3.30 \(\int (a \text{csch}^2(x))^{3/2} \, dx\)

Optimal. Leaf size=46 \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (x)}{\sqrt{a \text{csch}^2(x)}}\right )-\frac{1}{2} a \coth (x) \sqrt{a \text{csch}^2(x)} \]

[Out]

(a^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]])/2 - (a*Coth[x]*Sqrt[a*Csch[x]^2])/2

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Rubi [A]  time = 0.0251034, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4122, 195, 217, 206} \[ \frac{1}{2} a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (x)}{\sqrt{a \text{csch}^2(x)}}\right )-\frac{1}{2} a \coth (x) \sqrt{a \text{csch}^2(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Csch[x]^2)^(3/2),x]

[Out]

(a^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a*Csch[x]^2]])/2 - (a*Coth[x]*Sqrt[a*Csch[x]^2])/2

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

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

Mathematica [A]  time = 0.0636664, size = 30, normalized size = 0.65 \[ -\frac{1}{2} a \sinh (x) \sqrt{a \text{csch}^2(x)} \left (\log \left (\tanh \left (\frac{x}{2}\right )\right )+\coth (x) \text{csch}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Csch[x]^2)^(3/2),x]

[Out]

-(a*Sqrt[a*Csch[x]^2]*(Coth[x]*Csch[x] + Log[Tanh[x/2]])*Sinh[x])/2

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Maple [B]  time = 0.053, size = 103, normalized size = 2.2 \begin{align*} -{\frac{a \left ({{\rm e}^{2\,x}}+1 \right ) }{{{\rm e}^{2\,x}}-1}\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}-{\frac{a{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}-1 \right ) }{2}\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}+{\frac{a{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{x}}+1 \right ) }{2}\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*csch(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.63817, size = 81, normalized size = 1.76 \begin{align*} -\frac{1}{2} \, a^{\frac{3}{2}} \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, a^{\frac{3}{2}} \log \left (e^{\left (-x\right )} - 1\right ) - \frac{a^{\frac{3}{2}} e^{\left (-x\right )} + a^{\frac{3}{2}} e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.6269, size = 1007, normalized size = 21.89 \begin{align*} \frac{{\left (2 \, a \cosh \left (x\right )^{3} - 2 \,{\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{3} - 6 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) - 2 \,{\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} -{\left (a \cosh \left (x\right )^{4} -{\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{4} - 4 \,{\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} - a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, a \cosh \left (x\right )^{2} + 2 \,{\left (3 \, a \cosh \left (x\right )^{2} -{\left (3 \, a \cosh \left (x\right )^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{2} -{\left (a \cosh \left (x\right )^{4} - 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \,{\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right ) -{\left (a \cosh \left (x\right )^{3} - a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \log \left (\frac{\cosh \left (x\right ) + \sinh \left (x\right ) + 1}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) + 2 \,{\left (3 \, a \cosh \left (x\right )^{2} -{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \,{\left (4 \, \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{3} + e^{x} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} e^{x} \sinh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} e^{x} \sinh \left (x\right ) +{\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{x}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/2*(2*a*cosh(x)^3 - 2*(a*e^(2*x) - a)*sinh(x)^3 - 6*(a*cosh(x)*e^(2*x) - a*cosh(x))*sinh(x)^2 + 2*a*cosh(x) -
2*(a*cosh(x)^3 + a*cosh(x))*e^(2*x) - (a*cosh(x)^4 - (a*e^(2*x) - a)*sinh(x)^4 - 4*(a*cosh(x)*e^(2*x) - a*cos
h(x))*sinh(x)^3 - 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 - (3*a*cosh(x)^2 - a)*e^(2*x) - a)*sinh(x)^2 - (a*cosh(x)^4
- 2*a*cosh(x)^2 + a)*e^(2*x) + 4*(a*cosh(x)^3 - a*cosh(x) - (a*cosh(x)^3 - a*cosh(x))*e^(2*x))*sinh(x) + a)*l
og((cosh(x) + sinh(x) + 1)/(cosh(x) + sinh(x) - 1)) + 2*(3*a*cosh(x)^2 - (3*a*cosh(x)^2 + a)*e^(2*x) + a)*sinh
(x))*sqrt(a/(e^(4*x) - 2*e^(2*x) + 1))*e^x/(4*cosh(x)*e^x*sinh(x)^3 + e^x*sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*e^x*
sinh(x)^2 + 4*(cosh(x)^3 - cosh(x))*e^x*sinh(x) + (cosh(x)^4 - 2*cosh(x)^2 + 1)*e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)**2)**(3/2),x)

[Out]

Integral((a*csch(x)**2)**(3/2), x)

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Giac [A]  time = 1.13455, size = 78, normalized size = 1.7 \begin{align*} -\frac{1}{4} \, a^{\frac{3}{2}}{\left (\frac{4 \,{\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm{sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

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