### 3.412 $$\int x^3 \cosh ^2(a+b x) \coth (a+b x) \, dx$$

Optimal. Leaf size=180 $\frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4}$

[Out]

(3*x)/(8*b^3) + x^3/(4*b) - x^4/4 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*
b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4) - (3*Cosh[a + b*x]*
Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Si
nh[a + b*x]^2)/(2*b)

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Rubi [A]  time = 0.236801, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.667, Rules used = {5450, 5372, 3311, 30, 2635, 8, 3716, 2190, 2531, 6609, 2282, 6589} $\frac{3 x^2 \text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{3 x \text{PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac{3 \text{PolyLog}\left (4,e^{2 (a+b x)}\right )}{4 b^4}-\frac{3 x^2 \sinh (a+b x) \cosh (a+b x)}{4 b^2}+\frac{3 x \sinh ^2(a+b x)}{4 b^3}-\frac{3 \sinh (a+b x) \cosh (a+b x)}{8 b^4}+\frac{x^3 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac{x^3 \sinh ^2(a+b x)}{2 b}+\frac{3 x}{8 b^3}+\frac{x^3}{4 b}-\frac{x^4}{4}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*Cosh[a + b*x]^2*Coth[a + b*x],x]

[Out]

(3*x)/(8*b^3) + x^3/(4*b) - x^4/4 + (x^3*Log[1 - E^(2*(a + b*x))])/b + (3*x^2*PolyLog[2, E^(2*(a + b*x))])/(2*
b^2) - (3*x*PolyLog[3, E^(2*(a + b*x))])/(2*b^3) + (3*PolyLog[4, E^(2*(a + b*x))])/(4*b^4) - (3*Cosh[a + b*x]*
Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Si
nh[a + b*x]^2)/(2*b)

Rule 5450

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

Rule 5372

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.), x_Symbol] :> Simp[(x^(m -
n + 1)*Sinh[a + b*x^n]^(p + 1))/(b*n*(p + 1)), x] - Dist[(m - n + 1)/(b*n*(p + 1)), Int[x^(m - n)*Sinh[a + b*x
^n]^(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]

Rule 3311

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (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)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

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 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 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 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 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]

Rubi steps

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

Mathematica [A]  time = 2.52679, size = 236, normalized size = 1.31 $\frac{\sinh (a) (\sinh (a)+\cosh (a)) \left (-48 b^2 x^2 \text{PolyLog}\left (2,-e^{-a-b x}\right )-48 b^2 x^2 \text{PolyLog}\left (2,e^{-a-b x}\right )-96 b x \text{PolyLog}\left (3,-e^{-a-b x}\right )-96 b x \text{PolyLog}\left (3,e^{-a-b x}\right )-96 \text{PolyLog}\left (4,-e^{-a-b x}\right )-96 \text{PolyLog}\left (4,e^{-a-b x}\right )+16 b^3 x^3 \log \left (1-e^{-a-b x}\right )+16 b^3 x^3 \log \left (e^{-a-b x}+1\right )-6 b^2 x^2 \sinh (2 (a+b x))+4 b^3 x^3 \cosh (2 (a+b x))-3 \sinh (2 (a+b x))+6 b x \cosh (2 (a+b x))+4 b^4 x^4\right )}{8 \left (e^{2 a}-1\right ) b^4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*Cosh[a + b*x]^2*Coth[a + b*x],x]

[Out]

(Sinh[a]*(Cosh[a] + Sinh[a])*(4*b^4*x^4 + 6*b*x*Cosh[2*(a + b*x)] + 4*b^3*x^3*Cosh[2*(a + b*x)] + 16*b^3*x^3*L
og[1 - E^(-a - b*x)] + 16*b^3*x^3*Log[1 + E^(-a - b*x)] - 48*b^2*x^2*PolyLog[2, -E^(-a - b*x)] - 48*b^2*x^2*Po
lyLog[2, E^(-a - b*x)] - 96*b*x*PolyLog[3, -E^(-a - b*x)] - 96*b*x*PolyLog[3, E^(-a - b*x)] - 96*PolyLog[4, -E
^(-a - b*x)] - 96*PolyLog[4, E^(-a - b*x)] - 3*Sinh[2*(a + b*x)] - 6*b^2*x^2*Sinh[2*(a + b*x)]))/(8*b^4*(-1 +
E^(2*a)))

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Maple [A]  time = 0.087, size = 272, normalized size = 1.5 \begin{align*} -{\frac{{x}^{4}}{4}}+{\frac{ \left ( 4\,{x}^{3}{b}^{3}-6\,{x}^{2}{b}^{2}+6\,bx-3 \right ){{\rm e}^{2\,bx+2\,a}}}{32\,{b}^{4}}}+{\frac{ \left ( 4\,{x}^{3}{b}^{3}+6\,{x}^{2}{b}^{2}+6\,bx+3 \right ){{\rm e}^{-2\,bx-2\,a}}}{32\,{b}^{4}}}-2\,{\frac{{a}^{3}x}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{4}}}+2\,{\frac{{a}^{3}\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+6\,{\frac{{\it polylog} \left ( 4,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-{\frac{3\,{a}^{4}}{2\,{b}^{4}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{3}}{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{3}}{b}}+3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ){x}^{2}}{{b}^{2}}}-6\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{3}}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(b*x+a)^3*csch(b*x+a),x)

