3.453 \(\int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx\)

Optimal. Leaf size=201 \[ -\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}-\frac {3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b} \]

[Out]

-6*x*arctanh(exp(b*x+a))/b^3-x^3*arctanh(exp(b*x+a))/b-3/2*x^2*csch(b*x+a)/b^2-1/2*x^3*coth(b*x+a)*csch(b*x+a)
/b-3*polylog(2,-exp(b*x+a))/b^4-3/2*x^2*polylog(2,-exp(b*x+a))/b^2+3*polylog(2,exp(b*x+a))/b^4+3/2*x^2*polylog
(2,exp(b*x+a))/b^2+3*x*polylog(3,-exp(b*x+a))/b^3-3*x*polylog(3,exp(b*x+a))/b^3-3*polylog(4,-exp(b*x+a))/b^4+3
*polylog(4,exp(b*x+a))/b^4

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Rubi [A]  time = 0.36, antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5457, 4182, 2531, 6609, 2282, 6589, 4186, 2279, 2391} \[ -\frac {3 x^2 \text {PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac {3 x \text {PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac {3 \text {PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac {3 \text {PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac {3 \text {PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac {3 \text {PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*x*ArcTanh[E^(a + b*x)])/b^3 - (x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*Csch[a + b*x])/(2*b^2) - (x^3*Coth[a +
b*x]*Csch[a + b*x])/(2*b) - (3*PolyLog[2, -E^(a + b*x)])/b^4 - (3*x^2*PolyLog[2, -E^(a + b*x)])/(2*b^2) + (3*
PolyLog[2, E^(a + b*x)])/b^4 + (3*x^2*PolyLog[2, E^(a + b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(a + b*x)])/b^3 -
(3*x*PolyLog[3, E^(a + b*x)])/b^3 - (3*PolyLog[4, -E^(a + b*x)])/b^4 + (3*PolyLog[4, E^(a + b*x)])/b^4

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 4182

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

Rule 4186

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

Rule 5457

Int[Coth[(a_.) + (b_.)*(x_)]^(p_)*Csch[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d
*x)^m*Csch[a + b*x]*Coth[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csch[a + b*x]^3*Coth[a + b*x]^(p - 2), x] /; F
reeQ[{a, b, c, d, m}, x] && IGtQ[p/2, 0]

Rule 6589

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

Rule 6609

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

Rubi steps

\begin {align*} \int x^3 \coth ^2(a+b x) \text {csch}(a+b x) \, dx &=\int x^3 \text {csch}(a+b x) \, dx+\int x^3 \text {csch}^3(a+b x) \, dx\\ &=-\frac {2 x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \int x^3 \text {csch}(a+b x) \, dx+\frac {3 \int x \text {csch}(a+b x) \, dx}{b^2}-\frac {3 \int x^2 \log \left (1-e^{a+b x}\right ) \, dx}{b}+\frac {3 \int x^2 \log \left (1+e^{a+b x}\right ) \, dx}{b}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{b^2}-\frac {3 \int \log \left (1-e^{a+b x}\right ) \, dx}{b^3}+\frac {3 \int \log \left (1+e^{a+b x}\right ) \, dx}{b^3}+\frac {6 \int x \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}-\frac {6 \int x \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}+\frac {3 \int x^2 \log \left (1-e^{a+b x}\right ) \, dx}{2 b}-\frac {3 \int x^2 \log \left (1+e^{a+b x}\right ) \, dx}{2 b}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {6 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {6 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {6 \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}+\frac {6 \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}-\frac {3 \int x \text {Li}_2\left (-e^{a+b x}\right ) \, dx}{b^2}+\frac {3 \int x \text {Li}_2\left (e^{a+b x}\right ) \, dx}{b^2}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {6 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}+\frac {3 \int \text {Li}_3\left (-e^{a+b x}\right ) \, dx}{b^3}-\frac {3 \int \text {Li}_3\left (e^{a+b x}\right ) \, dx}{b^3}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {6 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {6 \text {Li}_4\left (e^{a+b x}\right )}{b^4}+\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}-\frac {3 \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{a+b x}\right )}{b^4}\\ &=-\frac {6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac {x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac {3 x^2 \text {csch}(a+b x)}{2 b^2}-\frac {x^3 \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {3 \text {Li}_2\left (-e^{a+b x}\right )}{b^4}-\frac {3 x^2 \text {Li}_2\left (-e^{a+b x}\right )}{2 b^2}+\frac {3 \text {Li}_2\left (e^{a+b x}\right )}{b^4}+\frac {3 x^2 \text {Li}_2\left (e^{a+b x}\right )}{2 b^2}+\frac {3 x \text {Li}_3\left (-e^{a+b x}\right )}{b^3}-\frac {3 x \text {Li}_3\left (e^{a+b x}\right )}{b^3}-\frac {3 \text {Li}_4\left (-e^{a+b x}\right )}{b^4}+\frac {3 \text {Li}_4\left (e^{a+b x}\right )}{b^4}\\ \end {align*}

