Optimal. Leaf size=72 \[ -x \left (a^2 A-2 a b B-A b^2\right )-\frac{a^2 A \cot (c+d x)}{d}+\frac{a (a B+2 A b) \log (\sin (c+d x))}{d}-\frac{b^2 B \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.133096, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3604, 3624, 3475} \[ -x \left (a^2 A-2 a b B-A b^2\right )-\frac{a^2 A \cot (c+d x)}{d}+\frac{a (a B+2 A b) \log (\sin (c+d x))}{d}-\frac{b^2 B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3604
Rule 3624
Rule 3475
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac{a^2 A \cot (c+d x)}{d}+\int \cot (c+d x) \left (a (2 A b+a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b^2 B \tan ^2(c+d x)\right ) \, dx\\ &=-\left (a^2 A-A b^2-2 a b B\right ) x-\frac{a^2 A \cot (c+d x)}{d}+\left (b^2 B\right ) \int \tan (c+d x) \, dx+(a (2 A b+a B)) \int \cot (c+d x) \, dx\\ &=-\left (a^2 A-A b^2-2 a b B\right ) x-\frac{a^2 A \cot (c+d x)}{d}-\frac{b^2 B \log (\cos (c+d x))}{d}+\frac{a (2 A b+a B) \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.262046, size = 100, normalized size = 1.39 \[ \frac{-2 a^2 A \cot (c+d x)+2 a (a B+2 A b) \log (\tan (c+d x))+i (a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)-(a-i b)^2 (B+i A) \log (\tan (c+d x)+i)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.066, size = 110, normalized size = 1.5 \begin{align*} -{a}^{2}Ax+A{b}^{2}x+2\,Babx-{\frac{{a}^{2}A\cot \left ( dx+c \right ) }{d}}+2\,{\frac{Aab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{2}c}{d}}+{\frac{A{b}^{2}c}{d}}+{\frac{{a}^{2}B\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{b}^{2}B\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+2\,{\frac{Babc}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48542, size = 126, normalized size = 1.75 \begin{align*} -\frac{2 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )} +{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (B a^{2} + 2 \, A a b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{2 \, A a^{2}}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01527, size = 274, normalized size = 3.81 \begin{align*} -\frac{B b^{2} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right ) + 2 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} d x \tan \left (d x + c\right ) + 2 \, A a^{2} -{\left (B a^{2} + 2 \, A a b\right )} \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 )}{2 \, d \tan \left (d x + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.86773, size = 167, normalized size = 2.32 \begin{align*} \begin{cases} \tilde{\infty } A a^{2} x & \text{for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\x \left (A + B \tan{\left (c \right )}\right ) \left (a + b \tan{\left (c \right )}\right )^{2} \cot ^{2}{\left (c \right )} & \text{for}\: d = 0 \\- A a^{2} x - \frac{A a^{2}}{d \tan{\left (c + d x \right )}} - \frac{A a b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{2 A a b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + A b^{2} x - \frac{B a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 2 B a b x + \frac{B b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.41451, size = 159, normalized size = 2.21 \begin{align*} -\frac{2 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )} +{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (B a^{2} + 2 \, A a b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) + \frac{2 \,{\left (B a^{2} \tan \left (d x + c\right ) + 2 \, A a b \tan \left (d x + c\right ) + A a^{2}\right )}}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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