Optimal. Leaf size=66 \[ -\frac {(a B-b C) \log (\cos (c+d x))}{d}-x (a C+b B)+\frac {C (a+b \tan (c+d x))^2}{2 b d}+\frac {b B \tan (c+d x)}{d} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3630, 3525, 3475} \[ -\frac {(a B-b C) \log (\cos (c+d x))}{d}-x (a C+b B)+\frac {C (a+b \tan (c+d x))^2}{2 b d}+\frac {b B \tan (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3630
Rubi steps
\begin {align*} \int (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\frac {C (a+b \tan (c+d x))^2}{2 b d}+\int (a+b \tan (c+d x)) (-C+B \tan (c+d x)) \, dx\\ &=-(b B+a C) x+\frac {b B \tan (c+d x)}{d}+\frac {C (a+b \tan (c+d x))^2}{2 b d}+(a B-b C) \int \tan (c+d x) \, dx\\ &=-(b B+a C) x-\frac {(a B-b C) \log (\cos (c+d x))}{d}+\frac {b B \tan (c+d x)}{d}+\frac {C (a+b \tan (c+d x))^2}{2 b d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 67, normalized size = 1.02 \[ \frac {-2 (a C+b B) \tan ^{-1}(\tan (c+d x))+2 (a C+b B) \tan (c+d x)+2 (b C-a B) \log (\cos (c+d x))+b C \tan ^2(c+d x)}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 66, normalized size = 1.00 \[ \frac {C b \tan \left (d x + c\right )^{2} - 2 \, {\left (C a + B b\right )} d x - {\left (B a - C b\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.92, size = 616, normalized size = 9.33 \[ -\frac {2 \, C a d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + 2 \, B b d x \tan \left (d x\right )^{2} \tan \relax (c)^{2} + B 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} - C b \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} - 4 \, C a d x \tan \left (d x\right ) \tan \relax (c) - 4 \, B b d x \tan \left (d x\right ) \tan \relax (c) - C b \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 2 \, B 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 \, C b \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 \, C a \tan \left (d x\right )^{2} \tan \relax (c) + 2 \, B b \tan \left (d x\right )^{2} \tan \relax (c) + 2 \, C a \tan \left (d x\right ) \tan \relax (c)^{2} + 2 \, B b \tan \left (d x\right ) \tan \relax (c)^{2} + 2 \, C a d x + 2 \, B b d x - C b \tan \left (d x\right )^{2} - C b \tan \relax (c)^{2} + B 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 ) - C b \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 \, C a \tan \left (d x\right ) - 2 \, B b \tan \left (d x\right ) - 2 \, C a \tan \relax (c) - 2 \, B b \tan \relax (c) - C b}{2 \, {\left (d \tan \left (d x\right )^{2} \tan \relax (c)^{2} - 2 \, 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 = 105, normalized size = 1.59 \[ \frac {C b \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {b B \tan \left (d x +c \right )}{d}+\frac {C \tan \left (d x +c \right ) a}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B}{2 d}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C b}{2 d}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b}{d}-\frac {C \arctan \left (\tan \left (d x +c \right )\right ) a}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 66, normalized size = 1.00 \[ \frac {C b \tan \left (d x + c\right )^{2} - 2 \, {\left (C a + B b\right )} {\left (d x + c\right )} + {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, {\left (C a + B b\right )} \tan \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.84, size = 63, normalized size = 0.95 \[ \frac {\mathrm {tan}\left (c+d\,x\right )\,\left (B\,b+C\,a\right )+\ln \left ({\mathrm {tan}\left (c+d\,x\right )}^2+1\right )\,\left (\frac {B\,a}{2}-\frac {C\,b}{2}\right )-d\,x\,\left (B\,b+C\,a\right )+\frac {C\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2}{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.24, size = 105, normalized size = 1.59 \[ \begin {cases} \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B b x + \frac {B b \tan {\left (c + d x \right )}}{d} - C a x + \frac {C a \tan {\left (c + d x \right )}}{d} - \frac {C b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {C b \tan ^{2}{\left (c + d x \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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