\(\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx\) [495]
Optimal result
Integrand size = 20, antiderivative size = 85 \[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \operatorname {PolyLog}\left (3,e^{4 (a+b x)}\right )}{8 b^4}
\]
[Out]
-2*x^3/b-2*x^3*coth(2*b*x+2*a)/b+3*x^2*ln(1-exp(4*b*x+4*a))/b^2+3/2*x*polylog(2,exp(4*b*x+4*a))/b^3-3/8*polylo
g(3,exp(4*b*x+4*a))/b^4
Rubi [A] (verified)
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of
steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {5569, 4269, 3797, 2221, 2611,
2320, 6724} \[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {3 \operatorname {PolyLog}\left (3,e^{4 (a+b x)}\right )}{8 b^4}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}-\frac {2 x^3 \coth (2 a+2 b x)}{b}-\frac {2 x^3}{b}
\]
[In]
Int[x^3*Csch[a + b*x]^2*Sech[a + b*x]^2,x]
[Out]
(-2*x^3)/b - (2*x^3*Coth[2*a + 2*b*x])/b + (3*x^2*Log[1 - E^(4*(a + b*x))])/b^2 + (3*x*PolyLog[2, E^(4*(a + b*
x))])/(2*b^3) - (3*PolyLog[3, E^(4*(a + b*x))])/(8*b^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 3797
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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]
Rule 4269
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 5569
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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]
Rubi steps \begin{align*}
\text {integral}& = 4 \int x^3 \text {csch}^2(2 a+2 b x) \, dx \\ & = -\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {6 \int x^2 \coth (2 a+2 b x) \, dx}{b} \\ & = -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}-\frac {12 \int \frac {e^{2 (2 a+2 b x)} x^2}{1-e^{2 (2 a+2 b x)}} \, dx}{b} \\ & = -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}-\frac {6 \int x \log \left (1-e^{2 (2 a+2 b x)}\right ) \, dx}{b^2} \\ & = -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \int \operatorname {PolyLog}\left (2,e^{2 (2 a+2 b x)}\right ) \, dx}{2 b^3} \\ & = -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{2 (2 a+2 b x)}\right )}{8 b^4} \\ & = -\frac {2 x^3}{b}-\frac {2 x^3 \coth (2 a+2 b x)}{b}+\frac {3 x^2 \log \left (1-e^{4 (a+b x)}\right )}{b^2}+\frac {3 x \operatorname {PolyLog}\left (2,e^{4 (a+b x)}\right )}{2 b^3}-\frac {3 \operatorname {PolyLog}\left (3,e^{4 (a+b x)}\right )}{8 b^4} \\
\end{align*}
Mathematica [B] (verified)
Leaf count is larger than twice the leaf count of optimal. \(307\) vs. \(2(85)=170\).
Time = 1.48 (sec) , antiderivative size = 307, normalized size of antiderivative = 3.61
\[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=4 \left (-\frac {e^{4 a} \left (8 b^3 e^{-4 a} x^3-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1-e^{-a-b x}\right )-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1+e^{-a-b x}\right )-6 b^2 \left (1-e^{-4 a}\right ) x^2 \log \left (1+e^{-2 (a+b x)}\right )+12 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-a-b x}\right )+12 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,e^{-a-b x}\right )+6 b \left (1-e^{-4 a}\right ) x \operatorname {PolyLog}\left (2,-e^{-2 (a+b x)}\right )+12 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,-e^{-a-b x}\right )+12 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,e^{-a-b x}\right )+3 \left (1-e^{-4 a}\right ) \operatorname {PolyLog}\left (3,-e^{-2 (a+b x)}\right )\right )}{8 b^4 \left (-1+e^{4 a}\right )}+\frac {x^3 \text {csch}(2 a) \text {csch}(2 a+2 b x) \sinh (2 b x)}{2 b}\right )
\]
[In]
Integrate[x^3*Csch[a + b*x]^2*Sech[a + b*x]^2,x]
[Out]
4*(-1/8*(E^(4*a)*((8*b^3*x^3)/E^(4*a) - 6*b^2*(1 - E^(-4*a))*x^2*Log[1 - E^(-a - b*x)] - 6*b^2*(1 - E^(-4*a))*
x^2*Log[1 + E^(-a - b*x)] - 6*b^2*(1 - E^(-4*a))*x^2*Log[1 + E^(-2*(a + b*x))] + 12*b*(1 - E^(-4*a))*x*PolyLog
[2, -E^(-a - b*x)] + 12*b*(1 - E^(-4*a))*x*PolyLog[2, E^(-a - b*x)] + 6*b*(1 - E^(-4*a))*x*PolyLog[2, -E^(-2*(
a + b*x))] + 12*(1 - E^(-4*a))*PolyLog[3, -E^(-a - b*x)] + 12*(1 - E^(-4*a))*PolyLog[3, E^(-a - b*x)] + 3*(1 -
E^(-4*a))*PolyLog[3, -E^(-2*(a + b*x))]))/(b^4*(-1 + E^(4*a))) + (x^3*Csch[2*a]*Csch[2*a + 2*b*x]*Sinh[2*b*x]
)/(2*b))
Maple [B] (verified)
Leaf count of result is larger than twice the leaf count of optimal. \(262\) vs. \(2(81)=162\).
