### 3.462 $$\int x \coth ^3(a+b x) \, dx$$

Optimal. Leaf size=82 $\frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{\coth (a+b x)}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x \coth ^2(a+b x)}{2 b}+\frac{x}{2 b}-\frac{x^2}{2}$

[Out]

x/(2*b) - x^2/2 - Coth[a + b*x]/(2*b^2) - (x*Coth[a + b*x]^2)/(2*b) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog
[2, E^(2*(a + b*x))]/(2*b^2)

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Rubi [A]  time = 0.12246, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.7, Rules used = {3720, 3473, 8, 3716, 2190, 2279, 2391} $\frac{\text{PolyLog}\left (2,e^{2 (a+b x)}\right )}{2 b^2}-\frac{\coth (a+b x)}{2 b^2}+\frac{x \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac{x \coth ^2(a+b x)}{2 b}+\frac{x}{2 b}-\frac{x^2}{2}$

Antiderivative was successfully veriﬁed.

[In]

Int[x*Coth[a + b*x]^3,x]

[Out]

x/(2*b) - x^2/2 - Coth[a + b*x]/(2*b^2) - (x*Coth[a + b*x]^2)/(2*b) + (x*Log[1 - E^(2*(a + b*x))])/b + PolyLog
[2, E^(2*(a + b*x))]/(2*b^2)

Rule 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

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

Mathematica [C]  time = 6.1231, size = 232, normalized size = 2.83 $\frac{\text{csch}(a) \text{sech}(a) \left (-b^2 x^2 e^{-\tanh ^{-1}(\tanh (a))}+\frac{i \tanh (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )-b x \left (-\pi +2 i \tanh ^{-1}(\tanh (a))\right )-2 \left (i \tanh ^{-1}(\tanh (a))+i b x\right ) \log \left (1-e^{2 i \left (i \tanh ^{-1}(\tanh (a))+i b x\right )}\right )+2 i \tanh ^{-1}(\tanh (a)) \log \left (i \sinh \left (\tanh ^{-1}(\tanh (a))+b x\right )\right )-\pi \log \left (e^{2 b x}+1\right )+\pi \log (\cosh (b x))\right )}{\sqrt{1-\tanh ^2(a)}}\right )}{2 b^2 \sqrt{\text{sech}^2(a) \left (\cosh ^2(a)-\sinh ^2(a)\right )}}+\frac{\text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{2 b^2}-\frac{x \text{csch}^2(a+b x)}{2 b}+\frac{1}{2} x^2 \coth (a)$

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Coth[a + b*x]^3,x]

[Out]

(x^2*Coth[a])/2 - (x*Csch[a + b*x]^2)/(2*b) + (Csch[a]*Csch[a + b*x]*Sinh[b*x])/(2*b^2) + (Csch[a]*Sech[a]*(-(
(b^2*x^2)/E^ArcTanh[Tanh[a]]) + (I*(-(b*x*(-Pi + (2*I)*ArcTanh[Tanh[a]])) - Pi*Log[1 + E^(2*b*x)] - 2*(I*b*x +
I*ArcTanh[Tanh[a]])*Log[1 - E^((2*I)*(I*b*x + I*ArcTanh[Tanh[a]]))] + Pi*Log[Cosh[b*x]] + (2*I)*ArcTanh[Tanh[
a]]*Log[I*Sinh[b*x + ArcTanh[Tanh[a]]]] + I*PolyLog[2, E^((2*I)*(I*b*x + I*ArcTanh[Tanh[a]]))])*Tanh[a])/Sqrt[
1 - Tanh[a]^2]))/(2*b^2*Sqrt[Sech[a]^2*(Cosh[a]^2 - Sinh[a]^2)])

