Optimal. Leaf size=74 \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{4 x}-\frac {1}{2 x} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4493, 4489} \[ \frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{\frac {1}{n}}}{4 x}-\frac {1}{2 x} \]
Antiderivative was successfully verified.
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Rule 4489
Rule 4493
Rubi steps
\begin {align*} \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx &=\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int x^{-1-\frac {1}{n}} \sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n x}\\ &=-\frac {\left (c x^n\right )^{\frac {1}{n}} \operatorname {Subst}\left (\int \left (\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n}}{x}-2 x^{-\frac {1+n}{n}}+e^{2 a \sqrt {-\frac {1}{n^2}} n} x^{-\frac {2+n}{n}}\right ) \, dx,x,c x^n\right )}{4 n x}\\ &=-\frac {1}{2 x}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}} \log (x)}{4 x}\\ \end {align*}
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Mathematica [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \]
Verification is Not applicable to the result.
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fricas [C] time = 0.43, size = 62, normalized size = 0.84 \[ -\frac {{\left (2 \, x^{2} \log \relax (x) + 4 \, x e^{\left (\frac {2 i \, a n - \log \relax (c)}{n}\right )} - e^{\left (\frac {2 \, {\left (2 i \, a n - \log \relax (c)\right )}}{n}\right )}\right )} e^{\left (-\frac {2 i \, a n - \log \relax (c)}{n}\right )}}{8 \, x^{2}} \]
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 \[ \int \frac {\sin \left (\frac {1}{2} \, \sqrt {-\frac {1}{n^{2}}} \log \left (c x^{n}\right ) + a\right )^{2}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sin ^{2}\left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 48, normalized size = 0.65 \[ -\frac {2 \, c^{\frac {2}{n}} x^{3} \cos \left (2 \, a\right ) \log \relax (x) + 4 \, c^{\left (\frac {1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac {1}{n}\right )} x^{3}} \]
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 \[ \int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{2}\right )}^2}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 28.49, size = 240, normalized size = 3.24 \[ \frac {i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} + \frac {i n \sqrt {\frac {1}{n^{2}}} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} + \frac {i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \sin {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} - \frac {\log {\relax (x )} \cos {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 x} - \frac {1}{2 x} - \frac {\log {\relax (c )} \cos {\left (2 a + i n \sqrt {\frac {1}{n^{2}}} \log {\relax (x )} + i \sqrt {\frac {1}{n^{2}}} \log {\relax (c )} \right )}}{4 n x} \]
Verification of antiderivative is not currently implemented for this CAS.
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