Optimal. Leaf size=110 \[ \frac {2 x \, _2F_1\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right );-\frac {2 i+b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4599, 4603,
371} \begin {gather*} \frac {2 x \, _2F_1\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right );-\frac {b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\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 4599
Rule 4603
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {x^{-1+\frac {1}{n}}}{\sec ^{\frac {5}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{\frac {5 i b}{2}-\frac {1}{n}}\right ) \text {Subst}\left (\int x^{-1-\frac {5 i b}{2}+\frac {1}{n}} \left (1+e^{2 i a} x^{2 i b}\right )^{5/2} \, dx,x,c x^n\right )}{n \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac {2 x \, _2F_1\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right );-\frac {2 i+b n}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\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. \(861\) vs. \(2(110)=220\).
time = 8.84, size = 861, normalized size = 7.83 \begin {gather*} \frac {30 i \sqrt {2} b^3 e^{-i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} n^3 x^{1-i b n} \sqrt {\frac {e^{i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{i b n}}{1+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}}} \left ((2 i+b n) \left (1+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )+\left (-2 i-b n+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} (-2 i+b n)\right ) \sqrt {1+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}} \, _2F_1\left (\frac {1}{2},-\frac {2 i+b n}{4 b n};\frac {3}{4}-\frac {i}{2 b n};-e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} x^{2 i b n}\right )\right )}{(-2 i+5 b n) (2 i+5 b n) \left (4+b^2 n^2\right ) \left (-2 i-b n+e^{2 i \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} (-2 i+b n)\right )}+\sqrt {\sec \left (a+b n \log (x)+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )} \left (-\frac {x \cos (b n \log (x)) \left (12+55 b^2 n^2+12 \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+65 b^2 n^2 \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+4 b n \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{4 (-2 i+5 b n) (2 i+5 b n) \left (-2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}+\frac {x \sin (b n \log (x)) \left (-16 b n-4 b n \cos \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+12 \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+65 b^2 n^2 \sin \left (2 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{4 (-2 i+5 b n) (2 i+5 b n) \left (-2 \cos \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )+b n \sin \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )}+\frac {x \sin (3 b n \log (x)) \left (5 b n \cos \left (3 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )-2 \sin \left (3 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{2 (-2 i+5 b n) (2 i+5 b n)}+\frac {x \cos (3 b n \log (x)) \left (2 \cos \left (3 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )+5 b n \sin \left (3 \left (a+b \left (-n \log (x)+\log \left (c x^n\right )\right )\right )\right )\right )}{2 (-2 i+5 b n) (2 i+5 b n)}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.11, size = 0, normalized size = 0.00 \[\int \frac {1}{\sec \left (a +b \ln \left (c \,x^{n}\right )\right )^{\frac {5}{2}}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {1}{{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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