Time = 6.98 (sec) , antiderivative size = 263, normalized size of antiderivative = 3.09
| | |
method | result | size |
| | |
risch |
\(-\frac {4 x^{3}}{b \left (1+{\mathrm e}^{2 b x +2 a}\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {8 a^{3}}{b^{4}}+\frac {12 x \,a^{2}}{b^{3}}-\frac {12 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{4}}-\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{4}}-\frac {4 x^{3}}{b}-\frac {6 \operatorname {polylog}\left (3, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {6 \operatorname {polylog}\left (3, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \operatorname {polylog}\left (3, -{\mathrm e}^{2 b x +2 a}\right )}{2 b^{4}}+\frac {3 x^{2} \ln \left (1+{\mathrm e}^{2 b x +2 a}\right )}{b^{2}}+\frac {3 x \operatorname {polylog}\left (2, -{\mathrm e}^{2 b x +2 a}\right )}{b^{3}}+\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right ) x^{2}}{b^{2}}+\frac {6 x \operatorname {polylog}\left (2, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b^{2}}+\frac {6 x \operatorname {polylog}\left (2, {\mathrm e}^{b x +a}\right )}{b^{3}}\) |
\(263\) |
| | |
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[In]
int(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x,method=_RETURNVERBOSE)
[Out]
-4*x^3/b/(1+exp(2*b*x+2*a))/(exp(2*b*x+2*a)-1)+8/b^4*a^3+12*x/b^3*a^2-12/b^4*a^2*ln(exp(b*x+a))+3/b^4*a^2*ln(e
xp(b*x+a)-1)-3/b^4*ln(1-exp(b*x+a))*a^2-4*x^3/b-6*polylog(3,-exp(b*x+a))/b^4-6*polylog(3,exp(b*x+a))/b^4-3/2*p
olylog(3,-exp(2*b*x+2*a))/b^4+3*x^2*ln(1+exp(2*b*x+2*a))/b^2+3*x*polylog(2,-exp(2*b*x+2*a))/b^3+3/b^2*ln(exp(b
*x+a)+1)*x^2+6*x*polylog(2,-exp(b*x+a))/b^3+3/b^2*ln(1-exp(b*x+a))*x^2+6*x*polylog(2,exp(b*x+a))/b^3
Fricas [C] (verification not implemented)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 1924, normalized size of antiderivative = 22.64
\[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\text {Too large to display}
\]
[In]
integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="fricas")
[Out]
-(4*(b^3*x^3 + a^3)*cosh(b*x + a)^4 + 16*(b^3*x^3 + a^3)*cosh(b*x + a)^3*sinh(b*x + a) + 24*(b^3*x^3 + a^3)*co
sh(b*x + a)^2*sinh(b*x + a)^2 + 16*(b^3*x^3 + a^3)*cosh(b*x + a)*sinh(b*x + a)^3 + 4*(b^3*x^3 + a^3)*sinh(b*x
+ a)^4 - 4*a^3 - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x
+ a)^2 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(cosh(b*x + a) + sinh(b*x + a)
) - 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b
*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(I*cosh(b*x + a) + I*sinh(b*x + a)) - 6*(b*
x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*x*cosh(b
*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(-I*cosh(b*x + a) - I*sinh(b*x + a)) - 6*(b*x*cosh(b
*x + a)^4 + 4*b*x*cosh(b*x + a)^3*sinh(b*x + a) + 6*b*x*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b*x*cosh(b*x + a)*
sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - b*x)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 3*(b^2*x^2*cosh(b*x + a)^
4 + 4*b^2*x^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*b^2*x^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*b^2*x^2*cosh(b*x +
a)*sinh(b*x + a)^3 + b^2*x^2*sinh(b*x + a)^4 - b^2*x^2)*log(cosh(b*x + a) + sinh(b*x + a) + 1) - 3*(a^2*cosh(
b*x + a)^4 + 4*a^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)
*sinh(b*x + a)^3 + a^2*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) + I) - 3*(a^2*cosh(b*x + a)^4
+ 4*a^2*cosh(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)*sinh(b*x +
a)^3 + a^2*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - I) - 3*(a^2*cosh(b*x + a)^4 + 4*a^2*cos
