3.170 \(\int \frac{\text{csch}^{\frac{5}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=111 \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right ),2\right )}{3 b n} \]

[Out]

(-2*Cosh[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^(3/2))/(3*b*n) + (((2*I)/3)*Sqrt[Csch[a + b*Log[c*x^n]]]*Ell
ipticF[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n)

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Rubi [A]  time = 0.068457, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {3768, 3771, 2641} \[ -\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{2 i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (i a+i b \log \left (c x^n\right )-\frac{\pi }{2}\right )\right |2\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(-2*Cosh[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^(3/2))/(3*b*n) + (((2*I)/3)*Sqrt[Csch[a + b*Log[c*x^n]]]*Ell
ipticF[(I*a - Pi/2 + I*b*Log[c*x^n])/2, 2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]])/(b*n)

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 2641

Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticF[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ
[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\text{csch}^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}^{\frac{5}{2}}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\operatorname{Subst}\left (\int \sqrt{\text{csch}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac{\left (\sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac{2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text{csch}^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac{2 i \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b \log \left (c x^n\right )\right )\right |2\right ) \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n}\\ \end{align*}

Mathematica [A]  time = 0.207335, size = 84, normalized size = 0.76 \[ -\frac{2 \sqrt{\text{csch}\left (a+b \log \left (c x^n\right )\right )} \left (\coth \left (a+b \log \left (c x^n\right )\right )+i \sqrt{i \sinh \left (a+b \log \left (c x^n\right )\right )} \text{EllipticF}\left (\frac{1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right ),2\right )\right )}{3 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^(5/2)/x,x]

[Out]

(-2*Sqrt[Csch[a + b*Log[c*x^n]]]*(Coth[a + b*Log[c*x^n]] + I*EllipticF[((-2*I)*a + Pi - (2*I)*b*Log[c*x^n])/4,
2]*Sqrt[I*Sinh[a + b*Log[c*x^n]]]))/(3*b*n)

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Maple [A]  time = 0.036, size = 144, normalized size = 1.3 \begin{align*} -{\frac{1}{3\,n\cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) b} \left ( i\sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }\sqrt{i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) },{\frac{\sqrt{2}}{2}} \right ) \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) +2\, \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2} \right ) \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(a+b*ln(c*x^n))^(5/2)/x,x)

[Out]

-1/3/n/sinh(a+b*ln(c*x^n))^(3/2)*(I*(1-I*sinh(a+b*ln(c*x^n)))^(1/2)*2^(1/2)*(1+I*sinh(a+b*ln(c*x^n)))^(1/2)*(I
*sinh(a+b*ln(c*x^n)))^(1/2)*EllipticF((1-I*sinh(a+b*ln(c*x^n)))^(1/2),1/2*2^(1/2))*sinh(a+b*ln(c*x^n))+2*cosh(
a+b*ln(c*x^n))^2)/cosh(a+b*ln(c*x^n))/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="maxima")

[Out]

integrate(csch(b*log(c*x^n) + a)^(5/2)/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="fricas")

[Out]

integral(csch(b*log(c*x^n) + a)^(5/2)/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{\frac{5}{2}}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*ln(c*x**n))**(5/2)/x,x)

[Out]

Integral(csch(a + b*log(c*x**n))**(5/2)/x, x)

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Giac [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(csch(a+b*log(c*x^n))^(5/2)/x,x, algorithm="giac")

[Out]

Timed out