3.4.38 \(\int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx\) [338]

Optimal. Leaf size=696 \[ \frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {PolyLog}\left (2,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {PolyLog}\left (3,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a \sqrt {a^2+b^2} f^3 \text {PolyLog}\left (4,-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d} \]

[Out]

3/4*e*f^2*x/b/d^2+3/8*f^3*x^2/b/d^2+1/4*a^2*(f*x+e)^4/b^3/f+1/8*(f*x+e)^4/b/f-6*a*f^2*(f*x+e)*cosh(d*x+c)/b^2/
d^3-a*(f*x+e)^3*cosh(d*x+c)/b^2/d-3/8*f^3*cosh(d*x+c)^2/b/d^4-3/4*f*(f*x+e)^2*cosh(d*x+c)^2/b/d^2+6*a*f^3*sinh
(d*x+c)/b^2/d^4+3*a*f*(f*x+e)^2*sinh(d*x+c)/b^2/d^2+3/4*f^2*(f*x+e)*cosh(d*x+c)*sinh(d*x+c)/b/d^3+1/2*(f*x+e)^
3*cosh(d*x+c)*sinh(d*x+c)/b/d-a*(f*x+e)^3*ln(1+b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d+a*(f*x+
e)^3*ln(1+b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d-3*a*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a-(
a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d^2+3*a*f*(f*x+e)^2*polylog(2,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2
)^(1/2)/b^3/d^2+6*a*f^2*(f*x+e)*polylog(3,-b*exp(d*x+c)/(a-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d^3-6*a*f^2*(
f*x+e)*polylog(3,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d^3-6*a*f^3*polylog(4,-b*exp(d*x+c)/(a
-(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2)/b^3/d^4+6*a*f^3*polylog(4,-b*exp(d*x+c)/(a+(a^2+b^2)^(1/2)))*(a^2+b^2)^(1/2
)/b^3/d^4

________________________________________________________________________________________

Rubi [A]
time = 0.80, antiderivative size = 696, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 14, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {5698, 3392, 32, 3391, 5684, 3377, 2717, 3403, 2296, 2221, 2611, 6744, 2320, 6724} \begin {gather*} \frac {a^2 (e+f x)^4}{4 b^3 f}-\frac {6 a f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sqrt {a^2+b^2} \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a f^2 \sqrt {a^2+b^2} (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {3 a f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a f \sqrt {a^2+b^2} (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}+1\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (\frac {b e^{c+d x}}{\sqrt {a^2+b^2}+a}+1\right )}{b^3 d}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}+\frac {3 f^2 (e+f x) \sinh (c+d x) \cosh (c+d x)}{4 b d^3}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \sinh (c+d x) \cosh (c+d x)}{2 b d}+\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {(e+f x)^4}{8 b f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*e*f^2*x)/(4*b*d^2) + (3*f^3*x^2)/(8*b*d^2) + (a^2*(e + f*x)^4)/(4*b^3*f) + (e + f*x)^4/(8*b*f) - (6*a*f^2*(
e + f*x)*Cosh[c + d*x])/(b^2*d^3) - (a*(e + f*x)^3*Cosh[c + d*x])/(b^2*d) - (3*f^3*Cosh[c + d*x]^2)/(8*b*d^4)
- (3*f*(e + f*x)^2*Cosh[c + d*x]^2)/(4*b*d^2) - (a*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a - Sq
rt[a^2 + b^2])])/(b^3*d) + (a*Sqrt[a^2 + b^2]*(e + f*x)^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])])/(b^3
*d) - (3*a*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^2) + (3*
a*Sqrt[a^2 + b^2]*f*(e + f*x)^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^2) + (6*a*Sqrt[a^
2 + b^2]*f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*a*Sqrt[a^2 + b^2]*
f^2*(e + f*x)*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^3) - (6*a*Sqrt[a^2 + b^2]*f^3*PolyL
og[4, -((b*E^(c + d*x))/(a - Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a*Sqrt[a^2 + b^2]*f^3*PolyLog[4, -((b*E^(c + d
*x))/(a + Sqrt[a^2 + b^2]))])/(b^3*d^4) + (6*a*f^3*Sinh[c + d*x])/(b^2*d^4) + (3*a*f*(e + f*x)^2*Sinh[c + d*x]
)/(b^2*d^2) + (3*f^2*(e + f*x)*Cosh[c + d*x]*Sinh[c + d*x])/(4*b*d^3) + ((e + f*x)^3*Cosh[c + d*x]*Sinh[c + d*
x])/(2*b*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

