Optimal. Leaf size=68 \[ \frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 68, 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 = {2716, 2720}
\begin {gather*} \frac {2 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 )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2720
Rubi steps
\begin {align*} \int \frac {1}{x \sin ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\sin ^{\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 )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\sin (a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{3 n}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (a-\frac {\pi }{2}+b \log \left (c x^n\right )\right )\right |2\right )}{3 b n}-\frac {2 \cos \left (a+b \log \left (c x^n\right )\right )}{3 b n \sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 61, normalized size = 0.90 \begin {gather*} \frac {2 \left (F\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 )}{\sin ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}\right )}{3 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 131, normalized size = 1.93
method | result | size |
derivativedivides | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 n \sin \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
default | \(\frac {\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}\, \sqrt {-2 \sin \left (a +b \ln \left (c \,x^{n}\right )\right )+2}\, \sqrt {-\sin \left (a +b \ln \left (c \,x^{n}\right )\right )}\, \EllipticF \left (\sqrt {\sin \left (a +b \ln \left (c \,x^{n}\right )\right )+1}, \frac {\sqrt {2}}{2}\right ) \sin \left (a +b \ln \left (c \,x^{n}\right )\right )-2 \left (\cos ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{3 n \sin \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {3}{2}} \cos \left (a +b \ln \left (c \,x^{n}\right )\right ) b}\) | \(131\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.46, size = 177, normalized size = 2.60 \begin {gather*} \frac {{\left (\sqrt {2} \sqrt {-i} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2} \sqrt {-i}\right )} {\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 ) + {\left (\sqrt {2} \sqrt {i} \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - \sqrt {2} \sqrt {i}\right )} {\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 ) + 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 )}}{3 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.96, size = 65, normalized size = 0.96 \begin {gather*} -\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 )}^{3/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {3}{2};\ {\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^2\right )}{b\,n\,{\sin \left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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