3.171 \(\int \frac {\text {csch}^{\frac {3}{2}}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=107 \[ -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

-2*cosh(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))^(1/2)/b/n+2*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))^2)^(1/2)/sin(1
/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))*EllipticE(cos(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))/b/n/csch(a+b*ln(c*x^
n))^(1/2)/(I*sinh(a+b*ln(c*x^n)))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 107, normalized size of antiderivative = 1.00, 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, 2639} \[ -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}}{b n}-\frac {2 i E\left (\left .\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

Rule 2639

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

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.12, size = 80, normalized size = 0.75 \[ -\frac {2 \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \left (\cosh \left (a+b \log \left (c x^n\right )\right )-\sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{4} \left (-2 i a-2 i b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]^(3/2)/x,x]

[Out]

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

________________________________________________________________________________________

fricas [F]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^(3/2)/x,x, algorithm="fricas")

[Out]

integral(csch(b*log(c*x^n) + a)^(3/2)/x, x)

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^(3/2)/x,x, algorithm="giac")

[Out]

Timed out

________________________________________________________________________________________

maple [A]  time = 0.55, size = 212, normalized size = 1.98 \[ \frac {2 \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(a+b*ln(c*x^n))^(3/2)/x,x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*log(c*x^n))^(3/2)/x,x, algorithm="maxima")

[Out]

integrate(csch(b*log(c*x^n) + a)^(3/2)/x, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/sinh(a + b*log(c*x^n)))^(3/2)/x,x)

[Out]

int((1/sinh(a + b*log(c*x^n)))^(3/2)/x, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+b*ln(c*x**n))**(3/2)/x,x)

[Out]

Integral(csch(a + b*log(c*x**n))**(3/2)/x, x)

________________________________________________________________________________________