Optimal. Leaf size=41 \[ -x \left (a^2-b^2\right )-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3542, 3531, 3475} \[ -x \left (a^2-b^2\right )-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3542
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 \, dx &=-\frac {a^2 \cot (c+d x)}{d}+\int \cot (c+d x) \left (2 a b-\left (a^2-b^2\right ) \tan (c+d x)\right ) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac {a^2 \cot (c+d x)}{d}+(2 a b) \int \cot (c+d x) \, dx\\ &=-\left (a^2-b^2\right ) x-\frac {a^2 \cot (c+d x)}{d}+\frac {2 a b \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.15, size = 82, normalized size = 2.00 \[ \frac {-2 a^2 \cot (c+d x)+i \left ((a-i b)^2 (-\log (\tan (c+d x)+i))+(a+i b)^2 \log (-\tan (c+d x)+i)-4 i a b \log (\tan (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 67, normalized size = 1.63 \[ -\frac {{\left (a^{2} - b^{2}\right )} d x \tan \left (d x + c\right ) - a b \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + a^{2}}{d \tan \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.43, size = 98, normalized size = 2.39 \[ -\frac {4 \, a b \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right ) - 4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \, {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {4 \, a b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.26, size = 58, normalized size = 1.41 \[ -a^{2} x +b^{2} x -\frac {a^{2} \cot \left (d x +c \right )}{d}+\frac {2 a b \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{2} c}{d}+\frac {c \,b^{2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 58, normalized size = 1.41 \[ -\frac {a b \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, a b \log \left (\tan \left (d x + c\right )\right ) + {\left (a^{2} - b^{2}\right )} {\left (d x + c\right )} + \frac {a^{2}}{\tan \left (d x + c\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.07, size = 79, normalized size = 1.93 \[ \frac {2\,a\,b\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {a^2\,\mathrm {cot}\left (c+d\,x\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,{\left (a-b\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,{\left (-b+a\,1{}\mathrm {i}\right )}^2\,1{}\mathrm {i}}{2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.83, size = 83, normalized size = 2.02 \[ \begin {cases} \tilde {\infty } a^{2} x & \text {for}\: c = 0 \wedge d = 0 \\x \left (a + b \tan {\relax (c )}\right )^{2} \cot ^{2}{\relax (c )} & \text {for}\: d = 0 \\\tilde {\infty } a^{2} x & \text {for}\: c = - d x \\- a^{2} x - \frac {a^{2}}{d \tan {\left (c + d x \right )}} - \frac {a b \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac {2 a b \log {\left (\tan {\left (c + d x \right )} \right )}}{d} + b^{2} x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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