Optimal. Leaf size=109 \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \text{Hypergeometric2F1}\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right ),\frac{1}{4} \left (9-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0717096, antiderivative size = 109, 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 = {4504, 4508, 364} \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \, _2F_1\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right );\frac{1}{4} \left (9-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )}{2+5 i b n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4504
Rule 4508
Rule 364
Rubi steps
\begin{align*} \int \csc ^{\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 x^{-1+\frac{1}{n}} \csc ^{\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}} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right )\right ) \operatorname{Subst}\left (\int \frac{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}\\ &=\frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{5/2} \csc ^{\frac{5}{2}}\left (a+b \log \left (c x^n\right )\right ) \, _2F_1\left (\frac{5}{2},\frac{1}{4} \left (5-\frac{2 i}{b n}\right );\frac{1}{4} \left (9-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{2+5 i b n}\\ \end{align*}
Mathematica [A] time = 1.82415, size = 174, normalized size = 1.6 \[ \frac{2 x^{1-2 i b n} e^{-2 i \left (a+b \log \left (c x^n\right )-b n \log (x)\right )} \sqrt{\csc \left (a+b \log \left (c x^n\right )\right )} \left ((2+i b n) \left (-1+e^{2 i a} \left (c x^n\right )^{2 i b}\right ) \text{Hypergeometric2F1}\left (1,\frac{3}{4}+\frac{i}{2 b n},\frac{5}{4}+\frac{i}{2 b n},e^{-2 i \left (a+b \log \left (c x^n\right )\right )}\right )-e^{2 i a} \left (c x^n\right )^{2 i b} \left (b n \cot \left (a+b \log \left (c x^n\right )\right )+2\right )\right )}{3 b^2 n^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.296, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \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 \csc \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]