Optimal. Leaf size=82 \[ \frac {1}{4} e^{-a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{\frac {1}{n}}-\frac {1}{2} e^{a \sqrt {-\frac {1}{n^2}} n} \sqrt {-\frac {1}{n^2}} n x \left (c x^n\right )^{-1/n} \log (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, 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 = {4571, 4577}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 4571
Rule 4577
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 ) \text {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 ) \text {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*}
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Mathematica [F]
time = 0.10, size = 0, normalized size = 0.00 \begin {gather*} \int \sin \left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right ) \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sin \left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 29, normalized size = 0.35 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 1.13, size = 42, normalized size = 0.51 \begin {gather*} \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 {gather*}
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 \begin {gather*} \int \sin {\left (a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 1, normalized size = 0.01 \begin {gather*} +\infty \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.73, size = 81, normalized size = 0.99 \begin {gather*} -\frac {x\,{\mathrm {e}}^{-a\,1{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}}{2\,n\,\sqrt {-\frac {1}{n^2}}+2{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,1{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}{2\,n\,\sqrt {-\frac {1}{n^2}}-2{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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