Optimal. Leaf size=29 \[ \frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x) \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3473, 8} \[ \frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\log (x) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \frac {\tan ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \tan ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\frac {\operatorname {Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\log (x)+\frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 51, normalized size = 1.76 \[ \frac {\tan \left (a d+b d \log \left (c x^n\right )\right )}{b d n}-\frac {\tan ^{-1}\left (\tan \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 85, normalized size = 2.93 \[ -\frac {b d n \cos \left (2 \, b d n \log \relax (x) + 2 \, b d \log \relax (c) + 2 \, a d\right ) \log \relax (x) + b d n \log \relax (x) - \sin \left (2 \, b d n \log \relax (x) + 2 \, b d \log \relax (c) + 2 \, a d\right )}{b d n \cos \left (2 \, b d n \log \relax (x) + 2 \, b d \log \relax (c) + 2 \, a d\right ) + b d n} \]
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 = 50, normalized size = 1.72 \[ \frac {\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b d n}-\frac {\arctan \left (\tan \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\right )}{b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 320, normalized size = 11.03 \[ -\frac {{\left (b d \cos \left (2 \, b d \log \relax (c)\right )^{2} + b d \sin \left (2 \, b d \log \relax (c)\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} \log \relax (x) + {\left (b d \cos \left (2 \, b d \log \relax (c)\right )^{2} + b d \sin \left (2 \, b d \log \relax (c)\right )^{2}\right )} n \log \relax (x) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + b d n \log \relax (x) + 2 \, {\left (b d n \cos \left (2 \, b d \log \relax (c)\right ) \log \relax (x) - \sin \left (2 \, b d \log \relax (c)\right )\right )} \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) - 2 \, {\left (b d n \log \relax (x) \sin \left (2 \, b d \log \relax (c)\right ) + \cos \left (2 \, b d \log \relax (c)\right )\right )} \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )}{2 \, b d n \cos \left (2 \, b d \log \relax (c)\right ) \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) - 2 \, b d n \sin \left (2 \, b d \log \relax (c)\right ) \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right ) + {\left (b d \cos \left (2 \, b d \log \relax (c)\right )^{2} + b d \sin \left (2 \, b d \log \relax (c)\right )^{2}\right )} n \cos \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + {\left (b d \cos \left (2 \, b d \log \relax (c)\right )^{2} + b d \sin \left (2 \, b d \log \relax (c)\right )^{2}\right )} n \sin \left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )^{2} + b d n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.84, size = 39, normalized size = 1.34 \[ -\ln \relax (x)+\frac {2{}\mathrm {i}}{b\,d\,n\,\left ({\mathrm {e}}^{a\,d\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,d\,2{}\mathrm {i}}+1\right )} \]
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 \[ \int \frac {\tan ^{2}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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