Optimal. Leaf size=82 \[ \frac{1}{4} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0517531, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {4483, 4489} \[ \frac{1}{4} \sqrt{-\frac{1}{n^2}} n x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} \sqrt{-\frac{1}{n^2}} n x e^{a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4483
Rule 4489
Rubi steps
\begin{align*} \int \sin \left (a+\sqrt{-\frac{1}{n^2}} \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}} \sin \left (a+\sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=-\left (\frac{1}{2} \left (\sqrt{-\frac{1}{n^2}} x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{a \sqrt{-\frac{1}{n^2}} n}}{x}-e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac{1}{4} e^{-a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{\frac{1}{n}}-\frac{1}{2} e^{a \sqrt{-\frac{1}{n^2}} n} \sqrt{-\frac{1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}
Mathematica [F] time = 0.0858786, size = 0, normalized size = 0. \[ \int \sin \left (a+\sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.034, size = 0, normalized size = 0. \begin{align*} \int \sin \left ( a+\ln \left ( c{x}^{n} \right ) \sqrt{-{n}^{-2}} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17032, size = 39, normalized size = 0.48 \begin{align*} \frac{c^{\frac{2}{n}} x^{2} \sin \left (a\right ) + 2 \, \log \left (x\right ) \sin \left (a\right )}{4 \, c^{\left (\frac{1}{n}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] time = 0.466307, size = 101, normalized size = 1.23 \begin{align*} \frac{1}{4} \,{\left (i \, x^{2} - 2 i \, e^{\left (\frac{2 \,{\left (i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{i \, a n - \log \left (c\right )}{n}\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 \sin{\left (a + \sqrt{- \frac{1}{n^{2}}} \log{\left (c x^{n} \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.27703, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]