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 3391

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^
2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[b*(c + d*x)*Cos[e + f*x
]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 3392

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*m*(c + d*x)^(m - 1)*((
b*Sin[e + f*x])^n/(f^2*n^2)), x] + (Dist[b^2*((n - 1)/n), Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[d^2*m*((m - 1)/(f^2*n^2)), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[b*(c + d*x)^m*Cos[e + f
*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3403

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

Rule 5684

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

Rule 5698

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

Rubi steps

\begin {align*} \int \frac {(e+f x)^3 \cosh ^2(c+d x) \sinh (c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {\int (e+f x)^3 \cosh ^2(c+d x) \, dx}{b}-\frac {a \int \frac {(e+f x)^3 \cosh ^2(c+d x)}{a+b \sinh (c+d x)} \, dx}{b}\\ &=-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {a^2 \int (e+f x)^3 \, dx}{b^3}-\frac {a \int (e+f x)^3 \sinh (c+d x) \, dx}{b^2}+\frac {\int (e+f x)^3 \, dx}{2 b}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \frac {(e+f x)^3}{a+b \sinh (c+d x)} \, dx}{b^3}+\frac {\left (3 f^2\right ) \int (e+f x) \cosh ^2(c+d x) \, dx}{2 b d^2}\\ &=\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (2 a \left (a^2+b^2\right )\right ) \int \frac {e^{c+d x} (e+f x)^3}{-b+2 a e^{c+d x}+b e^{2 (c+d x)}} \, dx}{b^3}+\frac {(3 a f) \int (e+f x)^2 \cosh (c+d x) \, dx}{b^2 d}+\frac {\left (3 f^2\right ) \int (e+f x) \, dx}{4 b d^2}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (2 a \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a-2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2}+\frac {\left (2 a \sqrt {a^2+b^2}\right ) \int \frac {e^{c+d x} (e+f x)^3}{2 a+2 \sqrt {a^2+b^2}+2 b e^{c+d x}} \, dx}{b^2}-\frac {\left (6 a f^2\right ) \int (e+f x) \sinh (c+d x) \, dx}{b^2 d^2}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\left (3 a \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}-\frac {\left (3 a \sqrt {a^2+b^2} f\right ) \int (e+f x)^2 \log \left (1+\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d}+\frac {\left (6 a f^3\right ) \int \cosh (c+d x) \, dx}{b^2 d^3}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\left (6 a \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}-\frac {\left (6 a \sqrt {a^2+b^2} f^2\right ) \int (e+f x) \text {Li}_2\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^2}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (6 a \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a-2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3}+\frac {\left (6 a \sqrt {a^2+b^2} f^3\right ) \int \text {Li}_3\left (-\frac {2 b e^{c+d x}}{2 a+2 \sqrt {a^2+b^2}}\right ) \, dx}{b^3 d^3}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {\left (6 a \sqrt {a^2+b^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^3 d^4}+\frac {\left (6 