Optimal. Leaf size=109 \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \text{Hypergeometric2F1}\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i}{b n}\right ),\frac{1}{4} \left (7-\frac{2 i}{b n}\right ),e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0703636, 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 = {4483, 4491, 364} \[ \frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i}{b n}\right );\frac{1}{4} \left (7-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4483
Rule 4491
Rule 364
Rubi steps
\begin{align*} \int \frac{1}{\sin ^{\frac{3}{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}}}{\sin ^{\frac{3}{2}}(a+b \log (x))} \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{3 i b}{2}-\frac{1}{n}} \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2}\right ) \operatorname{Subst}\left (\int \frac{x^{-1+\frac{3 i b}{2}+\frac{1}{n}}}{\left (1-e^{2 i a} x^{2 i b}\right )^{3/2}} \, dx,x,c x^n\right )}{n \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ &=\frac{2 x \left (1-e^{2 i a} \left (c x^n\right )^{2 i b}\right )^{3/2} \, _2F_1\left (\frac{3}{2},\frac{1}{4} \left (3-\frac{2 i}{b n}\right );\frac{1}{4} \left (7-\frac{2 i}{b n}\right );e^{2 i a} \left (c x^n\right )^{2 i b}\right )}{(2+3 i b n) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.917895, size = 96, normalized size = 0.88 \[ \frac{2 x \left (-1+e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right ) \text{Hypergeometric2F1}\left (1,\frac{1}{4}-\frac{i}{2 b n},\frac{7}{4}-\frac{i}{2 b n},e^{2 i \left (a+b \log \left (c x^n\right )\right )}\right )}{(-2-3 i b n) \sin ^{\frac{3}{2}}\left (a+b \log \left (c x^n\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.168, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{-{\frac{3}{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}{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin ^{\frac{3}{2}}{\left (a + b \log{\left (c x^{n} \right )} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sin \left (b \log \left (c x^{n}\right ) + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]