Optimal. Leaf size=71 \[ \frac{(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \csc ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0759444, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4504, 4508, 264} \[ \frac{(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \csc ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4504
Rule 4508
Rule 264
Rubi steps
\begin{align*} \int \csc ^p\left (a-\frac{i \log \left (c x^n\right )}{n (-2+p)}\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \csc ^p\left (a-\frac{i \log (x)}{n (-2+p)}\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-\frac{1}{n}-\frac{p}{n (-2+p)}} \left (1-e^{2 i a} \left (c x^n\right )^{\frac{2}{n (-2+p)}}\right )^p \csc ^p\left (a-\frac{i \log \left (c x^n\right )}{n (-2+p)}\right )\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}+\frac{p}{n (-2+p)}} \left (1-e^{2 i a} x^{\frac{2}{n (-2+p)}}\right )^{-p} \, dx,x,c x^n\right )}{n}\\ &=\frac{(2-p) x \left (1-e^{2 i a} \left (c x^n\right )^{-\frac{2}{n (2-p)}}\right ) \csc ^p\left (a+\frac{i \log \left (c x^n\right )}{n (2-p)}\right )}{2 (1-p)}\\ \end{align*}
Mathematica [A] time = 3.00541, size = 128, normalized size = 1.8 \[ \frac{2^{p-1} (p-2) x \left (\frac{i e^{i a} \left (c x^n\right )^{\frac{1}{n (p-2)}}}{-1+e^{2 i a} \left (c x^n\right )^{\frac{2}{n (p-2)}}}\right )^p \left (1+e^{2 i a} \left (c x^n\right )^{\frac{2}{n (p-2)}} \left (-1+\left (1-e^{-2 i a} \left (c x^n\right )^{-\frac{2}{n (p-2)}}\right )^p\right )\right )}{p-1} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.278, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( a-{\frac{i\ln \left ( c{x}^{n} \right ) }{n \left ( p-2 \right ) }} \right ) \right ) ^{p}\, 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 \left (-\csc \left (-a + \frac{i \, \log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 0.499228, size = 350, normalized size = 4.93 \begin{align*} \frac{{\left ({\left (p - 2\right )} x e^{\left (\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} -{\left (p - 2\right )} x\right )} \left (-\frac{2 i \, e^{\left (\frac{-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )}{n p - 2 \, n}\right )}}{e^{\left (\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )} - 1}\right )^{p} e^{\left (-\frac{2 \,{\left (-i \, a n p + 2 i \, a n - \log \left (c x^{n}\right )\right )}}{n p - 2 \, n}\right )}}{2 \,{\left (p - 1\right )}} \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 \csc ^{p}{\left (a - \frac{i \log{\left (c x^{n} \right )}}{n \left (p - 2\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 \csc \left (a - \frac{i \, \log \left (c x^{n}\right )}{n{\left (p - 2\right )}}\right )^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]