a \sqrt {a^2+b^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^3 d^4}\\ &=\frac {3 e f^2 x}{4 b d^2}+\frac {3 f^3 x^2}{8 b d^2}+\frac {a^2 (e+f x)^4}{4 b^3 f}+\frac {(e+f x)^4}{8 b f}-\frac {6 a f^2 (e+f x) \cosh (c+d x)}{b^2 d^3}-\frac {a (e+f x)^3 \cosh (c+d x)}{b^2 d}-\frac {3 f^3 \cosh ^2(c+d x)}{8 b d^4}-\frac {3 f (e+f x)^2 \cosh ^2(c+d x)}{4 b d^2}-\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d}+\frac {a \sqrt {a^2+b^2} (e+f x)^3 \log \left (1+\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d}-\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {3 a \sqrt {a^2+b^2} f (e+f x)^2 \text {Li}_2\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^2}+\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^2 (e+f x) \text {Li}_3\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^3}-\frac {6 a \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a-\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a \sqrt {a^2+b^2} f^3 \text {Li}_4\left (-\frac {b e^{c+d x}}{a+\sqrt {a^2+b^2}}\right )}{b^3 d^4}+\frac {6 a f^3 \sinh (c+d x)}{b^2 d^4}+\frac {3 a f (e+f x)^2 \sinh (c+d x)}{b^2 d^2}+\frac {3 f^2 (e+f x) \cosh (c+d x) \sinh (c+d x)}{4 b d^3}+\frac {(e+f x)^3 \cosh (c+d x) \sinh (c+d x)}{2 b d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 12.58, size = 2963, normalized size = 4.26 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^3*(c/d + x - (2*a*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]])/(Sqrt[-a^2 - b^2]*d)))/(4*b) + (3*e^2
*f*(x^2 + ((2*I)*a*Pi*ArcTanh[(-b + a*Tanh[(c + d*x)/2])/Sqrt[a^2 + b^2]])/(Sqrt[a^2 + b^2]*d^2) + (2*a*(2*((-
I)*c + ArcCos[((-I)*a)/b])*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] + ((-2*I)*c
+ Pi - (2*I)*d*x)*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (ArcCos[((-I)*a)/
b] + (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]])*Log[((I*a + b)*(a + I*(b +
Sqrt[-a^2 - b^2]))*(-I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c +
Pi + (2*I)*d*x)/4]))] - (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt
[-a^2 - b^2]])*Log[((I*a + b)*(I*a - b + Sqrt[-a^2 - b^2])*(I + Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(a - I*
b + Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))] + (ArcCos[((-I)*a)/b] - (2*I)*ArcTanh[((a + I*b)*Cot
[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]] - (2*I)*ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4]
)/Sqrt[-a^2 - b^2]])*Log[-(((-1)^(3/4)*Sqrt[-a^2 - b^2]*E^(-1/2*c - (d*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b
*Sinh[c + d*x]]))] + (ArcCos[((-I)*a)/b] + (2*I)*(ArcTanh[((a + I*b)*Cot[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-
a^2 - b^2]] + ArcTanh[((a - I*b)*Tan[((2*I)*c + Pi + (2*I)*d*x)/4])/Sqrt[-a^2 - b^2]]))*Log[((-1)^(1/4)*Sqrt[-
a^2 - b^2]*E^((c + d*x)/2))/(Sqrt[2]*Sqrt[(-I)*b]*Sqrt[a + b*Sinh[c + d*x]])] + I*(PolyLog[2, ((I*a + Sqrt[-a^
2 - b^2])*(I*a + b - I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*C
ot[((2*I)*c + Pi + (2*I)*d*x)/4]))] - PolyLog[2, ((a + I*Sqrt[-a^2 - b^2])*(-a + I*b + Sqrt[-a^2 - b^2]*Cot[((
