3.4.77 \(\int \frac {(e+f x)^3 \sinh (c+d x) \tanh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [377]

Optimal. Leaf size=1218 \[ -\frac {(e+f x)^4}{4 b f}-\frac {2 a (e+f x)^3 \text {ArcTan}\left (e^{c+d x}\right )}{b^2 d}+\frac {2 a^3 (e+f x)^3 \text {ArcTan}\left (e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {a^2 (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d}+\frac {(e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b d}-\frac {a^2 (e+f x)^3 \log \left (1+e^{2 (c+d x)}\right )}{b \left (a^2+b^2\right ) d}+\frac {3 i a f (e+f x)^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 d^2}-\frac {3 i a^3 f (e+f x)^2 \text {PolyLog}\left (2,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 i a f (e+f x)^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 d^2}+\frac {3 i a^3 f (e+f x)^2 \text {PolyLog}\left (2,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 a^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {3 a^2 f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^2}+\frac {3 f (e+f x)^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b d^2}-\frac {3 a^2 f (e+f x)^2 \text {PolyLog}\left (2,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^2}-\frac {6 i a f^2 (e+f x) \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 d^3}+\frac {6 i a^3 f^2 (e+f x) \text {PolyLog}\left (3,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i a f^2 (e+f x) \text {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 d^3}-\frac {6 i a^3 f^2 (e+f x) \text {PolyLog}\left (3,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^3}-\frac {3 f^2 (e+f x) \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b d^3}+\frac {3 a^2 f^2 (e+f x) \text {PolyLog}\left (3,-e^{2 (c+d x)}\right )}{2 b \left (a^2+b^2\right ) d^3}+\frac {6 i a f^3 \text {PolyLog}\left (4,-i e^{c+d x}\right )}{b^2 d^4}-\frac {6 i a^3 f^3 \text {PolyLog}\left (4,-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 i a f^3 \text {PolyLog}\left (4,i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \text {PolyLog}\left (4,i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b d^4}-\frac {3 a^2 f^3 \text {PolyLog}\left (4,-e^{2 (c+d x)}\right )}{4 b \left (a^2+b^2\right ) d^4} \]

[Out]

3*a^2*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2+3*a^2*f*(f*x+e)^2*polylog(2,-b*
exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^2-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))
/b/(a^2+b^2)/d^3-6*a^2*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^3-3/4*a^2*f^3*po
lylog(4,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^4-6*I*a*f^3*polylog(4,I*exp(d*x+c))/b^2/d^4+3*I*a^3*f*(f*x+e)^2*polylog
(2,I*exp(d*x+c))/b^2/(a^2+b^2)/d^2+6*I*a^3*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^3+(f*x+e)^3*ln
(1+exp(2*d*x+2*c))/b/d+2*a^3*(f*x+e)^3*arctan(exp(d*x+c))/b^2/(a^2+b^2)/d-1/4*(f*x+e)^4/b/f+6*a^2*f^3*polylog(
4,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d^4+6*a^2*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/
(a^2+b^2)/d^4-a^2*(f*x+e)^3*ln(1+exp(2*d*x+2*c))/b/(a^2+b^2)/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1
/2)))/b/(a^2+b^2)/d+a^2*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))/b/(a^2+b^2)/d-2*a*(f*x+e)^3*arctan(ex
p(d*x+c))/b^2/d+3/2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/d^2-3/2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b/
d^3+3/4*f^3*polylog(4,-exp(2*d*x+2*c))/b/d^4-3/2*a^2*f*(f*x+e)^2*polylog(2,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^2+3/
2*a^2*f^2*(f*x+e)*polylog(3,-exp(2*d*x+2*c))/b/(a^2+b^2)/d^3+6*I*a*f^3*polylog(4,-I*exp(d*x+c))/b^2/d^4-3*I*a*
f*(f*x+e)^2*polylog(2,I*exp(d*x+c))/b^2/d^2-6*I*a*f^2*(f*x+e)*polylog(3,-I*exp(d*x+c))/b^2/d^3-6*I*a^3*f^3*pol
ylog(4,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^4+3*I*a*f*(f*x+e)^2*polylog(2,-I*exp(d*x+c))/b^2/d^2+6*I*a*f^2*(f*x+e)*p
olylog(3,I*exp(d*x+c))/b^2/d^3+6*I*a^3*f^3*polylog(4,I*exp(d*x+c))/b^2/(a^2+b^2)/d^4-3*I*a^3*f*(f*x+e)^2*polyl
og(2,-I*exp(d*x+c))/b^2/(a^2+b^2)/d^2-6*I*a^3*f^2*(f*x+e)*polylog(3,I*exp(d*x+c))/b^2/(a^2+b^2)/d^3