h(b*x + a)^3*sinh(b*x + a) + 6*a^2*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*a^2*cosh(b*x + a)*sinh(b*x + a)^3 + a^2
*sinh(b*x + a)^4 - a^2)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 3*((b^2*x^2 - a^2)*cosh(b*x + a)^4 + 4*(b^2*x
^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*(b^2*x^2 - a^2
)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(I*cosh(b*x + a) + I*sin
h(b*x + a) + 1) - 3*((b^2*x^2 - a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^
2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^
2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(-I*cosh(b*x + a) - I*sinh(b*x + a) + 1) - 3*((b^2*x^2 - a^2)*cosh(b*x
+ a)^4 + 4*(b^2*x^2 - a^2)*cosh(b*x + a)^3*sinh(b*x + a) + 6*(b^2*x^2 - a^2)*cosh(b*x + a)^2*sinh(b*x + a)^2 +
4*(b^2*x^2 - a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - a^2)*sinh(b*x + a)^4 - b^2*x^2 + a^2)*log(-cosh(
b*x + a) - sinh(b*x + a) + 1) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(
b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, cosh(b*x + a) + sinh(b*x + a))
+ 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*s
inh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, I*cosh(b*x + a) + I*sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*c
osh(b*x + a)^3*sinh(b*x + a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x
+ a)^4 - 1)*polylog(3, -I*cosh(b*x + a) - I*sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)^3*sinh(b*x +
a) + 6*cosh(b*x + a)^2*sinh(b*x + a)^2 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 - 1)*polylog(3, -c
osh(b*x + a) - sinh(b*x + a)))/(b^4*cosh(b*x + a)^4 + 4*b^4*cosh(b*x + a)^3*sinh(b*x + a) + 6*b^4*cosh(b*x + a
)^2*sinh(b*x + a)^2 + 4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4 - b^4)
Sympy [F]
\[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int x^{3} \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{2}{\left (a + b x \right )}\, dx
\]
[In]
integrate(x**3*csch(b*x+a)**2*sech(b*x+a)**2,x)
[Out]
Integral(x**3*csch(a + b*x)**2*sech(a + b*x)**2, x)
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (80) = 160\).
Time = 0.21 (sec) , antiderivative size = 180, normalized size of antiderivative = 2.12
\[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=-\frac {4 \, x^{3}}{b e^{\left (4 \, b x + 4 \, a\right )} - b} - \frac {4 \, x^{3}}{b} + \frac {3 \, {\left (2 \, b^{2} x^{2} \log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (2 \, b x + 2 \, a\right )}\right ) - {\rm Li}_{3}(-e^{\left (2 \, b x + 2 \, a\right )})\right )}}{2 \, b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})\right )}}{b^{4}} + \frac {3 \, {\left (b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})\right )}}{b^{4}}
\]
[In]
integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="maxima")
[Out]
-4*x^3/(b*e^(4*b*x + 4*a) - b) - 4*x^3/b + 3/2*(2*b^2*x^2*log(e^(2*b*x + 2*a) + 1) + 2*b*x*dilog(-e^(2*b*x + 2
*a)) - polylog(3, -e^(2*b*x + 2*a)))/b^4 + 3*(b^2*x^2*log(e^(b*x + a) + 1) + 2*b*x*dilog(-e^(b*x + a)) - 2*pol
ylog(3, -e^(b*x + a)))/b^4 + 3*(b^2*x^2*log(-e^(b*x + a) + 1) + 2*b*x*dilog(e^(b*x + a)) - 2*polylog(3, e^(b*x
+ a)))/b^4
Giac [F]
\[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int { x^{3} \operatorname {csch}\left (b x + a\right )^{2} \operatorname {sech}\left (b x + a\right )^{2} \,d x }
\]
[In]
integrate(x^3*csch(b*x+a)^2*sech(b*x+a)^2,x, algorithm="giac")
[Out]
integrate(x^3*csch(b*x + a)^2*sech(b*x + a)^2, x)
Mupad [F(-1)]
Timed out. \[
\int x^3 \text {csch}^2(a+b x) \text {sech}^2(a+b x) \, dx=\int \frac {x^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \,d x
\]
[In]
int(x^3/(cosh(a + b*x)^2*sinh(a + b*x)^2),x)
[Out]
int(x^3/(cosh(a + b*x)^2*sinh(a + b*x)^2), x)