Optimal. Leaf size=64 \[ -\frac {2 E\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{b n \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}} \]
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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.
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Rule 2636
Rule 2639
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.
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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.
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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.
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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.
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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.
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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.
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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.
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