Optimal. Leaf size=68 \[ \frac {x}{2}+\frac {1}{8} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{4} e^{2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {4572, 4578}
\begin {gather*} \frac {1}{8} x e^{-2 a \sqrt {-\frac {1}{n^2}} n} \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{4} x e^{2 a \sqrt {-\frac {1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac {x}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4572
Rule 4578
Rubi steps
\begin {align*} \int \cos ^2\left (a+\frac {1}{2} \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}} \cos ^2\left (a+\frac {1}{2} \sqrt {-\frac {1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac {\left (x \left (c x^n\right )^{-1/n}\right ) \text {Subst}\left (\int \left (\frac {e^{2 a \sqrt {-\frac {1}{n^2}} n}}{x}+2 x^{-1+\frac {1}{n}}+e^{-2 a \sqrt {-\frac {1}{n^2}} n} x^{-1+\frac {2}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac {x}{2}+\frac {1}{8} e^{-2 a \sqrt {-\frac {1}{n^2}} n} x \left (c x^n\right )^{\frac {1}{n}}+\frac {1}{4} e^{2 a \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 = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^2\left (a+\frac {1}{2} \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 = 0.07, size = 0, normalized size = 0.00 \[\int \cos ^{2}\left (a +\frac {\ln \left (c \,x^{n}\right ) \sqrt {-\frac {1}{n^{2}}}}{2}\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 41, normalized size = 0.60 \begin {gather*} \frac {c^{\frac {2}{n}} x^{2} \cos \left (2 \, a\right ) + 4 \, c^{\left (\frac {1}{n}\right )} x + 2 \, \cos \left (2 \, a\right ) \log \left (x\right )}{8 \, 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 = 3.29, size = 57, normalized size = 0.84 \begin {gather*} \frac {1}{8} \, {\left (x^{2} + 4 \, x e^{\left (\frac {2 i \, a n - \log \left (c\right )}{n}\right )} + 2 \, e^{\left (\frac {2 \, {\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac {2 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 \cos ^{2}{\left (a + \frac {\sqrt {- \frac {1}{n^{2}}} \log {\left (c x^{n} \right )}}{2} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.68, 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 = 2.71, size = 86, normalized size = 1.26 \begin {gather*} \frac {x}{2}+\frac {x\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}}\,1{}\mathrm {i}}{4\,n\,\sqrt {-\frac {1}{n^2}}+4{}\mathrm {i}}-\frac {x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{\sqrt {-\frac {1}{n^2}}\,1{}\mathrm {i}}\,1{}\mathrm {i}}{4\,n\,\sqrt {-\frac {1}{n^2}}-4{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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