Optimal. Leaf size=119 \[ -x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac{a^2 (a B+3 A b) \log (\sin (c+d x))}{d}+\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{b^2 (3 a B+A b) \log (\cos (c+d x))}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d} \]
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Rubi [A] time = 0.262912, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {3605, 3637, 3624, 3475} \[ -x \left (a^3 A-3 a^2 b B-3 a A b^2+b^3 B\right )+\frac{a^2 (a B+3 A b) \log (\sin (c+d x))}{d}+\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{b^2 (3 a B+A b) \log (\cos (c+d x))}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d} \]
Antiderivative was successfully verified.
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Rule 3605
Rule 3637
Rule 3624
Rule 3475
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+b \tan (c+d x))^3 (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\int \cot (c+d x) (a+b \tan (c+d x)) \left (a (3 A b+a B)-\left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)+b (a A+b B) \tan ^2(c+d x)\right ) \, dx\\ &=\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d}-\int \cot (c+d x) \left (-a^2 (3 A b+a B)+\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \tan (c+d x)-b^2 (A b+3 a B) \tan ^2(c+d x)\right ) \, dx\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x+\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d}+\left (a^2 (3 A b+a B)\right ) \int \cot (c+d x) \, dx+\left (b^2 (A b+3 a B)\right ) \int \tan (c+d x) \, dx\\ &=-\left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) x-\frac{b^2 (A b+3 a B) \log (\cos (c+d x))}{d}+\frac{a^2 (3 A b+a B) \log (\sin (c+d x))}{d}+\frac{b^2 (a A+b B) \tan (c+d x)}{d}-\frac{a A \cot (c+d x) (a+b \tan (c+d x))^2}{d}\\ \end{align*}
Mathematica [C] time = 0.506833, size = 113, normalized size = 0.95 \[ \frac{2 a^2 (a B+3 A b) \log (\tan (c+d x))-2 a^3 A \cot (c+d x)+i (a+i b)^3 (A+i B) \log (-\tan (c+d x)+i)+(b+i a)^3 (A-i B) \log (\tan (c+d x)+i)+2 b^3 B \tan (c+d x)}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 168, normalized size = 1.4 \begin{align*} -A{a}^{3}x+3\,Aa{b}^{2}x+3\,B{a}^{2}bx-B{b}^{3}x-{\frac{A\cot \left ( dx+c \right ){a}^{3}}{d}}+3\,{\frac{A{a}^{2}b\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{A{b}^{3}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}-{\frac{A{a}^{3}c}{d}}+3\,{\frac{Aa{b}^{2}c}{d}}+{\frac{B{b}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{B{a}^{3}\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-3\,{\frac{Ba{b}^{2}\ln \left ( \cos \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{B{a}^{2}bc}{d}}-{\frac{B{b}^{3}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47403, size = 169, normalized size = 1.42 \begin{align*} \frac{2 \, B b^{3} \tan \left (d x + c\right ) - \frac{2 \, A a^{3}}{\tan \left (d x + c\right )} - 2 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )}{\left (d x + c\right )} -{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.16643, size = 347, normalized size = 2.92 \begin{align*} \frac{2 \, B b^{3} \tan \left (d x + c\right )^{2} - 2 \, A a^{3} - 2 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )} d x \tan \left (d x + c\right ) +{\left (B a^{3} + 3 \, A a^{2} 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 ) -{\left (3 \, B a b^{2} + A b^{3}\right )} \log \left (\frac{1}{\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 = 4.89423, size = 223, normalized size = 1.87 \begin{align*} \begin{cases} \tilde{\infty } A a^{3} 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 )^{3} \cot ^{2}{\left (c \right )} & \text{for}\: d = 0 \\- A a^{3} x - \frac{A a^{3}}{d \tan{\left (c + d x \right )}} - \frac{3 A a^{2} b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{3 A a^{2} b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 3 A a b^{2} x + \frac{A b^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - \frac{B a^{3} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a^{3} \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + 3 B a^{2} b x + \frac{3 B a b^{2} \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} - B b^{3} x + \frac{B b^{3} \tan{\left (c + d x \right )}}{d} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.90259, size = 205, normalized size = 1.72 \begin{align*} \frac{2 \, B b^{3} \tan \left (d x + c\right ) - 2 \,{\left (A a^{3} - 3 \, B a^{2} b - 3 \, A a b^{2} + B b^{3}\right )}{\left (d x + c\right )} -{\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right ) - \frac{2 \,{\left (B a^{3} \tan \left (d x + c\right ) + 3 \, A a^{2} b \tan \left (d x + c\right ) + A a^{3}\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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