[Out]

-1/4*x^4+1/32*(4*b^3*x^3-6*b^2*x^2+6*b*x-3)/b^4*exp(2*b*x+2*a)+1/32*(4*b^3*x^3+6*b^2*x^2+6*b*x+3)/b^4*exp(-2*b
*x-2*a)-2/b^3*a^3*x-1/b^4*a^3*ln(exp(b*x+a)-1)+2/b^4*a^3*ln(exp(b*x+a))+6/b^4*polylog(4,-exp(b*x+a))+6/b^4*pol
ylog(4,exp(b*x+a))-3/2/b^4*a^4+1/b^4*ln(1-exp(b*x+a))*a^3+3/b^2*polylog(2,exp(b*x+a))*x^2-6/b^3*polylog(3,exp(
b*x+a))*x+1/b*ln(1+exp(b*x+a))*x^3+3/b^2*polylog(2,-exp(b*x+a))*x^2-6/b^3*polylog(3,-exp(b*x+a))*x+1/b*ln(1-ex
p(b*x+a))*x^3

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Maxima [A]  time = 1.45046, size = 304, normalized size = 1.69 \begin{align*} -\frac{1}{2} \, x^{4} + \frac{{\left (8 \, b^{4} x^{4} e^{\left (2 \, a\right )} +{\left (4 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 6 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} +{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{32 \, b^{4}} + \frac{b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(-e^{\left (b x + a\right )})}{b^{4}} + \frac{b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2}{\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x{\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \,{\rm Li}_{4}(e^{\left (b x + a\right )})}{b^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="maxima")

[Out]

-1/2*x^4 + 1/32*(8*b^4*x^4*e^(2*a) + (4*b^3*x^3*e^(4*a) - 6*b^2*x^2*e^(4*a) + 6*b*x*e^(4*a) - 3*e^(4*a))*e^(2*
b*x) + (4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x))*e^(-2*a)/b^4 + (b^3*x^3*log(e^(b*x + a) + 1) + 3*b^2*x^
2*dilog(-e^(b*x + a)) - 6*b*x*polylog(3, -e^(b*x + a)) + 6*polylog(4, -e^(b*x + a)))/b^4 + (b^3*x^3*log(-e^(b*
x + a) + 1) + 3*b^2*x^2*dilog(e^(b*x + a)) - 6*b*x*polylog(3, e^(b*x + a)) + 6*polylog(4, e^(b*x + a)))/b^4

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Fricas [C]  time = 2.1776, size = 2325, normalized size = 12.92 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="fricas")

[Out]

1/32*(4*b^3*x^3 + (4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^4 + 4*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*
cosh(b*x + a)*sinh(b*x + a)^3 + (4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*sinh(b*x + a)^4 + 6*b^2*x^2 - 8*(b^4*x^4 -
2*a^4)*cosh(b*x + a)^2 - 2*(4*b^4*x^4 - 8*a^4 - 3*(4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^2)*sinh(b
*x + a)^2 + 6*b*x + 96*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a) + b^2*x^2*sinh(b*x + a
)^2)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 96*(b^2*x^2*cosh(b*x + a)^2 + 2*b^2*x^2*cosh(b*x + a)*sinh(b*x + a
) + b^2*x^2*sinh(b*x + a)^2)*dilog(-cosh(b*x + a) - sinh(b*x + a)) + 32*(b^3*x^3*cosh(b*x + a)^2 + 2*b^3*x^3*c
osh(b*x + a)*sinh(b*x + a) + b^3*x^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 32*(a^3*cosh(b*
x + a)^2 + 2*a^3*cosh(b*x + a)*sinh(b*x + a) + a^3*sinh(b*x + a)^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 3
2*((b^3*x^3 + a^3)*cosh(b*x + a)^2 + 2*(b^3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a) + (b^3*x^3 + a^3)*sinh(b*x
+ a)^2)*log(-cosh(b*x + a) - sinh(b*x + a) + 1) + 192*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(
b*x + a)^2)*polylog(4, cosh(b*x + a) + sinh(b*x + a)) + 192*(cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) +
sinh(b*x + a)^2)*polylog(4, -cosh(b*x + a) - sinh(b*x + a)) - 192*(b*x*cosh(b*x + a)^2 + 2*b*x*cosh(b*x + a)*
sinh(b*x + a) + b*x*sinh(b*x + a)^2)*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 192*(b*x*cosh(b*x + a)^2 + 2*
b*x*cosh(b*x + a)*sinh(b*x + a) + b*x*sinh(b*x + a)^2)*polylog(3, -cosh(b*x + a) - sinh(b*x + a)) + 4*((4*b^3*
x^3 - 6*b^2*x^2 + 6*b*x - 3)*cosh(b*x + a)^3 - 4*(b^4*x^4 - 2*a^4)*cosh(b*x + a))*sinh(b*x + a) + 3)/(b^4*cosh
(b*x + a)^2 + 2*b^4*cosh(b*x + a)*sinh(b*x + a) + b^4*sinh(b*x + a)^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cosh(b*x+a)**3*csch(b*x+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \cosh \left (b x + a\right )^{3} \operatorname{csch}\left (b x + a\right )\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^3*csch(b*x+a),x, algorithm="giac")

[Out]

integrate(x^3*cosh(b*x + a)^3*csch(b*x + a), x)