3.16 \(\int (1+\text{csch}^2(x))^{3/2} \, dx\)

Optimal. Leaf size=29 \[ \tanh (x) \sqrt{\coth ^2(x)} \log (\sinh (x))-\frac{1}{2} \tanh (x) \coth ^2(x)^{3/2} \]

[Out]

-((Coth[x]^2)^(3/2)*Tanh[x])/2 + Sqrt[Coth[x]^2]*Log[Sinh[x]]*Tanh[x]

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Rubi [A]  time = 0.0382828, antiderivative size = 29, 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 = {4121, 3658, 3473, 3475} \[ \tanh (x) \sqrt{\coth ^2(x)} \log (\sinh (x))-\frac{1}{2} \tanh (x) \coth ^2(x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(1 + Csch[x]^2)^(3/2),x]

[Out]

-((Coth[x]^2)^(3/2)*Tanh[x])/2 + Sqrt[Coth[x]^2]*Log[Sinh[x]]*Tanh[x]

Rule 4121

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0156797, size = 24, normalized size = 0.83 \[ -\frac{1}{2} \tanh (x) \sqrt{\coth ^2(x)} \left (\text{csch}^2(x)-2 \log (\sinh (x))\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Csch[x]^2)^(3/2),x]

[Out]

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

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Maple [B]  time = 0.107, size = 120, normalized size = 4.1 \begin{align*} -{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) x}{{{\rm e}^{2\,x}}+1}\sqrt{{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}-2\,{\frac{{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) \left ({{\rm e}^{2\,x}}-1 \right ) }\sqrt{{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}}+{\frac{ \left ({{\rm e}^{2\,x}}-1 \right ) \ln \left ({{\rm e}^{2\,x}}-1 \right ) }{{{\rm e}^{2\,x}}+1}\sqrt{{\frac{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+csch(x)^2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.7264, size = 59, normalized size = 2.03 \begin{align*} -x - \frac{2 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csch(x)^2)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.18325, size = 620, normalized size = 21.38 \begin{align*} -\frac{x \cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + x \sinh \left (x\right )^{4} - 2 \,{\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, x \cosh \left (x\right )^{2} - x + 1\right )} \sinh \left (x\right )^{2} -{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \,{\left (x \cosh \left (x\right )^{3} -{\left (x - 1\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + x}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csch(x)^2)^(3/2),x, algorithm="fricas")

[Out]

-(x*cosh(x)^4 + 4*x*cosh(x)*sinh(x)^3 + x*sinh(x)^4 - 2*(x - 1)*cosh(x)^2 + 2*(3*x*cosh(x)^2 - x + 1)*sinh(x)^
2 - (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3
- cosh(x))*sinh(x) + 1)*log(2*sinh(x)/(cosh(x) - sinh(x))) + 4*(x*cosh(x)^3 - (x - 1)*cosh(x))*sinh(x) + x)/(c
osh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh
(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\operatorname{csch}^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csch(x)**2)**(3/2),x)

[Out]

Integral((csch(x)**2 + 1)**(3/2), x)

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Giac [B]  time = 1.19135, size = 99, normalized size = 3.41 \begin{align*} -x \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac{3 \, e^{\left (4 \, x\right )} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - 2 \, e^{\left (2 \, x\right )} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 3 \, \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{2 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+csch(x)^2)^(3/2),x, algorithm="giac")

[Out]

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