________________________________________________________________________________________

Rubi [A]
time = 1.30, antiderivative size = 1218, normalized size of antiderivative = 1.00, number of steps used = 46, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5700, 3799, 2221, 2611, 6744, 2320, 6724, 5686, 4265, 5692, 5680, 6874} \begin {gather*} -\frac {(e+f x)^4}{4 b f}+\frac {2 a^3 \text {ArcTan}\left (e^{c+d x}\right ) (e+f x)^3}{b^2 \left (a^2+b^2\right ) d}-\frac {2 a \text {ArcTan}\left (e^{c+d x}\right ) (e+f x)^3}{b^2 d}+\frac {a^2 \log \left (\frac {e^{c+d x} b}{a-\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}+\frac {a^2 \log \left (\frac {e^{c+d x} b}{a+\sqrt {a^2+b^2}}+1\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}+\frac {\log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{b d}-\frac {a^2 \log \left (1+e^{2 (c+d x)}\right ) (e+f x)^3}{b \left (a^2+b^2\right ) d}-\frac {3 i a^3 f \text {Li}_2\left (-i e^{c+d x}\right ) (e+f x)^2}{b^2 \left (a^2+b^2\right ) d^2}+\frac {3 i a f \text {Li}_2\left (-i e^{c+d x}\right ) (e+f x)^2}{b^2 d^2}+\frac {3 i a^3 f \text {Li}_2\left (i e^{c+d x}\right ) (e+f x)^2}{b^2 \left (a^2+b^2\right ) d^2}-\frac {3 i a f \text {Li}_2\left (i e^{c+d x}\right ) (e+f x)^2}{b^2 d^2}+\frac {3 a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b \left (a^2+b^2\right ) d^2}+\frac {3 a^2 f \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)^2}{b \left (a^2+b^2\right ) d^2}+\frac {3 f \text {Li}_2\left (-e^{2 (c+d x)}\right ) (e+f x)^2}{2 b d^2}-\frac {3 a^2 f \text {Li}_2\left (-e^{2 (c+d x)}\right ) (e+f x)^2}{2 b \left (a^2+b^2\right ) d^2}+\frac {6 i a^3 f^2 \text {Li}_3\left (-i e^{c+d x}\right ) (e+f x)}{b^2 \left (a^2+b^2\right ) d^3}-\frac {6 i a f^2 \text {Li}_3\left (-i e^{c+d x}\right ) (e+f x)}{b^2 d^3}-\frac {6 i a^3 f^2 \text {Li}_3\left (i e^{c+d x}\right ) (e+f x)}{b^2 \left (a^2+b^2\right ) d^3}+\frac {6 i a f^2 \text {Li}_3\left (i e^{c+d x}\right ) (e+f x)}{b^2 d^3}-\frac {6 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right ) (e+f x)}{b \left (a^2+b^2\right ) d^3}-\frac {6 a^2 f^2 \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right ) (e+f x)}{b \left (a^2+b^2\right ) d^3}-\frac {3 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) (e+f x)}{2 b d^3}+\frac {3 a^2 f^2 \text {Li}_3\left (-e^{2 (c+d x)}\right ) (e+f x)}{2 b \left (a^2+b^2\right ) d^3}-\frac {6 i a^3 f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}+\frac {6 i a f^3 \text {Li}_4\left (-i e^{c+d x}\right )}{b^2 d^4}+\frac {6 i a^3 f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^2 \left (a^2+b^2\right ) d^4}-\frac {6 i a f^3 \text {Li}_4\left (i e^{c+d x}\right )}{b^2 d^4}+\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {6 a^2 f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b \left (a^2+b^2\right ) d^4}+\frac {3 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b d^4}-\frac {3 a^2 f^3 \text {Li}_4\left (-e^{2 (c+d x)}\right )}{4 b \left (a^2+b^2\right ) d^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2320