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Maple [B]  time = 0.083, size = 164, normalized size = 2. \begin{align*} -{\frac{{x}^{2}}{2}}-{\frac{2\,bx{{\rm e}^{2\,bx+2\,a}}+{{\rm e}^{2\,bx+2\,a}}-1}{{b}^{2} \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) ^{2}}}-2\,{\frac{ax}{b}}-{\frac{{a}^{2}}{{b}^{2}}}+{\frac{\ln \left ( 1+{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{{\it polylog} \left ( 2,-{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) x}{b}}+{\frac{\ln \left ( 1-{{\rm e}^{bx+a}} \right ) a}{{b}^{2}}}+{\frac{{\it polylog} \left ( 2,{{\rm e}^{bx+a}} \right ) }{{b}^{2}}}-{\frac{a\ln \left ({{\rm e}^{bx+a}}-1 \right ) }{{b}^{2}}}+2\,{\frac{a\ln \left ({{\rm e}^{bx+a}} \right ) }{{b}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*x^2-(2*b*x*exp(2*b*x+2*a)+exp(2*b*x+2*a)-1)/b^2/(exp(2*b*x+2*a)-1)^2-2/b*a*x-a^2/b^2+1/b*ln(1+exp(b*x+a))
*x+1/b^2*polylog(2,-exp(b*x+a))+1/b*ln(1-exp(b*x+a))*x+1/b^2*ln(1-exp(b*x+a))*a+1/b^2*polylog(2,exp(b*x+a))-1/
b^2*a*ln(exp(b*x+a)-1)+2/b^2*a*ln(exp(b*x+a))

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Maxima [B]  time = 1.35998, size = 201, normalized size = 2.45 \begin{align*} -x^{2} + \frac{b^{2} x^{2} e^{\left (4 \, b x + 4 \, a\right )} + b^{2} x^{2} - 2 \,{\left (b^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b x e^{\left (2 \, a\right )} + e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 2}{2 \,{\left (b^{2} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, b^{2} e^{\left (2 \, b x + 2 \, a\right )} + b^{2}\right )}} + \frac{b x \log \left (e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (b x + a\right )}\right )}{b^{2}} + \frac{b x \log \left (-e^{\left (b x + a\right )} + 1\right ) +{\rm Li}_2\left (e^{\left (b x + a\right )}\right )}{b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.52685, size = 2569, normalized size = 31.33 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*((b^2*x^2 - 2*a^2)*cosh(b*x + a)^4 + 4*(b^2*x^2 - 2*a^2)*cosh(b*x + a)*sinh(b*x + a)^3 + (b^2*x^2 - 2*a^2
)*sinh(b*x + a)^4 + b^2*x^2 - 2*(b^2*x^2 - 2*a^2 - 2*b*x - 1)*cosh(b*x + a)^2 - 2*(b^2*x^2 - 3*(b^2*x^2 - 2*a^
2)*cosh(b*x + a)^2 - 2*a^2 - 2*b*x - 1)*sinh(b*x + a)^2 - 2*a^2 - 2*(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)*dilog(cosh(b*x + a) + sinh(b*x + a)) - 2*(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*(cos
h(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*dilog(-cosh(b*x + a) - sinh(b*x + a)) - 2*(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))*log(cosh(b*x + a)
+ sinh(b*x + a) + 1) + 2*(a*cosh(b*x + a)^4 + 4*a*cosh(b*x + a)*sinh(b*x + a)^3 + a*sinh(b*x + a)^4 - 2*a*cos
h(b*x + a)^2 + 2*(3*a*cosh(b*x + a)^2 - a)*sinh(b*x + a)^2 + 4*(a*cosh(b*x + a)^3 - a*cosh(b*x + a))*sinh(b*x
+ a) + a)*log(cosh(b*x + a) + sinh(b*x + a) - 1) - 2*((b*x + a)*cosh(b*x + a)^4 + 4*(b*x + a)*cosh(b*x + a)*si
nh(b*x + a)^3 + (b*x + a)*sinh(b*x + a)^4 - 2*(b*x + a)*cosh(b*x + a)^2 + 2*(3*(b*x + a)*cosh(b*x + a)^2 - b*x
- a)*sinh(b*x + a)^2 + b*x + 4*((b*x + a)*cosh(b*x + a)^3 - (b*x + a)*cosh(b*x + a))*sinh(b*x + a) + a)*log(-
cosh(b*x + a) - sinh(b*x + a) + 1) + 4*((b^2*x^2 - 2*a^2)*cosh(b*x + a)^3 - (b^2*x^2 - 2*a^2 - 2*b*x - 1)*cosh
(b*x + a))*sinh(b*x + a) - 2)/(b^2*cosh(b*x + a)^4 + 4*b^2*cosh(b*x + a)*sinh(b*x + a)^3 + b^2*sinh(b*x + a)^4
- 2*b^2*cosh(b*x + a)^2 + 2*(3*b^2*cosh(b*x + a)^2 - b^2)*sinh(b*x + a)^2 + b^2 + 4*(b^2*cosh(b*x + a)^3 - b^
2*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*cosh(b*x+a)**3*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 \cosh \left (b x + a\right )^{3} \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*cosh(b*x+a)^3*csch(b*x+a)^3,x, algorithm="giac")

[Out]

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