2*I)*c + Pi + (2*I)*d*x)/4]))/(b*(I*a + b + I*Sqrt[-a^2 - b^2]*Cot[((2*I)*c + Pi + (2*I)*d*x)/4]))])))/(Sqrt[-
a^2 - b^2]*d^2)))/(8*b) + (e*f^2*(x^3 - (3*a*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2
*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2
*d*x*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2]
)] + 2*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3)))/(4*b) + (f^3*(x^4 - (4*a
*(d^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2
])] + 3*d^2*x^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 3*d^2*x^2*PolyLog[2, -((b*E^(c + d*x))/(a
+ Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((b*E^(c
+ d*x))/(a + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((b*E^(
c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4)))/(16*b) + (e*f^2*(2*(4*a^2 + b^2)*x^3 - (6*a*(4*a^2
+ 3*b^2)*(d^2*x^2*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])] - d^2*x^2*Log[1 + (b*E^(c + d*x))/(a + Sqrt[
a^2 + b^2])] + 2*d*x*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] - 2*d*x*PolyLog[2, -((b*E^(c + d*x))/(
a + Sqrt[a^2 + b^2]))] - 2*PolyLog[3, (b*E^(c + d*x))/(-a + Sqrt[a^2 + b^2])] + 2*PolyLog[3, -((b*E^(c + d*x))
/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^3) - (24*a*b*Cosh[d*x]*((2 + d^2*x^2)*Cosh[c] - 2*d*x*Sinh[c]))/
d^3 + (3*b^2*Cosh[2*d*x]*(-2*d*x*Cosh[2*c] + (1 + 2*d^2*x^2)*Sinh[2*c]))/d^3 - (24*a*b*(-2*d*x*Cosh[c] + (2 +
d^2*x^2)*Sinh[c])*Sinh[d*x])/d^3 + (3*b^2*((1 + 2*d^2*x^2)*Cosh[2*c] - 2*d*x*Sinh[2*c])*Sinh[2*d*x])/d^3))/(8*
b^3) + (f^3*((4*a^2 + b^2)*x^4 - (4*a*(4*a^2 + 3*b^2)*(d^3*x^3*Log[1 + (b*E^(c + d*x))/(a - Sqrt[a^2 + b^2])]
- d^3*x^3*Log[1 + (b*E^(c + d*x))/(a + Sqrt[a^2 + b^2])] + 3*d^2*x^2*PolyLog[2, (b*E^(c + d*x))/(-a + Sqrt[a^2
+ b^2])] - 3*d^2*x^2*PolyLog[2, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] - 6*d*x*PolyLog[3, (b*E^(c + d*x))/
(-a + Sqrt[a^2 + b^2])] + 6*d*x*PolyLog[3, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))] + 6*PolyLog[4, (b*E^(c +
d*x))/(-a + Sqrt[a^2 + b^2])] - 6*PolyLog[4, -((b*E^(c + d*x))/(a + Sqrt[a^2 + b^2]))]))/(Sqrt[a^2 + b^2]*d^4)
- (16*a*b*Cosh[d*x]*(d*x*(6 + d^2*x^2)*Cosh[c] - 3*(2 + d^2*x^2)*Sinh[c]))/d^4 + (b^2*Cosh[2*d*x]*(-3*(1 + 2*
d^2*x^2)*Cosh[2*c] + 2*d*x*(3 + 2*d^2*x^2)*Sinh[2*c]))/d^4 - (16*a*b*(-3*(2 + d^2*x^2)*Cosh[c] + d*x*(6 + d^2*
x^2)*Sinh[c])*Sinh[d*x])/d^4 + (b^2*(2*d*x*(3 + 2*d^2*x^2)*Cosh[2*c] - 3*(1 + 2*d^2*x^2)*Sinh[2*c])*Sinh[2*d*x
])/d^4))/(16*b^3) + (e^3*((4*a^2 + b^2)*(c + d*x) - (2*a*(4*a^2 + 3*b^2)*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt
[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*b*Cosh[c + d*x] + b^2*Sinh[2*(c + d*x)]))/(4*b^3*d) + (3*e^2*f*((4*a^2 +
b^2)*(-c + d*x)*(c + d*x) - 8*a*b*d*x*Cosh[c + d*x] - b^2*Cosh[2*(c + d*x)] - (2*a*(4*a^2 + 3*b^2)*(2*c*ArcTa
nh[(a + b*Cosh[c + d*x] + b*Sinh[c + d*x])/Sqrt[a^2 + b^2]] + (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d
*x]))/(a - Sqrt[a^2 + b^2])] - (c + d*x)*Log[1 + (b*(Cosh[c + d*x] + Sinh[c + d*x]))/(a + Sqrt[a^2 + b^2])] +
PolyLog[2, (b*(Cosh[c + d*x] + Sinh[c + d*x]))/...

