Optimal. Leaf size=176 \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{8 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.13176, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ -\frac{\sqrt{-\frac{1}{n^2}} n e^{3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 \sqrt{-\frac{1}{n^2}} n e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 \sqrt{-\frac{1}{n^2}} n e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{\sqrt{-\frac{1}{n^2}} n e^{-3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac{1}{n}}}{8 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4493
Rule 4489
Rubi steps
\begin{align*} \int \frac{\sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac{\left (c x^n\right )^{\frac{1}{n}} \operatorname{Subst}\left (\int x^{-1-\frac{1}{n}} \sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac{\left (\sqrt{-\frac{1}{n^2}} \left (c x^n\right )^{\frac{1}{n}}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{-3 a \sqrt{-\frac{1}{n^2}} n}}{x}+3 e^{a \sqrt{-\frac{1}{n^2}} n} x^{-1-\frac{4}{3 n}}-3 e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1-\frac{2}{3 n}}-e^{3 a \sqrt{-\frac{1}{n^2}} n} x^{-\frac{2+n}{n}}\right ) \, dx,x,c x^n\right )}{8 x}\\ &=-\frac{e^{3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{-1/n}}{16 x}+\frac{9 e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}}{32 x}-\frac{9 e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\left .\frac{1}{3}\right /n}}{16 x}-\frac{e^{-3 a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n \left (c x^n\right )^{\frac{1}{n}} \log (x)}{8 x}\\ \end{align*}
Mathematica [F] time = 0.170141, size = 0, normalized size = 0. \[ \int \frac{\sin ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( \sin \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{3}\sqrt{-{n}^{-2}}} \right ) \right ) ^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.14978, size = 165, normalized size = 0.94 \begin{align*} -\frac{{\left (4 \, c^{\frac{7}{3 \, n}} x e^{\left (\frac{\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \log \left (x\right ) \sin \left (3 \, a\right ) - 2 \, c^{\frac{1}{3 \, n}} x{\left (x^{n}\right )}^{\frac{1}{3 \, n}} \sin \left (3 \, a\right ) + 9 \, c^{\left (\frac{1}{n}\right )} x^{2} \sin \left (a\right ) + 18 \, c^{\frac{5}{3 \, n}} e^{\left (\frac{2 \, \log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )} \sin \left (a\right )\right )} e^{\left (-\frac{\log \left (x^{n}\right )}{3 \, n} - 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac{4}{3 \, n}} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 0.478729, size = 244, normalized size = 1.39 \begin{align*} \frac{{\left (-12 i \, x^{2} \log \left (x^{\frac{1}{3}}\right ) - 18 i \, x^{\frac{4}{3}} e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 9 i \, x^{\frac{2}{3}} e^{\left (\frac{4 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} - 2 i \, e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac{3 i \, a n - \log \left (c\right )}{n}\right )}}{32 \, x^{2}} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin \left (\frac{1}{3} \, \sqrt{-\frac{1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]