Optimal. Leaf size=35 \[ \frac {a^2 \log (\sin (c+d x))}{d}+2 a b x-\frac {b^2 \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3541, 3475} \[ \frac {a^2 \log (\sin (c+d x))}{d}+2 a b x-\frac {b^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3541
Rubi steps
\begin {align*} \int \cot (c+d x) (a+b \tan (c+d x))^2 \, dx &=2 a b x+a^2 \int \cot (c+d x) \, dx+b^2 \int \tan (c+d x) \, dx\\ &=2 a b x-\frac {b^2 \log (\cos (c+d x))}{d}+\frac {a^2 \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 43, normalized size = 1.23 \[ \frac {a^2 (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+2 a b x-\frac {b^2 \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 56, normalized size = 1.60 \[ \frac {4 \, a b d x + a^{2} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - b^{2} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.50, size = 50, normalized size = 1.43 \[ \frac {4 \, {\left (d x + c\right )} a b + 2 \, a^{2} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 44, normalized size = 1.26 \[ 2 a b x -\frac {b^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {2 a b c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 49, normalized size = 1.40 \[ \frac {4 \, {\left (d x + c\right )} a b + 2 \, a^{2} \log \left (\tan \left (d x + c\right )\right ) - {\left (a^{2} - b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.19, size = 61, normalized size = 1.74 \[ \frac {a^2\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (b+a\,1{}\mathrm {i}\right )}^2}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (a+b\,1{}\mathrm {i}\right )}^2}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.42, size = 70, normalized size = 2.00 \[ \begin {cases} - \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + 2 a b x + \frac {b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text {for}\: d \neq 0 \\x \left (a + b \tan {\relax (c )}\right )^{2} \cot {\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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