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

Optimal. Leaf size=64 \[ -\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]

[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)

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Rubi [A]  time = 0.04, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2636, 2639} \[ -\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]

Antiderivative was successfully verified.

[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 2636

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 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]

Rubi steps

\begin {align*} \int \frac {1}{x \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\operatorname {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 {\operatorname {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*}

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Mathematica [A]  time = 0.18, size = 57, normalized size = 0.89 \[ \frac {2 \left (E\left (\left .\frac {1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )-\frac {\cos \left (a+b \log \left (c x^n\right )\right )}{\sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}\right )}{b n} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {\sin \left (b \log \left (c x^{n}\right ) + a\right )}}{x \cos \left (b \log \left (c x^{n}\right ) + a\right )^{2} - x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(sin(b*log(c*x^n) + a))/(x*cos(b*log(c*x^n) + a)^2 - x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.06, size = 190, normalized size = 2.97 \[ \frac {2 \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticE \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{n \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 2.73, size = 65, normalized size = 1.02 \[ -\frac {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left ({\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,\sqrt {\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \sin ^{\frac {3}{2}}{\left (a + b \log {\left (c x^{n} \right )} \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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