Optimal. Leaf size=74 \[ -\frac {1}{2 x}+\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{-1/n}}{8 x}-\frac {e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}} \log (x)}{4 x} \]
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Rubi [A]
time = 0.04, 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 = {4581, 4577}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 4577
Rule 4581
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}} \text {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}} \text {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 = 0.17, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sin ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log \left (c x^n\right )\right )}{x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.06, 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 = 0.30, size = 48, normalized size = 0.65 \begin {gather*} -\frac {2 \, c^{\frac {2}{n}} x^{3} \cos \left (2 \, a\right ) \log \left (x\right ) + 4 \, c^{\left (\frac {1}{n}\right )} x^{2} - x \cos \left (2 \, a\right )}{8 \, c^{\left (\frac {1}{n}\right )} x^{3}} \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 = 1.58, size = 62, normalized size = 0.84 \begin {gather*} -\frac {{\left (2 \, x^{2} \log \left (x\right ) + 4 \, x e^{\left (\frac {2 i \, a n - \log \left (c\right )}{n}\right )} - e^{\left (\frac {2 \, {\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )}\right )} e^{\left (-\frac {2 i \, a n - \log \left (c\right )}{n}\right )}}{8 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 14.41, size = 117, normalized size = 1.58 \begin {gather*} \frac {n \sqrt {- \frac {1}{n^{2}}} \sin {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x} + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \sin {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 x} - \frac {1}{2 x} - \frac {\log {\left (c x^{n} \right )} \cos {\left (2 a + \sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )} \right )}}{4 n x} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\sin \left (a+\frac {\ln \left (c\,x^n\right )\,\sqrt {-\frac {1}{n^2}}}{2}\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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