\(\int \frac {1}{x \sin ^{\frac {3}{2}}(a+b \log (c x^n))} \, dx\) [66]

Optimal result

Integrand size = 19, antiderivative size = 64 \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2716, 2719} \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Int[1/(x*Sin[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

(-2*EllipticE[(a - Pi/2 + b*Log[c*x^n])/2, 2])/(b*n) - (2*Cos[a + b*Log[c*x^n]])/(b*n*Sqrt[Sin[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 2719

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

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sin ^{\frac {3}{2}}(a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}-\frac {\text {Subst}\left (\int \sqrt {\sin (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 E\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {2 \left (E\left (\left .\frac {1}{4} \left (-2 a+\pi -2 b \log \left (c x^n\right )\right )\right |2\right )-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}\right )}{b n} \]

[In]

Integrate[1/(x*Sin[a + b*Log[c*x^n]]^(3/2)),x]

[Out]

(2*(EllipticE[(-2*a + Pi - 2*b*Log[c*x^n])/4, 2] - Cos[a + b*Log[c*x^n]]/Sqrt[Sin[a + b*Log[c*x^n]]]))/(b*n)

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.97

method	result	size
derivativedivides	\(\frac {2 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(190\)
default	\(\frac {2 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticE}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \operatorname {EllipticF}\left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-2 {\cos \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b}\)	\(190\)

[In]

int(1/x/sin(a+b*ln(c*x^n))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/n*(2*(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*ln(c*x^n)))^(1/2)*EllipticE((sin
(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))-(sin(a+b*ln(c*x^n))+1)^(1/2)*(-2*sin(a+b*ln(c*x^n))+2)^(1/2)*(-sin(a+b*l
n(c*x^n)))^(1/2)*EllipticF((sin(a+b*ln(c*x^n))+1)^(1/2),1/2*2^(1/2))-2*cos(a+b*ln(c*x^n))^2)/cos(a+b*ln(c*x^n)
)/sin(a+b*ln(c*x^n))^(1/2)/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 = 156, normalized size of antiderivative = 2.44 \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-i \, \sqrt {2} \sqrt {-i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) + i \, \sqrt {2} \sqrt {i} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - i \, \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right ) - 2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sqrt {\sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}{b n \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(3/2),x, algorithm="fricas")

[Out]

(-I*sqrt(2)*sqrt(-I)*sin(b*n*log(x) + b*log(c) + a)*weierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*n*lo
g(x) + b*log(c) + a) + I*sin(b*n*log(x) + b*log(c) + a))) + I*sqrt(2)*sqrt(I)*sin(b*n*log(x) + b*log(c) + a)*w
eierstrassZeta(4, 0, weierstrassPInverse(4, 0, cos(b*n*log(x) + b*log(c) + a) - I*sin(b*n*log(x) + b*log(c) +
a))) - 2*cos(b*n*log(x) + b*log(c) + a)*sqrt(sin(b*n*log(x) + b*log(c) + a)))/(b*n*sin(b*n*log(x) + b*log(c) +
a))

Sympy [F]

\[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {1}{x \sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

[In]

integrate(1/x/sin(a+b*ln(c*x**n))**(3/2),x)

[Out]

Integral(1/(x*sin(a + b*log(c*x**n))**(3/2)), x)

Maxima [F]

\[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}} \,d x } \]

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \]

[In]

integrate(1/x/sin(a+b*log(c*x^n))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 27.62 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=-\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,\sqrt {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \]

[In]

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

[Out]

-(cos(a + b*log(c*x^n))*(sin(a + b*log(c*x^n))^2)^(1/4)*hypergeom([1/2, 5/4], 3/2, cos(a + b*log(c*x^n))^2))/(
b*n*sin(a + b*log(c*x^n))^(1/2))