Optimal. Leaf size=27 \[ \frac {\tan ^2(a+b x)}{2 b}+\frac {\log (\cos (a+b x))}{b} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3473, 3475} \[ \frac {\tan ^2(a+b x)}{2 b}+\frac {\log (\cos (a+b x))}{b} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rubi steps
\begin {align*} \int \tan ^3(a+b x) \, dx &=\frac {\tan ^2(a+b x)}{2 b}-\int \tan (a+b x) \, dx\\ &=\frac {\log (\cos (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.93 \[ \frac {\tan ^2(a+b x)+2 \log (\cos (a+b x))}{2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 34, normalized size = 1.26 \[ \frac {2 \, \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right )\right ) + 1}{2 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 42, normalized size = 1.56 \[ \frac {\log \left (\frac {\cos \left (b x + a\right )^{2}}{b^{2}}\right )}{2 \, b} - \frac {\cos \left (b x + a\right )^{2} - 1}{2 \, b \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 26, normalized size = 0.96 \[ \frac {\ln \left (\cos \left (b x +a \right )\right )}{b}+\frac {\tan ^{2}\left (b x +a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 31, normalized size = 1.15 \[ -\frac {\frac {1}{\sin \left (b x + a\right )^{2} - 1} - \log \left (\sin \left (b x + a\right )^{2} - 1\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.38, size = 27, normalized size = 1.00 \[ -\frac {\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )-{\mathrm {tan}\left (a+b\,x\right )}^2}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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