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 2611

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 3799

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 4265

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 5680

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 5686

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*Sech
[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 5692

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 5700

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 6724

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 6744

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 6874

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

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2891\) vs. \(2(1218)=2436\).
time = 19.93, size = 2891, normalized size = 2.37 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 2.66, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \sinh \left (d x +c \right ) \tanh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^3*sinh(d*x+c)*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 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^2*log(-2*a*e^(-d*x - c) + b*e^(-2*d*x - 2*c) - b)/((a^2*b + b^3)*d) + 2*a*arctan(e^(-d*x - c))/((a^2 + b^2)
*d) + b*log(e^(-2*d*x - 2*c) + 1)/((a^2 + b^2)*d) + (d*x + c)/(b*d))*e^3 + 1/4*(f^3*x^4 + 4*f^2*x^3*e + 6*f*x^
2*e^2)/b - integrate(2*(a^2*b*f^3*x^3 + 3*a^2*b*f^2*x^2*e + 3*a^2*b*f*x*e^2 - (a^3*f^3*x^3*e^c + 3*a^3*f^2*x^2
*e^(c + 1) + 3*a^3*f*x*e^(c + 2))*e^(d*x))/(a^2*b^2 + b^4 - (a^2*b^2*e^(2*c) + b^4*e^(2*c))*e^(2*d*x) - 2*(a^3
*b*e^c + a*b^3*e^c)*e^(d*x)), x) - integrate(2*(b*f^3*x^3 + 3*b*f^2*x^2*e + 3*b*f*x*e^2 + (a*f^3*x^3*e^c + 3*a
*f^2*x^2*e^(c + 1) + 3*a*f*x*e^(c + 2))*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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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3310 vs. \(2 (1133) = 2266\).
time = 0.52, size = 3310, normalized size = 2.72 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*((a^2 + b^2)*d^4*f^3*x^4 + 4*(a^2 + b^2)*d^4*f^2*x^3*cosh(1) + 6*(a^2 + b^2)*d^4*f*x^2*cosh(1)^2 + 4*(a^2
+ b^2)*d^4*x*cosh(1)^3 + 4*(a^2 + b^2)*d^4*x*sinh(1)^3 - 24*a^2*f^3*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^2*f^3*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) + 6*((a^2 + b^2)*d^4*f*x^2 +
