Optimal. Leaf size=43 \[ \frac{(a B+A b) \log (\sin (c+d x))}{d}+x (-(a A-b B))-\frac{a A \cot (c+d x)}{d} \]
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Rubi [A] time = 0.082227, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3591, 3531, 3475} \[ \frac{(a B+A b) \log (\sin (c+d x))}{d}+x (-(a A-b B))-\frac{a A \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3591
Rule 3531
Rule 3475
Rubi steps
\begin{align*} \int \cot ^2(c+d x) (a+b \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=-\frac{a A \cot (c+d x)}{d}+\int \cot (c+d x) (A b+a B-(a A-b B) \tan (c+d x)) \, dx\\ &=-(a A-b B) x-\frac{a A \cot (c+d x)}{d}+(A b+a B) \int \cot (c+d x) \, dx\\ &=-(a A-b B) x-\frac{a A \cot (c+d x)}{d}+\frac{(A b+a B) \log (\sin (c+d x))}{d}\\ \end{align*}
Mathematica [C] time = 0.174416, size = 78, normalized size = 1.81 \[ -\frac{a A \cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-\tan ^2(c+d x)\right )}{d}+\frac{a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac{A b (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+b B x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 65, normalized size = 1.5 \begin{align*} -Axa+Bbx-{\frac{Aa\cot \left ( dx+c \right ) }{d}}+{\frac{Ab\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-{\frac{Aac}{d}}+{\frac{aB\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}+{\frac{Bbc}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.45873, size = 92, normalized size = 2.14 \begin{align*} -\frac{2 \,{\left (A a - B b\right )}{\left (d x + c\right )} +{\left (B a + A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \,{\left (B a + A b\right )} \log \left (\tan \left (d x + c\right )\right ) + \frac{2 \, A a}{\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 = 1.98403, size = 178, normalized size = 4.14 \begin{align*} -\frac{2 \,{\left (A a - B b\right )} d x \tan \left (d x + c\right ) -{\left (B 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 \, A a}{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 = 1.5604, size = 122, normalized size = 2.84 \begin{align*} \begin{cases} \tilde{\infty } A a 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 ) \cot ^{2}{\left (c \right )} & \text{for}\: d = 0 \\- A a x - \frac{A a}{d \tan{\left (c + d x \right )}} - \frac{A b \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{A b \log{\left (\tan{\left (c + d x \right )} \right )}}{d} - \frac{B a \log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac{B a \log{\left (\tan{\left (c + d x \right )} \right )}}{d} + B b x & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.27816, size = 161, normalized size = 3.74 \begin{align*} \frac{A a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 2 \,{\left (A a - B b\right )}{\left (d x + c\right )} - 2 \,{\left (B a + A b\right )} \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) + 2 \,{\left (B a + A b\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - \frac{2 \, B a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, A b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + A a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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