Optimal. Leaf size=37 \[ x (a C+b B)+\frac {a B \log (\sin (c+d x))}{d}-\frac {b C \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.11, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3632, 3589, 3475, 3531} \[ x (a C+b B)+\frac {a B \log (\sin (c+d x))}{d}-\frac {b C \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3589
Rule 3632
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \tan (c+d x)) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\int \cot (c+d x) (a+b \tan (c+d x)) (B+C \tan (c+d x)) \, dx\\ &=(b C) \int \tan (c+d x) \, dx+\int \cot (c+d x) (a B+(b B+a C) \tan (c+d x)) \, dx\\ &=(b B+a C) x-\frac {b C \log (\cos (c+d x))}{d}+(a B) \int \cot (c+d x) \, dx\\ &=(b B+a C) x-\frac {b C \log (\cos (c+d x))}{d}+\frac {a B \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 44, normalized size = 1.19 \[ \frac {a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+a C x+b B x-\frac {b C \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 59, normalized size = 1.59 \[ \frac {2 \, {\left (C a + B b\right )} d x + B a \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - C b \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 3.67, size = 53, normalized size = 1.43 \[ \frac {2 \, B a \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + 2 \, {\left (C a + B b\right )} {\left (d x + c\right )} - {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.57, size = 51, normalized size = 1.38 \[ B x b +a C x +\frac {a B \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {B b c}{d}-\frac {b C \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {C a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 52, normalized size = 1.41 \[ \frac {2 \, B a \log \left (\tan \left (d x + c\right )\right ) + 2 \, {\left (C a + B b\right )} {\left (d x + c\right )} - {\left (B a - C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.96, size = 69, normalized size = 1.86 \[ \frac {B\,a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B+C\,1{}\mathrm {i}\right )\,\left (a+b\,1{}\mathrm {i}\right )}{2\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )\,\left (b+a\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.98, size = 85, normalized size = 2.30 \[ \begin {cases} - \frac {B a \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {B a \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + B b x + C a x + \frac {C b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right ) \left (B \tan {\relax (c )} + C \tan ^{2}{\relax (c )}\right ) \cot ^{2}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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