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Mathematica [A]  time = 6.92, size = 280, normalized size = 1.39 \[ -\frac {-4 b^3 x^3 \log \left (1-e^{a+b x}\right )+4 b^3 x^3 \log \left (e^{a+b x}+1\right )+b^3 x^3 \text {csch}^2\left (\frac {1}{2} (a+b x)\right )+b^3 x^3 \text {sech}^2\left (\frac {1}{2} (a+b x)\right )+12 \left (b^2 x^2+2\right ) \text {Li}_2\left (-e^{a+b x}\right )-12 \left (b^2 x^2+2\right ) \text {Li}_2\left (e^{a+b x}\right )+12 b^2 x^2 \text {csch}(a)-6 b^2 x^2 \text {csch}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {csch}\left (\frac {1}{2} (a+b x)\right )-6 b^2 x^2 \text {sech}\left (\frac {a}{2}\right ) \sinh \left (\frac {b x}{2}\right ) \text {sech}\left (\frac {1}{2} (a+b x)\right )-24 b x \text {Li}_3\left (-e^{a+b x}\right )+24 b x \text {Li}_3\left (e^{a+b x}\right )+24 \text {Li}_4\left (-e^{a+b x}\right )-24 \text {Li}_4\left (e^{a+b x}\right )-24 b x \log \left (1-e^{a+b x}\right )+24 b x \log \left (e^{a+b x}+1\right )}{8 b^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/8*(12*b^2*x^2*Csch[a] + b^3*x^3*Csch[(a + b*x)/2]^2 - 24*b*x*Log[1 - E^(a + b*x)] - 4*b^3*x^3*Log[1 - E^(a
+ b*x)] + 24*b*x*Log[1 + E^(a + b*x)] + 4*b^3*x^3*Log[1 + E^(a + b*x)] + 12*(2 + b^2*x^2)*PolyLog[2, -E^(a + b
*x)] - 12*(2 + b^2*x^2)*PolyLog[2, E^(a + b*x)] - 24*b*x*PolyLog[3, -E^(a + b*x)] + 24*b*x*PolyLog[3, E^(a + b
*x)] + 24*PolyLog[4, -E^(a + b*x)] - 24*PolyLog[4, E^(a + b*x)] + b^3*x^3*Sech[(a + b*x)/2]^2 - 6*b^2*x^2*Csch
[a/2]*Csch[(a + b*x)/2]*Sinh[(b*x)/2] - 6*b^2*x^2*Sech[a/2]*Sech[(a + b*x)/2]*Sinh[(b*x)/2])/b^4

