Optimal. Leaf size=88 \[ \frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]
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Rubi [A] time = 0.05, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4493, 4489} \[ \frac {1}{8} \sqrt {-\frac {1}{n^2}} n x^2 e^{-a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n}-\frac {1}{2} \sqrt {-\frac {1}{n^2}} n x^2 e^{a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-2/n} \]
Antiderivative was successfully verified.
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Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac {\left (x^2 \left (c x^n\right )^{-2/n}\right ) \operatorname {Subst}\left (\int x^{-1+\frac {2}{n}} \sin \left (a+2 \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^2 \left (c x^n\right )^{-2/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 {4}{n}}\right ) \, dx,x,c x^n\right )\right )\\ &=\frac {1}{8} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{2/n}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x^2 \left (c x^n\right )^{-2/n} \log (x)\\ \end {align*}
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Mathematica [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int x \sin \left (a+2 \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.51, size = 42, normalized size = 0.48 \[ \frac {1}{8} \, {\left (i \, x^{4} - 4 i \, e^{\left (\frac {2 \, {\left (i \, a n - 2 \, \log \relax (c)\right )}}{n}\right )} \log \relax (x)\right )} e^{\left (-\frac {i \, a n - 2 \, \log \relax (c)}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 1, normalized size = 0.01 \[ +\infty \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int x \sin \left (a +2 \ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 31, normalized size = 0.35 \[ \frac {c^{\frac {4}{n}} x^{4} \sin \relax (a) + 4 \, \log \relax (x) \sin \relax (a)}{8 \, c^{\frac {2}{n}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.80, size = 85, normalized size = 0.97 \[ -\frac {x^2\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}-\frac {x^2\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,2{}\mathrm {i}}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \sin {\left (a + 2 \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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