Optimal. Leaf size=88 \[ -\frac{\left (a^2 A-2 a b B-A b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}+x \left (b^2 B-a (a B+2 A b)\right )-\frac{a (a B+2 A b) \cot (c+d x)}{d} \]
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Rubi [A] time = 0.191368, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3604, 3628, 3531, 3475} \[ -\frac{\left (a^2 A-2 a b B-A b^2\right ) \log (\sin (c+d x))}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}+x \left (b^2 B-a (a B+2 A b)\right )-\frac{a (a B+2 A b) \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3604
Rule 3628
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx &=-\frac{a^2 A \cot ^2(c+d x)}{2 d}+\int \cot ^2(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\\ &=-\frac{a (2 A b+a B) \cot (c+d x)}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) \left (-a^2 A+A b^2+2 a b B+\left (b^2 B-a (2 A b+a B)\right ) \tan (c+d x)\right ) \, dx\\ &=\left (b^2 B-a (2 A b+a B)\right ) x-\frac{a (2 A b+a B) \cot (c+d x)}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}+\left (-a^2 A+A b^2+2 a b B\right ) \int \cot (c+d x) \, dx\\ &=\left (b^2 B-a (2 A b+a B)\right ) x-\frac{a (2 A b+a B) \cot (c+d x)}{d}-\frac{a^2 A \cot ^2(c+d x)}{2 d}-\frac{\left (a^2 A-A b^2-2 a b B\right ) \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.344597, size = 123, normalized size = 1.4 \[ \frac{-2 \left (a^2 A-2 a b B-A b^2\right ) \log (\tan (c+d x))-a^2 A \cot ^2(c+d x)-2 a (a B+2 A b) \cot (c+d x)+(a-i b)^2 (A-i B) \log (\tan (c+d x)+i)+(a+i b)^2 (A+i B) \log (-\tan (c+d x)+i)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 141, normalized size = 1.6 \begin{align*}{\frac{A{b}^{2}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{b}^{2}Bx+{\frac{B{b}^{2}c}{d}}-2\,Axab-2\,{\frac{A\cot \left ( dx+c \right ) ab}{d}}-2\,{\frac{Aabc}{d}}+2\,{\frac{Bab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{{a}^{2}A \left ( \cot \left ( dx+c \right ) \right ) ^{2}}{2\,d}}-{\frac{{a}^{2}A\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{a}^{2}Bx-{\frac{B\cot \left ( dx+c \right ){a}^{2}}{d}}-{\frac{B{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.50988, size = 162, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )}{\left (d x + c\right )} -{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.96709, size = 285, normalized size = 3.24 \begin{align*} -\frac{{\left (A a^{2} - 2 \, B a b - A b^{2}\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} + A a^{2} +{\left (A a^{2} + 2 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )} d x\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \tan \left (d x + c\right )}{2 \, d \tan \left (d x + c\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.77574, size = 214, normalized size = 2.43 \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 ^{3}{\left (c \right )} & \text{for}\: d = 0 \\\frac{A a^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{A a^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{A a^{2}}{2 d \tan ^{2}{\left (c + d x \right )}} - 2 A a b x - \frac{2 A a b}{d \tan{\left (c + d x \right )}} - \frac{A b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{A b^{2} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - B a^{2} x - \frac{B a^{2}}{d \tan{\left (c + d x \right )}} - \frac{B a b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{d} + \frac{2 B a b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + B b^{2} x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.47828, size = 320, normalized size = 3.64 \begin{align*} -\frac{A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \,{\left (B a^{2} + 2 \, A a b - B b^{2}\right )}{\left (d x + c\right )} - 8 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 8 \,{\left (A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{12 \, A a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 24 \, B a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, A b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 4 \, B a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8 \, A a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - A a^{2}}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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