Optimal. Leaf size=110 \[ \frac {2 x \, _2F_1\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right );-\frac {b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.07, 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 = {4503, 4507, 364} \[ \frac {2 x \, _2F_1\left (-\frac {5}{2},\frac {1}{4} \left (-5-\frac {2 i}{b n}\right );-\frac {b n+2 i}{4 b n};-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 364
Rule 4503
Rule 4507
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 ) \operatorname {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 ) \operatorname {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] time = 8.68, size = 867, normalized size = 7.88 \[ \frac {30 b^3 e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x \left ((b n+2 i) \, _2F_1\left (\frac {1}{2},\frac {3}{4}-\frac {i}{2 b n};\frac {7}{4}-\frac {i}{2 b n};-e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}\right ) x^{2 i b n}+(3 b n-2 i) \, _2F_1\left (\frac {1}{2},-\frac {b n+2 i}{4 b n};\frac {3}{4}-\frac {i}{2 b n};-e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}\right )\right ) n^3}{(2-5 i b n) (b n+2 i) (3 b n-2 i) (5 b n-2 i) \left (-b n+e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} (b n-2 i)-2 i\right ) \sqrt {e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}+1} \sqrt {\frac {e^{i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{i b n}}{2 e^{2 i \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} x^{2 i b n}+2}}}+\sqrt {\sec \left (a+b n \log (x)+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )} \left (-\frac {x \cos (b n \log (x)) \left (55 b^2 n^2+65 b^2 \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n^2+4 b \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n+12 \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+12\right )}{4 (5 b n-2 i) (5 b n+2 i) \left (b n \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )-2 \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}+\frac {x \sin (b n \log (x)) \left (65 b^2 \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n^2-16 b n-4 b \cos \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right ) n+12 \sin \left (2 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{4 (5 b n-2 i) (5 b n+2 i) \left (b n \sin \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )-2 \cos \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )}+\frac {x \sin (3 b n \log (x)) \left (5 b n \cos \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )-2 \sin \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{2 (5 b n-2 i) (5 b n+2 i)}+\frac {x \cos (3 b n \log (x)) \left (2 \cos \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )+5 b n \sin \left (3 \left (a+b \left (\log \left (c x^n\right )-n \log (x)\right )\right )\right )\right )}{2 (5 b n-2 i) (5 b n+2 i)}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {1}{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, 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 \[ \int \frac {1}{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {1}{\cos \left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}^{5/2}} \,d x \]
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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