Optimal. Leaf size=98 \[ \frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
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Rubi [A] time = 0.06, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3768, 3771, 2641} \[ \frac {2 \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )} \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a+b \log \left (c x^n\right )-\frac {\pi }{2}\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3768
Rule 3771
Rubi steps
\begin {align*} \int \frac {\csc ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \csc ^{\frac {5}{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 ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\operatorname {Subst}\left (\int \sqrt {\csc (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {\left (\sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right ) \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}{3 b n}+\frac {2 \sqrt {\csc \left (a+b \log \left (c x^n\right )\right )} F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\sin \left (a+b \log \left (c x^n\right )\right )}}{3 b n}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 73, normalized size = 0.74 \[ -\frac {2 \csc ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) \left (\cos \left (a+b \log \left (c x^n\right )\right )+\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right ) F\left (\left .\frac {1}{4} \left (-2 a-2 b \log \left (c x^n\right )+\pi \right )\right |2\right )\right )}{3 b n} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.08, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 131, normalized size = 1.34 \[ \frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 n \sin \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b} \]
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 \[ \int \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (\frac {1}{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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