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(b*x+a)/b-1/3*csch(b*x+a)^3/b

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Rubi [A]  time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ -\frac {\text {csch}^3(a+b x)}{3 b}-\frac {\text {csch}(a+b x)}{b} \]

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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fricas [B]  time = 0.39, size = 171, normalized size = 6.33 \[ -\frac {2 \, {\left (3 \, \cosh \left (b x + a\right )^{3} + 9 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 3 \, \sinh \left (b x + a\right )^{3} + {\left (9 \, \cosh \left (b x + a\right )^{2} - 5\right )} \sinh \left (b x + a\right ) + \cosh \left (b x + a\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 4 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - 2 \, b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3 \, b\right )}} \]

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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giac [A]  time = 0.15, size = 49, normalized size = 1.81 \[ -\frac {2 \, {\left (3 \, e^{\left (5 \, b x + 5 \, a\right )} - 2 \, e^{\left (3 \, b x + 3 \, a\right )} + 3 \, e^{\left (b x + a\right )}\right )}}{3 \, b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} \]

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)

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maple [A]  time = 0.18, size = 34, normalized size = 1.26 \[ \frac {-\frac {\cosh ^{2}\left (b x +a \right )}{\sinh \left (b x +a \right )^{3}}+\frac {2}{3 \sinh \left (b x +a \right )^{3}}}{b} \]

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

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maxima [B]  time = 0.32, size = 148, normalized size = 5.48 \[ \frac {2 \, e^{\left (-b x - a\right )}}{b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} - \frac {4 \, e^{\left (-3 \, b x - 3 \, a\right )}}{3 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} + \frac {2 \, e^{\left (-5 \, b x - 5 \, a\right )}}{b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} - 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} - 1\right )}} \]

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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mupad [B]  time = 1.41, size = 48, normalized size = 1.78 \[ -\frac {2\,{\mathrm {e}}^{a+b\,x}\,\left (3\,{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\right )}{3\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + b*x)^3/sinh(a + b*x),x)

[Out]

-(2*exp(a + b*x)*(3*exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 3))/(3*b*(exp(2*a + 2*b*x) - 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{3}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \]

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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