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Maple [F]
time = 1.94, size = 0, normalized size = 0.00 \[\int \frac {\left (f x +e \right )^{3} \left (\cosh ^{2}\left (d x +c \right )\right ) \sinh \left (d x +c \right )}{a +b \sinh \left (d x +c \right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

-1/8*((4*a*e^(-d*x - c) - b)*e^(2*d*x + 2*c)/(b^2*d) + 8*sqrt(a^2 + b^2)*a*log((b*e^(-d*x - c) - a - sqrt(a^2
+ b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(b^3*d) - 4*(2*a^2 + b^2)*(d*x + c)/(b^3*d) + (4*a*e^(-d*x - c
) + b*e^(-2*d*x - 2*c))/(b^2*d))*e^3 + 1/32*(4*(2*a^2*d^4*f^3*e^(2*c) + b^2*d^4*f^3*e^(2*c))*x^4 + 16*(2*a^2*d
^4*f^2*e^(2*c) + b^2*d^4*f^2*e^(2*c))*x^3*e + 24*(2*a^2*d^4*f*e^(2*c) + b^2*d^4*f*e^(2*c))*x^2*e^2 + (4*b^2*d^
3*f^3*x^3*e^(4*c) - 3*b^2*f^3*e^(4*c) - 6*b^2*d^2*f*e^(4*c + 2) + 6*b^2*d*f^2*e^(4*c + 1) - 6*(b^2*d^2*f^3*e^(
4*c) - 2*b^2*d^3*f^2*e^(4*c + 1))*x^2 + 6*(b^2*d*f^3*e^(4*c) + 2*b^2*d^3*f*e^(4*c + 2) - 2*b^2*d^2*f^2*e^(4*c
+ 1))*x)*e^(2*d*x) - 16*(a*b*d^3*f^3*x^3*e^(3*c) - 6*a*b*f^3*e^(3*c) - 3*a*b*d^2*f*e^(3*c + 2) + 6*a*b*d*f^2*e
^(3*c + 1) - 3*(a*b*d^2*f^3*e^(3*c) - a*b*d^3*f^2*e^(3*c + 1))*x^2 + 3*(2*a*b*d*f^3*e^(3*c) + a*b*d^3*f*e^(3*c
+ 2) - 2*a*b*d^2*f^2*e^(3*c + 1))*x)*e^(d*x) - 16*(a*b*d^3*f^3*x^3*e^c + 3*a*b*d^2*f*e^(c + 2) + 6*a*b*d*f^2*
e^(c + 1) + 6*a*b*f^3*e^c + 3*(a*b*d^3*f^2*e^(c + 1) + a*b*d^2*f^3*e^c)*x^2 + 3*(a*b*d^3*f*e^(c + 2) + 2*a*b*d
^2*f^2*e^(c + 1) + 2*a*b*d*f^3*e^c)*x)*e^(-d*x) - (4*b^2*d^3*f^3*x^3 + 6*b^2*d^2*f*e^2 + 6*b^2*d*f^2*e + 3*b^2
*f^3 + 6*(2*b^2*d^3*f^2*e + b^2*d^2*f^3)*x^2 + 6*(2*b^2*d^3*f*e^2 + 2*b^2*d^2*f^2*e + b^2*d*f^3)*x)*e^(-2*d*x)
)*e^(-2*c)/(b^3*d^4) - integrate(2*((a^3*f^3*e^c + a*b^2*f^3*e^c)*x^3 + 3*(a^3*f^2*e^c + a*b^2*f^2*e^c)*x^2*e
+ 3*(a^3*f*e^c + a*b^2*f*e^c)*x*e^2)*e^(d*x)/(b^4*e^(2*d*x + 2*c) + 2*a*b^3*e^(d*x + c) - b^4), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6807 vs. \(2 (653) = 1306\).
time = 0.45, size = 6807, normalized size = 9.78 \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*cosh(d*x+c)^2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/32*(4*b^2*d^3*f^3*x^3 + 6*b^2*d^2*f^3*x^2 + 4*b^2*d^3*cosh(1)^3 + 4*b^2*d^3*sinh(1)^3 + 6*b^2*d*f^3*x + 3*b
^2*f^3 - (4*b^2*d^3*f^3*x^3 - 6*b^2*d^2*f^3*x^2 + 4*b^2*d^3*cosh(1)^3 + 4*b^2*d^3*sinh(1)^3 + 6*b^2*d*f^3*x -
3*b^2*f^3 + 6*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1)^2 + 6*(2*b^2*d^3*f*x + 2*b^2*d^3*cosh(1) - b^2*d^2*f)*sinh(1
)^2 + 6*(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + b^2*d*f^2)*cosh(1) + 6*(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + 2
