {\left (d x + c\right )} a}{b^{2} d} - \frac {e^{\left (d x + c\right )}}{b d} + \frac {e^{\left (-d x - c\right )}}{b d}\right )} e^{3} - \frac {{\left (a d^{4} f^{3} x^{4} e^{c} + 4 \, a d^{4} e f^{2} x^{3} e^{c} + 6 \, a d^{4} e^{2} f x^{2} e^{c} - 2 \, {\left (b d^{3} f^{3} x^{3} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e f^{2} - d^{2} f^{3}\right )} b x^{2} e^{\left (2 \, c\right )} + 3 \, {\left (d^{3} e^{2} f - 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} b x e^{\left (2 \, c\right )} - 3 \, {\left (d^{2} e^{2} f - 2 \, d e f^{2} + 2 \, f^{3}\right )} b e^{\left (2 \, c\right )}\right )} e^{\left (d x\right )} + 2 \, {\left (b d^{3} f^{3} x^{3} + 3 \, {\left (d^{3} e f^{2} + d^{2} f^{3}\right )} b x^{2} + 3 \, {\left (d^{3} e^{2} f + 2 \, d^{2} e f^{2} + 2 \, d f^{3}\right )} b x + 3 \, {\left (d^{2} e^{2} f + 2 \, d e f^{2} + 2 \, f^{3}\right )} b\right )} e^{\left (-d x\right )}\right )} e^{\left (-c\right )}}{4 \, b^{2} d^{4}} + \int \frac {2 \, {\left (a^{3} b f^{3} x^{3} + 3 \, a^{3} b e f^{2} x^{2} + 3 \, a^{3} b e^{2} f x - {\left (a^{4} f^{3} x^{3} e^{c} + 3 \, a^{4} e f^{2} x^{2} e^{c} + 3 \, a^{4} e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} b^{3} + b^{5} - {\left (a^{2} b^{3} e^{\left (2 \, c\right )} + b^{5} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )} - 2 \, {\left (a^{3} b^{2} e^{c} + a b^{4} e^{c}\right )} e^{\left (d x\right )}}\,{d x} - \int -\frac {2 \, {\left (a f^{3} x^{3} + 3 \, a e f^{2} x^{2} + 3 \, a e^{2} f x - {\left (b f^{3} x^{3} e^{c} + 3 \, b e f^{2} x^{2} e^{c} + 3 \, b e^{2} f x e^{c}\right )} e^{\left (d x\right )}\right )}}{a^{2} + b^{2} + {\left (a^{2} e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^3*sinh(d*x+c)^2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(2*a^3*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b^2 + b^4)*d) - 4*b*arctan(e^(-d*x - c))/((a
^2 + b^2)*d) + 2*a*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + 2*(d*x + c)*a/(b^2*d) - e^(d*x + c)/(b*d) + e^(
-d*x - c)/(b*d))*e^3 - 1/4*(a*d^4*f^3*x^4*e^c + 4*a*d^4*e*f^2*x^3*e^c + 6*a*d^4*e^2*f*x^2*e^c - 2*(b*d^3*f^3*x
^3*e^(2*c) + 3*(d^3*e*f^2 - d^2*f^3)*b*x^2*e^(2*c) + 3*(d^3*e^2*f - 2*d^2*e*f^2 + 2*d*f^3)*b*x*e^(2*c) - 3*(d^
2*e^2*f - 2*d*e*f^2 + 2*f^3)*b*e^(2*c))*e^(d*x) + 2*(b*d^3*f^3*x^3 + 3*(d^3*e*f^2 + d^2*f^3)*b*x^2 + 3*(d^3*e^
2*f + 2*d^2*e*f^2 + 2*d*f^3)*b*x + 3*(d^2*e^2*f + 2*d*e*f^2 + 2*f^3)*b)*e^(-d*x))*e^(-c)/(b^2*d^4) + integrate
(2*(a^3*b*f^3*x^3 + 3*a^3*b*e*f^2*x^2 + 3*a^3*b*e^2*f*x - (a^4*f^3*x^3*e^c + 3*a^4*e*f^2*x^2*e^c + 3*a^4*e^2*f
*x*e^c)*e^(d*x))/(a^2*b^3 + b^5 - (a^2*b^3*e^(2*c) + b^5*e^(2*c))*e^(2*d*x) - 2*(a^3*b^2*e^c + a*b^4*e^c)*e^(d
*x)), x) - integrate(-2*(a*f^3*x^3 + 3*a*e*f^2*x^2 + 3*a*e^2*f*x - (b*f^3*x^3*e^c + 3*b*e*f^2*x^2*e^c + 3*b*e^
2*f*x*e^c)*e^(d*x))/(a^2 + b^2 + (a^2*e^(2*c) + b^2*e^(2*c))*e^(2*d*x)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2\,\mathrm {tanh}\left (c+d\,x\right )\,{\left (e+f\,x\right )}^3}{a+b\,\mathrm {sinh}\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)),x)

[Out]

int((sinh(c + d*x)^2*tanh(c + d*x)*(e + f*x)^3)/(a + b*sinh(c + d*x)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{3} \sinh ^{2}{\left (c + d x \right )} \tanh {\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**3*sinh(d*x+c)**2*tanh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral((e + f*x)**3*sinh(c + d*x)**2*tanh(c + d*x)/(a + b*sinh(c + d*x)), x)

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