Optimal. Leaf size=60 \[ \frac {a \tan ^2(c+d x)}{2 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]
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Rubi [A] time = 0.06, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3528, 3525, 3475} \[ \frac {a \tan ^2(c+d x)}{2 d}+\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d}+b x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \tan ^3(c+d x) (a+b \tan (c+d x)) \, dx &=\frac {b \tan ^3(c+d x)}{3 d}+\int \tan ^2(c+d x) (-b+a \tan (c+d x)) \, dx\\ &=\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}+\int \tan (c+d x) (-a-b \tan (c+d x)) \, dx\\ &=b x-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}-a \int \tan (c+d x) \, dx\\ &=b x+\frac {a \log (\cos (c+d x))}{d}-\frac {b \tan (c+d x)}{d}+\frac {a \tan ^2(c+d x)}{2 d}+\frac {b \tan ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 67, normalized size = 1.12 \[ \frac {a \left (\tan ^2(c+d x)+2 \log (\cos (c+d x))\right )}{2 d}+\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan ^3(c+d x)}{3 d}-\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 58, normalized size = 0.97 \[ \frac {2 \, b \tan \left (d x + c\right )^{3} + 6 \, b d x + 3 \, a \tan \left (d x + c\right )^{2} + 3 \, a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.06, size = 515, normalized size = 8.58 \[ \frac {6 \, b d x \tan \left (d x\right )^{3} \tan \relax (c)^{3} + 3 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 18 \, b d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 3 \, a \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 9 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 6 \, b \tan \left (d x\right )^{3} \tan \relax (c)^{2} + 6 \, b \tan \left (d x\right )^{2} \tan \relax (c)^{3} + 18 \, b d x \tan \left (d x\right ) \tan \relax (c) + 3 \, a \tan \left (d x\right )^{3} \tan \relax (c) - 3 \, a \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 3 \, a \tan \left (d x\right ) \tan \relax (c)^{3} - 2 \, b \tan \left (d x\right )^{3} + 9 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right ) \tan \relax (c) - 18 \, b \tan \left (d x\right )^{2} \tan \relax (c) - 18 \, b \tan \left (d x\right ) \tan \relax (c)^{2} - 2 \, b \tan \relax (c)^{3} - 6 \, b d x - 3 \, a \tan \left (d x\right )^{2} + 3 \, a \tan \left (d x\right ) \tan \relax (c) - 3 \, a \tan \relax (c)^{2} - 3 \, a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) + 6 \, b \tan \left (d x\right ) + 6 \, b \tan \relax (c) - 3 \, a}{6 \, {\left (d \tan \left (d x\right )^{3} \tan \relax (c)^{3} - 3 \, d \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 3 \, d \tan \left (d x\right ) \tan \relax (c) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 71, normalized size = 1.18 \[ \frac {b \left (\tan ^{3}\left (d x +c \right )\right )}{3 d}+\frac {a \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \tan \left (d x +c \right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 59, normalized size = 0.98 \[ \frac {2 \, b \tan \left (d x + c\right )^{3} + 3 \, a \tan \left (d x + c\right )^{2} + 6 \, {\left (d x + c\right )} b - 3 \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 6 \, b \tan \left (d x + c\right )}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.04, size = 54, normalized size = 0.90 \[ \frac {\frac {a\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}-\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-b\,\mathrm {tan}\left (c+d\,x\right )+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3}+b\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.31, size = 70, normalized size = 1.17 \[ \begin {cases} - \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a \tan ^{2}{\left (c + d x \right )}}{2 d} + b x + \frac {b \tan ^{3}{\left (c + d x \right )}}{3 d} - \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \tan ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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