2*(a^2 + b^2)*d^4*x*cosh(1))*sinh(1)^2 - 12*(a^2*d^2*f^3*x^2 + 2*a^2*d^2*f^2*x*cosh(1) + a^2*d^2*f*cosh(1)^2
+ a^2*d^2*f*sinh(1)^2 + 2*(a^2*d^2*f^2*x + a^2*d^2*f*cosh(1))*sinh(1))*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^2*d^2*f^3*x^2 + 2*a^2*d^2*f^
2*x*cosh(1) + a^2*d^2*f*cosh(1)^2 + a^2*d^2*f*sinh(1)^2 + 2*(a^2*d^2*f^2*x + a^2*d^2*f*cosh(1))*sinh(1))*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*(I*a*b*d^2*f^3*x^2 - b^2*d^2*f^3*x^2 + 2*I*a*b*d^2*f^2*x*cosh(1) - 2*b^2*d^2*f^2*x*cosh(1) + I*a*b*d^2*f*co
sh(1)^2 - b^2*d^2*f*cosh(1)^2 + I*a*b*d^2*f*sinh(1)^2 - b^2*d^2*f*sinh(1)^2 + 2*I*(a*b*d^2*f^2*x + a*b*d^2*f*c
osh(1))*sinh(1) - 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*dilog(I*cosh(d*x + c) + I*sinh(d*x + c)) + 12
*(-I*a*b*d^2*f^3*x^2 - b^2*d^2*f^3*x^2 - 2*I*a*b*d^2*f^2*x*cosh(1) - 2*b^2*d^2*f^2*x*cosh(1) - I*a*b*d^2*f*cos
h(1)^2 - b^2*d^2*f*cosh(1)^2 - I*a*b*d^2*f*sinh(1)^2 - b^2*d^2*f*sinh(1)^2 - 2*I*(a*b*d^2*f^2*x + a*b*d^2*f*co
sh(1))*sinh(1) - 2*(b^2*d^2*f^2*x + b^2*d^2*f*cosh(1))*sinh(1))*dilog(-I*cosh(d*x + c) - I*sinh(d*x + c)) + 4*
(a^2*c^3*f^3 - 3*a^2*c^2*d*f^2*cosh(1) + 3*a^2*c*d^2*f*cosh(1)^2 - a^2*d^3*cosh(1)^3 - a^2*d^3*sinh(1)^3 + 3*(
a^2*c*d^2*f - a^2*d^3*cosh(1))*sinh(1)^2 - 3*(a^2*c^2*d*f^2 - 2*a^2*c*d^2*f*cosh(1) + a^2*d^3*cosh(1)^2)*sinh(
1))*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^2*c^3*f^3 - 3*a^2*c^2*
d*f^2*cosh(1) + 3*a^2*c*d^2*f*cosh(1)^2 - a^2*d^3*cosh(1)^3 - a^2*d^3*sinh(1)^3 + 3*(a^2*c*d^2*f - a^2*d^3*cos
h(1))*sinh(1)^2 - 3*(a^2*c^2*d*f^2 - 2*a^2*c*d^2*f*cosh(1) + a^2*d^3*cosh(1)^2)*sinh(1))*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^2*d^3*f^3*x^3 + a^2*c^3*f^3 + 3*(a^2*d^3*f*x +
a^2*c*d^2*f)*cosh(1)^2 + 3*(a^2*d^3*f*x + a^2*c*d^2*f)*sinh(1)^2 + 3*(a^2*d^3*f^2*x^2 - a^2*c^2*d*f^2)*cosh(1)
+ 3*(a^2*d^3*f^2*x^2 - a^2*c^2*d*f^2 + 2*(a^2*d^3*f*x + a^2*c*d^2*f)*cosh(1))*sinh(1))*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^2*d^3*f^3*x^3 + a
^2*c^3*f^3 + 3*(a^2*d^3*f*x + a^2*c*d^2*f)*cosh(1)^2 + 3*(a^2*d^3*f*x + a^2*c*d^2*f)*sinh(1)^2 + 3*(a^2*d^3*f^
2*x^2 - a^2*c^2*d*f^2)*cosh(1) + 3*(a^2*d^3*f^2*x^2 - a^2*c^2*d*f^2 + 2*(a^2*d^3*f*x + a^2*c*d^2*f)*cosh(1))*s
