Optimal. Leaf size=88 \[ \frac {1}{12} \sqrt {-\frac {1}{n^2}} n x^3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^3 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4493, 4489} \[ \frac {1}{12} \sqrt {-\frac {1}{n^2}} n x^3 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{3/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^3 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-3/n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^3 \left (c x^n\right )^{-3/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {3}{n}} \sin \left (a+3 \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^3 \left (c x^n\right )^{-3/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 {6}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac {1}{12} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{3/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^3 \left (c x^n\right )^{-3/n} \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int x^2 \sin \left (a+3 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.53, size = 42, normalized size = 0.48 \[ \frac {1}{12} \, {\left (i \, x^{6} - 6 i \, e^{\left (\frac {2 \, {\left (i \, a n - 3 \, \log \relax (c)\right )}}{n}\right )} \log \relax (x)\right )} e^{\left (-\frac {i \, a n - 3 \, \log \relax (c)}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.57, size = 1, normalized size = 0.01 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int x^{2} \sin \left (a +3 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 31, normalized size = 0.35 \[ \frac {c^{\frac {6}{n}} x^{6} \sin \relax (a) + 6 \, \log \relax (x) \sin \relax (a)}{12 \, c^{\frac {3}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.02, size = 85, normalized size = 0.97 \[ -\frac {x^3\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}}{6\,n\,\sqrt {-\frac {1}{n^2}}+6{}\mathrm {i}}-\frac {x^3\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,3{}\mathrm {i}}}{6\,n\,\sqrt {-\frac {1}{n^2}}-6{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \sin {\left (a + 3 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________