Optimal. Leaf size=76 \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{4 x^2}-\frac {1}{4 x^2} \]
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Rubi [A] time = 0.06, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {4493, 4489} \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{2/n}}{4 x^2}-\frac {1}{4 x^2} \]
Antiderivative was successfully verified.
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Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx &=\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int x^{-1-\frac {2}{n}} \sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x^2}\\ &=-\frac {\left (c x^n\right )^{2/n} \operatorname {Subst}\left (\int \left (\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-\frac {2+n}{n}}+e^{2 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {4+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x^2}\\ &=-\frac {1}{4 x^2}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-2/n}}{16 x^2}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{2/n} \log (x)}{4 x^2}\\ \end {align*}
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Mathematica [F] time = 0.17, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2\left (a+\sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^3} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.43, size = 65, normalized size = 0.86 \[ -\frac {{\left (4 \, x^{4} \log \relax (x) + 4 \, x^{2} e^{\left (\frac {2 \, {\left (i \, a n - \log \relax (c)\right )}}{n}\right )} - e^{\left (\frac {4 \, {\left (i \, a n - \log \relax (c)\right )}}{n}\right )}\right )} e^{\left (-\frac {2 \, {\left (i \, a n - \log \relax (c)\right )}}{n}\right )}}{16 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (\sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^{2}\left (a +\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 54, normalized size = 0.71 \[ -\frac {4 \, c^{\frac {4}{n}} x^{6} \cos \left (2 \, a\right ) \log \relax (x) + 4 \, c^{\frac {2}{n}} x^{4} - x^{2} \cos \left (2 \, a\right )}{16 \, c^{\frac {2}{n}} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\sin \left (a+\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}\right )}^2}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 16.77, size = 462, normalized size = 6.08 \[ \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} \sin {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )} \cos {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{2 x^{2}} + \frac {3 i n \sqrt {\frac {1}{n^{2}}} \sin {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )} \cos {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x^{2}} + \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \sin {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )} \cos {\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{2 x^{2}} + \frac {\log {\relax (x )} \sin ^{2}{\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x^{2}} - \frac {\log {\relax (x )} \cos ^{2}{\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x^{2}} - \frac {\cos ^{2}{\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{2 x^{2}} + \frac {\log {\relax (c )} \sin ^{2}{\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 n x^{2}} - \frac {\log {\relax (c )} \cos ^{2}{\left (a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 n x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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