Optimal. Leaf size=43 \[ \frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
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Rubi [A] time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3473, 3475} \[ \frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}+\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \frac {\tan ^3\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \tan ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\operatorname {Subst}\left (\int \tan (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{b n}+\frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 38, normalized size = 0.88 \[ \frac {\tan ^2\left (a+b \log \left (c x^n\right )\right )+2 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )}{2 b n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 69, normalized size = 1.60 \[ \frac {{\left (\cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + \frac {1}{2}\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \relax (x) + 2 \, b \log \relax (c) + 2 \, a\right ) + b n\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 1.09 \[ \frac {\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )}{2 b n}-\frac {\ln \left (1+\tan ^{2}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{2 n b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.38, size = 1242, normalized size = 28.88 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.72, size = 105, normalized size = 2.44 \[ -\ln \relax (x)\,1{}\mathrm {i}-\frac {2}{b\,n\,\left (2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+1\right )}+\frac {2}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}+1\right )}{b\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.82, size = 70, normalized size = 1.63 \[ \begin {cases} \log {\relax (x )} \tan ^{3}{\relax (a )} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\log {\relax (x )} \tan ^{3}{\left (a + b \log {\relax (c )} \right )} & \text {for}\: n = 0 \\- \frac {\log {\left (\tan ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )} + 1 \right )}}{2 b n} + \frac {\tan ^{2}{\left (a + b n \log {\relax (x )} + b \log {\relax (c )} \right )}}{2 b n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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