Optimal. Leaf size=110 \[ \frac{2 x \text{Hypergeometric2F1}\left (-\frac{5}{2},\frac{1}{4} \left (-5-\frac{2 i}{b n}\right ),-\frac{b n+2 i}{4 b n},-e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2-5 i b n) \left (1+e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \sec ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0724982, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, 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.
[In]
[Out]
Rule 4503
Rule 4507
Rule 364
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*}
Mathematica [B] time = 8.65317, 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) \text{Hypergeometric2F1}\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) \text{Hypergeometric2F1}\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.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.289, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sec \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]