Optimal. Leaf size=130 \[ \frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac {5}{2},-\frac {2 i+2 i m-5 b n}{4 b n};-\frac {2 i+2 i m-9 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+5 i b n) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A]
time = 0.07, antiderivative size = 126, normalized size of antiderivative = 0.97, number of steps
used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {4582, 4580,
371} \begin {gather*} \frac {2 x^{m+1} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (m+1)}{b n}\right );-\frac {2 i m-9 b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(5 i b n+2 m+2) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 371
Rule 4580
Rule 4582
Rubi steps
\begin {align*} \int \frac {x^m}{\cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1+m}{n}}}{\cos ^{\frac {5}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x^{1+m} \left (c x^n\right )^{-\frac {5 i b}{2}-\frac {1+m}{n}} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {5 i b}{2}+\frac {1+m}{n}}}{\left (1+e^{2 i a} x^{2 i b}\right )^{5/2}} \, dx,x,c x^n\right )}{n \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac {2 x^{1+m} \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac {5}{2},\frac {1}{4} \left (5-\frac {2 i (1+m)}{b n}\right );-\frac {2 i+2 i m-9 b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+2 m+5 i b n) \cos ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(263\) vs. \(2(130)=260\).
time = 3.46, size = 263, normalized size = 2.02 \begin {gather*} \frac {x^{1+m} \left (-4 (1+m) \cos \left (a+b \log \left (c x^n\right )\right )+\frac {(2+2 m-i b n) \sqrt {1+e^{2 i a} \left (c x^n\right )^{2 i b}} \, _2F_1\left (\frac {1}{2},\frac {-2 i-2 i m+b n}{4 b n};-\frac {2 i+2 i m-5 b n}{4 b n};-e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \left (e^{-i a} \left (\left (1+e^{2 i a}\right ) \cos \left (b \log \left (c x^n\right )\right )+i \left (-1+e^{2 i a}\right ) \sin \left (b \log \left (c x^n\right )\right )\right )\right )^{3/2}}{\sqrt {e^{-i a} \left (c x^n\right )^{-i b}+e^{i a} \left (c x^n\right )^{i b}}}+2 b n \sin \left (a+b \log \left (c x^n\right )\right )\right )}{3 b^2 n^2 \cos ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\cos \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {5}{2}}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m}{{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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