inh(1))*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*(-I*a*b*c^3*f^3 + b^2*c^3*f^3 + 3*I*a*b*c^2*d*f^2*cosh(1) - 3*b^2*c^2*d*f^2*cosh(1) - 3*I*a*b*c*d^2*
f*cosh(1)^2 + 3*b^2*c*d^2*f*cosh(1)^2 + I*a*b*d^3*cosh(1)^3 - b^2*d^3*cosh(1)^3 + I*a*b*d^3*sinh(1)^3 - b^2*d^
3*sinh(1)^3 - 3*I*(a*b*c*d^2*f - a*b*d^3*cosh(1))*sinh(1)^2 + 3*(b^2*c*d^2*f - b^2*d^3*cosh(1))*sinh(1)^2 + 3*
I*(a*b*c^2*d*f^2 - 2*a*b*c*d^2*f*cosh(1) + a*b*d^3*cosh(1)^2)*sinh(1) - 3*(b^2*c^2*d*f^2 - 2*b^2*c*d^2*f*cosh(
1) + b^2*d^3*cosh(1)^2)*sinh(1))*log(cosh(d*x + c) + sinh(d*x + c) + I) + 4*(I*a*b*c^3*f^3 + b^2*c^3*f^3 - 3*I
*a*b*c^2*d*f^2*cosh(1) - 3*b^2*c^2*d*f^2*cosh(1) + 3*I*a*b*c*d^2*f*cosh(1)^2 + 3*b^2*c*d^2*f*cosh(1)^2 - I*a*b
*d^3*cosh(1)^3 - b^2*d^3*cosh(1)^3 - I*a*b*d^3*sinh(1)^3 - b^2*d^3*sinh(1)^3 + 3*I*(a*b*c*d^2*f - a*b*d^3*cosh
(1))*sinh(1)^2 + 3*(b^2*c*d^2*f - b^2*d^3*cosh(1))*sinh(1)^2 - 3*I*(a*b*c^2*d*f^2 - 2*a*b*c*d^2*f*cosh(1) + a*
b*d^3*cosh(1)^2)*sinh(1) - 3*(b^2*c^2*d*f^2 - 2*b^2*c*d^2*f*cosh(1) + b^2*d^3*cosh(1)^2)*sinh(1))*log(cosh(d*x
+ c) + sinh(d*x + c) - I) + 4*(-I*a*b*d^3*f^3*x^3 - b^2*d^3*f^3*x^3 - I*a*b*c^3*f^3 - b^2*c^3*f^3 - 3*I*(a*b*
d^3*f*x + a*b*c*d^2*f)*cosh(1)^2 - 3*(b^2*d^3*f*x + b^2*c*d^2*f)*cosh(1)^2 - 3*I*(a*b*d^3*f*x + a*b*c*d^2*f)*s
inh(1)^2 - 3*(b^2*d^3*f*x + b^2*c*d^2*f)*sinh(1)^2 - 3*I*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*cosh(1) - 3*(b^2*d^
3*f^2*x^2 - b^2*c^2*d*f^2)*cosh(1) - 3*I*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2 + 2*(a*b*d^3*f*x + a*b*c*d^2*f)*cosh
(1))*sinh(1) - 3*(b^2*d^3*f^2*x^2 - b^2*c^2*d*f^2 + 2*(b^2*d^3*f*x + b^2*c*d^2*f)*cosh(1))*sinh(1))*log(I*cosh
(d*x + c) + I*sinh(d*x + c) + 1) + 4*(I*a*b*d^3*f^3*x^3 - b^2*d^3*f^3*x^3 + I*a*b*c^3*f^3 - b^2*c^3*f^3 + 3*I*
(a*b*d^3*f*x + a*b*c*d^2*f)*cosh(1)^2 - 3*(b^2*d^3*f*x + b^2*c*d^2*f)*cosh(1)^2 + 3*I*(a*b*d^3*f*x + a*b*c*d^2
*f)*sinh(1)^2 - 3*(b^2*d^3*f*x + b^2*c*d^2*f)*sinh(1)^2 + 3*I*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2)*cosh(1) - 3*(b
^2*d^3*f^2*x^2 - b^2*c^2*d*f^2)*cosh(1) + 3*I*(a*b*d^3*f^2*x^2 - a*b*c^2*d*f^2 + 2*(a*b*d^3*f*x + a*b*c*d^2*f)
*cosh(1))*sinh(1) - 3*(b^2*d^3*f^2*x^2 - b^2*c^...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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