\(\int \frac {1}{x \sinh ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [284]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.04 (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 = {2716, 2721, 2720} \[ \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {2 i \sqrt {i \sinh \left (a+b \log \left (c x^n\right )\right )} \operatorname {EllipticF}\left (\frac {1}{2} \left (i a+i b \log \left (c x^n\right )-\frac {\pi }{2}\right ),2\right )}{3 b n \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/(x*Sinh[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

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

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*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 2721

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sinh ^{\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 )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\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 )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}-\frac {\sqrt {i \sinh \left (a+b \log \left (c x^n\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 \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \\ & = -\frac {2 \cosh \left (a+b \log \left (c x^n\right )\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {2 i \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 \sqrt {\sinh \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.08 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.10 \[ \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \left (\cosh \left (a+b \log \left (c x^n\right )\right )+\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right ) \sqrt {1-\cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-\sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )}\right )}{3 b n \sinh ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]

[In]

Integrate[1/(x*Sinh[a + b*Log[c*x^n]]^(5/2)),x]

[Out]

(-2*(Cosh[a + b*Log[c*x^n]] + Hypergeometric2F1[1/4, 1/2, 5/4, Cosh[2*(a + b*Log[c*x^n])] + Sinh[2*(a + b*Log[
c*x^n])]]*Sinh[a + b*Log[c*x^n]]*Sqrt[1 - Cosh[2*(a + b*Log[c*x^n])] - Sinh[2*(a + b*Log[c*x^n])]]))/(3*b*n*Si
nh[a + b*Log[c*x^n]]^(3/2))

Maple [A] (verified)

Time = 0.86 (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(1/x/sinh(a+b*ln(c*x^n))^(5/2),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 = 0.10 (sec) , antiderivative size = 504, normalized size of antiderivative = 4.54 \[ \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, \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 )^{3} + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2}\right )} \sinh \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 )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - \sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \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 ) + 2 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, \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 )^{2} + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (3 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\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 )\right )} \sqrt {\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, 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 )^{3} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + b n + 4 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \]

[In]

integrate(1/x/sinh(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

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

[In]

integrate(1/x/sinh(a+b*ln(c*x**n))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/sinh(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {1}{x \sinh ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sinh \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/x/sinh(a+b*log(c*x^n))^(5/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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