\(\int \frac {\text {csch}^{\frac {5}{2}}(a+b \log (c x^n))}{x} \, dx\) [170]

Optimal result

Integrand size = 19, antiderivative size = 111 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 i \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \]

[Out]

-2/3*cosh(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))^(3/2)/b/n-2/3*I*(sin(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))^2)^(1/2)/s
in(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n))*EllipticF(cos(1/2*I*a+1/4*Pi+1/2*I*b*ln(c*x^n)),2^(1/2))*csch(a+b*ln(c*x^
n))^(1/2)*(I*sinh(a+b*ln(c*x^n)))^(1/2)/b/n

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3853, 3856, 2720} \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right ) \text {csch}^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b n} \]

[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 2720

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

Rule 3853

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 3856

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*} \text {integral}& = \frac {\text {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 {\text {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 ) \text {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 )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a-\frac {\pi }{2}+i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}}{3 b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \sqrt {\text {csch}\left (a+b \log \left (c x^n\right )\right )} \left (\coth \left (a+b \log \left (c x^n\right )\right )+i \operatorname {EllipticF}\left (\frac {1}{4} \left (-2 i a+\pi -2 i b \log \left (c x^n\right )\right ),2\right ) \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \]

[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)

Maple [A] (verified)

Time = 1.59 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.30

method	result	size
derivativedivides	\(-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\)	\(144\)
default	\(-\frac {i \sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \sqrt {i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (a +b \ln \left (c \,x^{n}\right )\right )+2 {\cosh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3 n {\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}} \cosh \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\)	\(144\)

[In]

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

[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

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 318, normalized size of antiderivative = 2.86 \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \, {\left (\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1}} + {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2}\right )} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \]

[In]

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

[Out]

-2/3*(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c
) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 + 1)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*lo
g(c) + a))/(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) +
a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 1)) + (sqrt(2)*cosh(b*n*log(x) + b*log(c) + a)^2 + 2*sqrt(2)*cosh(b*
n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*log(x) + b*log(c) + a)^2 - sqrt(2)
)*weierstrassPInverse(4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)))/(b*n*cosh(b*n*
log(x) + b*log(c) + a)^2 + 2*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a) + b*n*sinh(b*
n*log(x) + b*log(c) + a)^2 - b*n)

Sympy [F]

\[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{\frac {5}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[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)

Maxima [F]

\[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\operatorname {csch}\left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \]

[In]

int((1/sinh(a + b*log(c*x^n)))^(5/2)/x,x)

[Out]

int((1/sinh(a + b*log(c*x^n)))^(5/2)/x, x)