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

Optimal. Leaf size=201 $-\frac{3 x^2 \text{PolyLog}\left (2,-e^{a+b x}\right )}{2 b^2}+\frac{3 x^2 \text{PolyLog}\left (2,e^{a+b x}\right )}{2 b^2}+\frac{3 x \text{PolyLog}\left (3,-e^{a+b x}\right )}{b^3}-\frac{3 x \text{PolyLog}\left (3,e^{a+b x}\right )}{b^3}-\frac{3 \text{PolyLog}\left (2,-e^{a+b x}\right )}{b^4}+\frac{3 \text{PolyLog}\left (2,e^{a+b x}\right )}{b^4}-\frac{3 \text{PolyLog}\left (4,-e^{a+b x}\right )}{b^4}+\frac{3 \text{PolyLog}\left (4,e^{a+b x}\right )}{b^4}-\frac{3 x^2 \text{csch}(a+b x)}{2 b^2}-\frac{6 x \tanh ^{-1}\left (e^{a+b x}\right )}{b^3}-\frac{x^3 \tanh ^{-1}\left (e^{a+b x}\right )}{b}-\frac{x^3 \coth (a+b x) \text{csch}(a+b x)}{2 b}$

[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

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Rubi [A]  time = 0.35973, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 9, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, 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 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 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 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]

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 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 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]

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*}

Mathematica [A]  time = 6.84958, size = 280, normalized size = 1.39 $-\frac{12 \left (b^2 x^2+2\right ) \text{PolyLog}\left (2,-e^{a+b x}\right )-12 \left (b^2 x^2+2\right ) \text{PolyLog}\left (2,e^{a+b x}\right )-24 b x \text{PolyLog}\left (3,-e^{a+b x}\right )+24 b x \text{PolyLog}\left (3,e^{a+b x}\right )+24 \text{PolyLog}\left (4,-e^{a+b x}\right )-24 \text{PolyLog}\left (4,e^{a+b x}\right )-4 b^3 x^3 \log \left (1-e^{a+b x}\right )+4 b^3 x^3 \log \left (e^{a+b x}+1\right )+b^3 x^3 \text{csch}^2\left (\frac{1}{2} (a+b x)\right )+12 b^2 x^2 \text{csch}(a)+b^3 x^3 \text{sech}^2\left (\frac{1}{2} (a+b x)\right )-6 b^2 x^2 \text{csch}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{csch}\left (\frac{1}{2} (a+b x)\right )-6 b^2 x^2 \text{sech}\left (\frac{a}{2}\right ) \sinh \left (\frac{b x}{2}\right ) \text{sech}\left (\frac{1}{2} (a+b x)\right )-24 b x \log \left (1-e^{a+b x}\right )+24 b x \log \left (e^{a+b x}+1\right )}{8 b^4}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(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])/(8*b^4)

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Maple [A]  time = 0.078, size = 340, normalized size = 1.7 \begin{align*} -{\frac{{x}^{2}{{\rm e}^{bx+a}} \left ( bx{{\rm e}^{2\,bx+2\,a}}+bx+3\,{{\rm e}^{2\,bx+2\,a}}-3 \right ) }{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}-3\,{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{{\it polylog} \left ( 4,-{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+6\,{\frac{a{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+{\frac{{a}^{3}{\it Artanh} \left ({{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 4,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}-3\,{\frac{a\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{{b}^{4}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{4}}}-{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ){x}^{3}}{2\,b}}-{\frac{3\,{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ){x}^{2}}{2\,{b}^{2}}}+3\,{\frac{{\it polylog} \left ( 3,-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){x}^{3}}{2\,b}}+{\frac{3\,{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ){x}^{2}}{2\,{b}^{2}}}-3\,{\frac{{\it polylog} \left ( 3,{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}-3\,{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}+3\,{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{{b}^{3}}}-{\frac{{a}^{3}\ln \left ( 1+{{\rm e}^{bx+a}} \right ) }{2\,{b}^{4}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ){a}^{3}}{2\,{b}^{4}}} \end{align*}

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*polylog(2,-exp(b*x+
a))-3/b^4*polylog(4,-exp(b*x+a))+6/b^4*a*arctanh(exp(b*x+a))+1/b^4*a^3*arctanh(exp(b*x+a))+3/b^4*polylog(4,exp
(b*x+a))+3/b^4*polylog(2,exp(b*x+a))-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/b^2*polylog(2,-exp(b*x+a))*x^2+3/b^3*polylog(3,-exp(b*x+a))*x+1/2/b*ln(1-exp(b*x+a))*x^3+3/2/b^2*polyl
og(2,exp(b*x+a))*x^2-3/b^3*polylog(3,exp(b*x+a))*x-3/b^3*ln(1+exp(b*x+a))*x+3/b^3*ln(1-exp(b*x+a))*x-1/2/b^4*l
n(1+exp(b*x+a))*a^3+1/2/b^4*ln(1-exp(b*x+a))*a^3

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Maxima [A]  time = 1.62837, size = 354, normalized size = 1.76 \begin{align*} -\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}} \end{align*}

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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Fricas [C]  time = 2.48115, size = 4602, normalized size = 22.9 \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)^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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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)**2*csch(b*x+a)**3,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 )^{2} \operatorname{csch}\left (b x + a\right )^{3}\,{d x} \end{align*}

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)