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fricas [C]  time = 0.46, size = 1802, normalized size = 8.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*csch(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/2*(2*(b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^3 + 6*(b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)*sinh(b*x + a)^2 + 2*(b^3
*x^3 + 3*b^2*x^2)*sinh(b*x + a)^3 + 2*(b^3*x^3 - 3*b^2*x^2)*cosh(b*x + a) - 3*((b^2*x^2 + 2)*cosh(b*x + a)^4 +
4*(b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 2)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 + 2)*cos
h(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*((b^2*x^2 + 2)*cosh(b*x +
a)^3 - (b^2*x^2 + 2)*cosh(b*x + a))*sinh(b*x + a) + 2)*dilog(cosh(b*x + a) + sinh(b*x + a)) + 3*((b^2*x^2 + 2
)*cosh(b*x + a)^4 + 4*(b^2*x^2 + 2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 + 2)*sinh(b*x + a)^4 + b^2*x^2 -
2*(b^2*x^2 + 2)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 + 2)*cosh(b*x + a)^2 + 2)*sinh(b*x + a)^2 + 4*((b^2*
x^2 + 2)*cosh(b*x + a)^3 - (b^2*x^2 + 2)*cosh(b*x + a))*sinh(b*x + a) + 2)*dilog(-cosh(b*x + a) - sinh(b*x + a
)) + (b^3*x^3 + (b^3*x^3 + 6*b*x)*cosh(b*x + a)^4 + 4*(b^3*x^3 + 6*b*x)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x
^3 + 6*b*x)*sinh(b*x + a)^4 - 2*(b^3*x^3 + 6*b*x)*cosh(b*x + a)^2 - 2*(b^3*x^3 - 3*(b^3*x^3 + 6*b*x)*cosh(b*x
+ a)^2 + 6*b*x)*sinh(b*x + a)^2 + 6*b*x + 4*((b^3*x^3 + 6*b*x)*cosh(b*x + a)^3 - (b^3*x^3 + 6*b*x)*cosh(b*x +
a))*sinh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + ((a^3 + 6*a)*cosh(b*x + a)^4 + 4*(a^3 + 6*a)*cosh(
b*x + a)*sinh(b*x + a)^3 + (a^3 + 6*a)*sinh(b*x + a)^4 + a^3 - 2*(a^3 + 6*a)*cosh(b*x + a)^2 - 2*(a^3 - 3*(a^3
+ 6*a)*cosh(b*x + a)^2 + 6*a)*sinh(b*x + a)^2 + 4*((a^3 + 6*a)*cosh(b*x + a)^3 - (a^3 + 6*a)*cosh(b*x + a))*s
inh(b*x + a) + 6*a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - (b^3*x^3 + (b^3*x^3 + a^3 + 6*b*x + 6*a)*cosh(b*x
+ a)^4 + 4*(b^3*x^3 + a^3 + 6*b*x + 6*a)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^3*x^3 + a^3 + 6*b*x + 6*a)*sinh(b
*x + a)^4 + a^3 - 2*(b^3*x^3 + a^3 + 6*b*x + 6*a)*cosh(b*x + a)^2 - 2*(b^3*x^3 + a^3 - 3*(b^3*x^3 + a^3 + 6*b*
x + 6*a)*cosh(b*x + a)^2 + 6*b*x + 6*a)*sinh(b*x + a)^2 + 6*b*x + 4*((b^3*x^3 + a^3 + 6*b*x + 6*a)*cosh(b*x +
a)^3 - (b^3*x^3 + a^3 + 6*b*x + 6*a)*cosh(b*x + a))*sinh(b*x + a) + 6*a)*log(-cosh(b*x + a) - sinh(b*x + a) +
1) - 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b
*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*polylog(4, cosh(b*x + a
) + sinh(b*x + a)) + 6*(cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x +
a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*polylog
(4, -cosh(b*x + a) - sinh(b*x + a)) + 6*(b*x*cosh(b*x + a)^4 + 4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(
b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x +
a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*polylog(3, cosh(b*x + a) + sinh(b*x + a)) - 6*(b*x*cosh(b*x + a)^4 +
4*b*x*cosh(b*x + a)*sinh(b*x + a)^3 + b*x*sinh(b*x + a)^4 - 2*b*x*cosh(b*x + a)^2 + 2*(3*b*x*cosh(b*x + a)^2
- b*x)*sinh(b*x + a)^2 + b*x + 4*(b*x*cosh(b*x + a)^3 - b*x*cosh(b*x + a))*sinh(b*x + a))*polylog(3, -cosh(b*x
+ a) - sinh(b*x + a)) + 2*(b^3*x^3 - 3*b^2*x^2 + 3*(b^3*x^3 + 3*b^2*x^2)*cosh(b*x + a)^2)*sinh(b*x + a))/(b^4
*cosh(b*x + a)^4 + 4*b^4*cosh(b*x + a)*sinh(b*x + a)^3 + b^4*sinh(b*x + a)^4 - 2*b^4*cosh(b*x + a)^2 + b^4 + 2
*(3*b^4*cosh(b*x + a)^2 - b^4)*sinh(b*x + a)^2 + 4*(b^4*cosh(b*x + a)^3 - b^4*cosh(b*x + a))*sinh(b*x + a))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \cosh \left (b x + a\right )^{2} \operatorname {csch}\left (b x + a\right )^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*csch(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^3*cosh(b*x + a)^2*csch(b*x + a)^3, x)