*b^2*d^3*cosh(1)^2 + b^2*d*f^2 + 2*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^4 - (4*b^2*d^3*
f^3*x^3 - 6*b^2*d^2*f^3*x^2 + 4*b^2*d^3*cosh(1)^3 + 4*b^2*d^3*sinh(1)^3 + 6*b^2*d*f^3*x - 3*b^2*f^3 + 6*(2*b^2
*d^3*f*x - b^2*d^2*f)*cosh(1)^2 + 6*(2*b^2*d^3*f*x + 2*b^2*d^3*cosh(1) - b^2*d^2*f)*sinh(1)^2 + 6*(2*b^2*d^3*f
^2*x^2 - 2*b^2*d^2*f^2*x + b^2*d*f^2)*cosh(1) + 6*(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + 2*b^2*d^3*cosh(1)^2 +
b^2*d*f^2 + 2*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^4 + 16*(a*b*d^3*f^3*x^3 - 3*a*b*d^2
*f^3*x^2 + a*b*d^3*cosh(1)^3 + a*b*d^3*sinh(1)^3 + 6*a*b*d*f^3*x - 6*a*b*f^3 + 3*(a*b*d^3*f*x - a*b*d^2*f)*cos
h(1)^2 + 3*(a*b*d^3*f*x + a*b*d^3*cosh(1) - a*b*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + 2*a*
b*d*f^2)*cosh(1) + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + a*b*d^3*cosh(1)^2 + 2*a*b*d*f^2 + 2*(a*b*d^3*f*x - a
*b*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^3 + 4*(4*a*b*d^3*f^3*x^3 - 12*a*b*d^2*f^3*x^2 + 4*a*b*d^3*cosh(1)^3
+ 4*a*b*d^3*sinh(1)^3 + 24*a*b*d*f^3*x - 24*a*b*f^3 + 12*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1)^2 + 12*(a*b*d^3*f*x
+ a*b*d^3*cosh(1) - a*b*d^2*f)*sinh(1)^2 + 12*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + 2*a*b*d*f^2)*cosh(1) - (4*
b^2*d^3*f^3*x^3 - 6*b^2*d^2*f^3*x^2 + 4*b^2*d^3*cosh(1)^3 + 4*b^2*d^3*sinh(1)^3 + 6*b^2*d*f^3*x - 3*b^2*f^3 +
6*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1)^2 + 6*(2*b^2*d^3*f*x + 2*b^2*d^3*cosh(1) - b^2*d^2*f)*sinh(1)^2 + 6*(2*b
^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + b^2*d*f^2)*cosh(1) + 6*(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + 2*b^2*d^3*cos
h(1)^2 + b^2*d*f^2 + 2*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c) + 12*(a*b*d^3*f^2*x^2 - 2*a
*b*d^2*f^2*x + a*b*d^3*cosh(1)^2 + 2*a*b*d*f^2 + 2*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1))*sinh(1))*sinh(d*x + c)^3
+ 6*(2*b^2*d^3*f*x + b^2*d^2*f)*cosh(1)^2 - 4*((2*a^2 + b^2)*d^4*f^3*x^4 + 4*(2*a^2 + b^2)*d^4*f^2*x^3*cosh(1
) + 6*(2*a^2 + b^2)*d^4*f*x^2*cosh(1)^2 + 4*(2*a^2 + b^2)*d^4*x*cosh(1)^3 + 4*(2*a^2 + b^2)*d^4*x*sinh(1)^3 +
6*((2*a^2 + b^2)*d^4*f*x^2 + 2*(2*a^2 + b^2)*d^4*x*cosh(1))*sinh(1)^2 + 4*((2*a^2 + b^2)*d^4*f^2*x^3 + 3*(2*a^
2 + b^2)*d^4*f*x^2*cosh(1) + 3*(2*a^2 + b^2)*d^4*x*cosh(1)^2)*sinh(1))*cosh(d*x + c)^2 + 6*(2*b^2*d^3*f*x + 2*
b^2*d^3*cosh(1) + b^2*d^2*f)*sinh(1)^2 - 2*(2*(2*a^2 + b^2)*d^4*f^3*x^4 + 8*(2*a^2 + b^2)*d^4*f^2*x^3*cosh(1)
+ 12*(2*a^2 + b^2)*d^4*f*x^2*cosh(1)^2 + 8*(2*a^2 + b^2)*d^4*x*cosh(1)^3 + 8*(2*a^2 + b^2)*d^4*x*sinh(1)^3 + 3
