Optimal. Leaf size=29 \[ -\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-b x \]
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Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3525, 3475} \[ -\frac {a \log (\cos (c+d x))}{d}+\frac {b \tan (c+d x)}{d}-b x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rubi steps
\begin {align*} \int \tan (c+d x) (a+b \tan (c+d x)) \, dx &=-b x+\frac {b \tan (c+d x)}{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}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 38, normalized size = 1.31 \[ -\frac {a \log (\cos (c+d x))}{d}-\frac {b \tan ^{-1}(\tan (c+d x))}{d}+\frac {b \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 35, normalized size = 1.21 \[ -\frac {2 \, b d x + a \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.60, size = 174, normalized size = 6.00 \[ -\frac {2 \, b d x \tan \left (d x\right ) \tan \relax (c) + a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) \tan \left (d x\right ) \tan \relax (c) - 2 \, b d x - a \log \left (\frac {4 \, {\left (\tan \left (d x\right )^{4} \tan \relax (c)^{2} - 2 \, \tan \left (d x\right )^{3} \tan \relax (c) + \tan \left (d x\right )^{2} \tan \relax (c)^{2} + \tan \left (d x\right )^{2} - 2 \, \tan \left (d x\right ) \tan \relax (c) + 1\right )}}{\tan \relax (c)^{2} + 1}\right ) + 2 \, b \tan \left (d x\right ) + 2 \, b \tan \relax (c)}{2 \, {\left (d \tan \left (d x\right ) \tan \relax (c) - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 43, normalized size = 1.48 \[ \frac {b \tan \left (d x +c \right )}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {b \arctan \left (\tan \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.60, size = 37, normalized size = 1.28 \[ -\frac {2 \, {\left (d x + c\right )} b - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, b \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.06, size = 32, normalized size = 1.10 \[ \frac {b\,\mathrm {tan}\left (c+d\,x\right )+\frac {a\,\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )}{2}-b\,d\,x}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 41, normalized size = 1.41 \[ \begin {cases} \frac {a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - b x + \frac {b \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \tan {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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