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maple [A]  time = 0.76, size = 340, normalized size = 1.69 \[ -\frac {x^{2} {\mathrm e}^{b x +a} \left (b x \,{\mathrm e}^{2 b x +2 a}+b x +3 \,{\mathrm e}^{2 b x +2 a}-3\right )}{b^{2} \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}-\frac {3 a \ln \left (1+{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) a}{b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}-\frac {3 x^{2} \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{2 b^{2}}+\frac {3 x \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{3}}{2 b}+\frac {3 x^{2} \polylog \left (2, {\mathrm e}^{b x +a}\right )}{2 b^{2}}-\frac {3 x \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {3 \ln \left (1-{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 \ln \left (1+{\mathrm e}^{b x +a}\right ) x}{b^{3}}-\frac {3 \polylog \left (2, -{\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {3 \polylog \left (4, -{\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (2, {\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {3 \polylog \left (4, {\mathrm e}^{b x +a}\right )}{b^{4}}-\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{3}}{2 b^{4}}+\frac {6 a \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}}+\frac {a^{3} \arctanh \left ({\mathrm e}^{b x +a}\right )}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*cosh(b*x+a)^2*csch(b*x+a)^3,x)

[Out]

-x^2*exp(b*x+a)*(b*x*exp(2*b*x+2*a)+b*x+3*exp(2*b*x+2*a)-3)/b^2/(exp(2*b*x+2*a)-1)^2-3/b^4*a*ln(1+exp(b*x+a))+
3/b^4*ln(1-exp(b*x+a))*a-1/2/b*ln(1+exp(b*x+a))*x^3-3/2*x^2*polylog(2,-exp(b*x+a))/b^2+3*x*polylog(3,-exp(b*x+
a))/b^3+1/2/b*ln(1-exp(b*x+a))*x^3+3/2*x^2*polylog(2,exp(b*x+a))/b^2-3*x*polylog(3,exp(b*x+a))/b^3+3/b^3*ln(1-
exp(b*x+a))*x-3/b^3*ln(1+exp(b*x+a))*x-3*polylog(2,-exp(b*x+a))/b^4-3*polylog(4,-exp(b*x+a))/b^4+3*polylog(2,e
xp(b*x+a))/b^4+3*polylog(4,exp(b*x+a))/b^4-1/2/b^4*ln(1+exp(b*x+a))*a^3+1/2/b^4*ln(1-exp(b*x+a))*a^3+6/b^4*a*a
rctanh(exp(b*x+a))+1/b^4*a^3*arctanh(exp(b*x+a))

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maxima [A]  time = 0.54, size = 262, normalized size = 1.30 \[ -\frac {{\left (b x^{3} e^{\left (3 \, a\right )} + 3 \, x^{2} e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )} + {\left (b x^{3} e^{a} - 3 \, x^{2} e^{a}\right )} e^{\left (b x\right )}}{b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}} - \frac {b^{3} x^{3} \log \left (e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(-e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(-e^{\left (b x + a\right )})}{2 \, b^{4}} + \frac {b^{3} x^{3} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 3 \, b^{2} x^{2} {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 6 \, b x {\rm Li}_{3}(e^{\left (b x + a\right )}) + 6 \, {\rm Li}_{4}(e^{\left (b x + a\right )})}{2 \, b^{4}} - \frac {3 \, {\left (b x \log \left (e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (-e^{\left (b x + a\right )}\right )\right )}}{b^{4}} + \frac {3 \, {\left (b x \log \left (-e^{\left (b x + a\right )} + 1\right ) + {\rm Li}_2\left (e^{\left (b x + a\right )}\right )\right )}}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*cosh(b*x+a)^2*csch(b*x+a)^3,x, algorithm="maxima")

[Out]

-((b*x^3*e^(3*a) + 3*x^2*e^(3*a))*e^(3*b*x) + (b*x^3*e^a - 3*x^2*e^a)*e^(b*x))/(b^2*e^(4*b*x + 4*a) - 2*b^2*e^
(2*b*x + 2*a) + b^2) - 1/2*(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 + 1/2*(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 - 3*(b*x*log(e^(b*x + a) + 1) + dilog(-e^
(b*x + a)))/b^4 + 3*(b*x*log(-e^(b*x + a) + 1) + dilog(e^(b*x + a)))/b^4

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3*cosh(a + b*x)^2)/sinh(a + b*x)^3,x)

[Out]

int((x^3*cosh(a + b*x)^2)/sinh(a + b*x)^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*cosh(b*x+a)**2*csch(b*x+a)**3,x)

[Out]

Timed out

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