*(4*b^2*d^3*f^3*x^3 - 6*b^2*d^2*f^3*x^2 + 4*b^2*d^3*cosh(1)^3 + 4*b^2*d^3*sinh(1)^3 + 6*b^2*d*f^3*x - 3*b^2*f^
3 + 6*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1)^2 + 6*(2*b^2*d^3*f*x + 2*b^2*d^3*cosh(1) - b^2*d^2*f)*sinh(1)^2 + 6*
(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + b^2*d*f^2)*cosh(1) + 6*(2*b^2*d^3*f^2*x^2 - 2*b^2*d^2*f^2*x + 2*b^2*d^3
*cosh(1)^2 + b^2*d*f^2 + 2*(2*b^2*d^3*f*x - b^2*d^2*f)*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 12*((2*a^2 + b^2)*d
^4*f*x^2 + 2*(2*a^2 + b^2)*d^4*x*cosh(1))*sinh(1)^2 - 24*(a*b*d^3*f^3*x^3 - 3*a*b*d^2*f^3*x^2 + a*b*d^3*cosh(1
)^3 + a*b*d^3*sinh(1)^3 + 6*a*b*d*f^3*x - 6*a*b*f^3 + 3*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1)^2 + 3*(a*b*d^3*f*x +
a*b*d^3*cosh(1) - a*b*d^2*f)*sinh(1)^2 + 3*(a*b*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + 2*a*b*d*f^2)*cosh(1) + 3*(a*b
*d^3*f^2*x^2 - 2*a*b*d^2*f^2*x + a*b*d^3*cosh(1)^2 + 2*a*b*d*f^2 + 2*(a*b*d^3*f*x - a*b*d^2*f)*cosh(1))*sinh(1
))*cosh(d*x + c) + 8*((2*a^2 + b^2)*d^4*f^2*x^3 + 3*(2*a^2 + b^2)*d^4*f*x^2*cosh(1) + 3*(2*a^2 + b^2)*d^4*x*co
sh(1)^2)*sinh(1))*sinh(d*x + c)^2 + 96*((a*b*d^2*f^3*x^2 + 2*a*b*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a*b
*d^2*f*sinh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b
*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a*b*d^2*f*sinh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^2*f*cosh(1))*sinh(1)
)*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^3*x^2 + 2*a*b*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a*b*d^2*f*s
inh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^2*f*cosh(1))*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2)/b^2)*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) - 96*((a*b
*d^2*f^3*x^2 + 2*a*b*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a*b*d^2*f*sinh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^
2*f*cosh(1))*sinh(1))*cosh(d*x + c)^2 + 2*(a*b*d^2*f^3*x^2 + 2*a*b*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a
*b*d^2*f*sinh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^2*f*cosh(1))*sinh(1))*cosh(d*x + c)*sinh(d*x + c) + (a*b*d^2*f^3
*x^2 + 2*a*b*d^2*f^2*x*cosh(1) + a*b*d^2*f*cosh(1)^2 + a*b*d^2*f*sinh(1)^2 + 2*(a*b*d^2*f^2*x + a*b*d^2*f*cosh
(1))*sinh(1))*sinh(d*x + c)^2)*sqrt((a^2 + b^2)...

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Sympy [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*cosh(d*x+c)**2*sinh(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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