### 3.118 $$\int \coth ^3(a+b x) \text{csch}(a+b x) \, dx$$

Optimal. Leaf size=27 $-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b}$

[Out]

-(Csch[a + b*x]/b) - Csch[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0206844, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.067, Rules used = {2606} $-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[a + b*x]^3*Csch[a + b*x],x]

[Out]

-(Csch[a + b*x]/b) - Csch[a + b*x]^3/(3*b)

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps

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

Mathematica [A]  time = 0.0137484, size = 27, normalized size = 1. $-\frac{\text{csch}^3(a+b x)}{3 b}-\frac{\text{csch}(a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Csch[a + b*x]/b) - Csch[a + b*x]^3/(3*b)

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Maple [A]  time = 0.014, size = 50, normalized size = 1.9 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\, \left ( \sinh \left ( bx+a \right ) \right ) ^{3}}}-{\frac{2\, \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3\,\sinh \left ( bx+a \right ) }}+{\frac{2\,\sinh \left ( bx+a \right ) }{3}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/3/sinh(b*x+a)^3*cosh(b*x+a)^2-2/3/sinh(b*x+a)*cosh(b*x+a)^2+2/3*sinh(b*x+a))

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Maxima [B]  time = 1.05293, size = 200, normalized size = 7.41 \begin{align*} \frac{2 \, e^{\left (-b x - a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} - \frac{4 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} + \frac{2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*e^(-b*x - a)/(b*(3*e^(-2*b*x - 2*a) - 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) - 1)) - 4/3*e^(-3*b*x - 3*a)/(b*
(3*e^(-2*b*x - 2*a) - 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) - 1)) + 2*e^(-5*b*x - 5*a)/(b*(3*e^(-2*b*x - 2*a)
- 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) - 1))

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Fricas [B]  time = 1.7614, size = 464, normalized size = 17.19 \begin{align*} -\frac{2 \,{\left (3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \sinh \left (b x + a\right )^{3} +{\left (9 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )}}{3 \,{\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2} + 2 \,{\left (3 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \,{\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*cosh(b*x + a)^3 + 9*cosh(b*x + a)*sinh(b*x + a)^2 + 3*sinh(b*x + a)^3 + (9*cosh(b*x + a)^2 - 5)*sinh(b
*x + a) + cosh(b*x + a))/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*sinh(b*x + a)^3 + b*sinh(b*x + a)^4 - 4*b*cosh
(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 - 2*b)*sinh(b*x + a)^2 + 4*(b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x
+ a) + 3*b)

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + b*x)**3*csch(a + b*x), x)

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Giac [A]  time = 1.21634, size = 66, normalized size = 2.44 \begin{align*} -\frac{2 \,{\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{3 \, b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*e^(5*b*x + 5*a) - 2*e^(3*b*x + 3*a) + 3*e^(b*x + a))/(b*(e^(2*b*x + 2*a) - 1)^3)