Optimal. Leaf size=111 \[ \frac{A b-a B}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac{\left (a^2 (-B)+2 a A b+b^2 B\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.149454, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3531, 3530} \[ \frac{A b-a B}{d \left (a^2+b^2\right ) (a+b \cot (c+d x))}-\frac{\left (a^2 (-B)+2 a A b+b^2 B\right ) \log (a \sin (c+d x)+b \cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{A+B \cot (c+d x)}{(a+b \cot (c+d x))^2} \, dx &=\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}+\frac{\int \frac{a A+b B-(A b-a B) \cot (c+d x)}{a+b \cot (c+d x)} \, dx}{a^2+b^2}\\ &=\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (2 a A b-a^2 B+b^2 B\right ) \int \frac{-b+a \cot (c+d x)}{a+b \cot (c+d x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{\left (a^2 A-A b^2+2 a b B\right ) x}{\left (a^2+b^2\right )^2}+\frac{A b-a B}{\left (a^2+b^2\right ) d (a+b \cot (c+d x))}-\frac{\left (2 a A b-a^2 B+b^2 B\right ) \log (b \cos (c+d x)+a \sin (c+d x))}{\left (a^2+b^2\right )^2 d}\\ \end{align*}
Mathematica [C] time = 1.79675, size = 144, normalized size = 1.3 \[ \frac{\frac{2 b (a B-A b)}{a \left (a^2+b^2\right ) (a \tan (c+d x)+b)}+\frac{2 \left (a^2 B-2 a A b-b^2 B\right ) \log (a \tan (c+d x)+b)}{\left (a^2+b^2\right )^2}-\frac{(B+i A) \log (-\tan (c+d x)+i)}{(a-i b)^2}+\frac{i (A+i B) \log (\tan (c+d x)+i)}{(a+i b)^2}}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.031, size = 356, normalized size = 3.2 \begin{align*}{\frac{Ab}{d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\cot \left ( dx+c \right ) \right ) }}-{\frac{Ba}{d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\cot \left ( dx+c \right ) \right ) }}-2\,{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) Aab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) B{a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( a+b\cot \left ( dx+c \right ) \right ) B{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) Aab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) B{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{\ln \left ( \left ( \cot \left ( dx+c \right ) \right ) ^{2}+1 \right ) B{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A\pi \,{a}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{A\pi \,{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{B\pi \,ab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+{\frac{A{\rm arccot} \left (\cot \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{A{\rm arccot} \left (\cot \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}+2\,{\frac{B{\rm arccot} \left (\cot \left ( dx+c \right ) \right )ab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.75696, size = 250, normalized size = 2.25 \begin{align*} \frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (a \tan \left (d x + c\right ) + b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a b - A b^{2}\right )}}{a^{3} b + a b^{3} +{\left (a^{4} + a^{2} b^{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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Fricas [B] time = 1.69335, size = 748, normalized size = 6.74 \begin{align*} \frac{2 \, B a^{2} b - 2 \, A a b^{2} + 2 \,{\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} d x + 2 \,{\left (B a^{2} b - A a b^{2} +{\left (A a^{2} b + 2 \, B a b^{2} - A b^{3}\right )} d x\right )} \cos \left (2 \, d x + 2 \, c\right ) +{\left (B a^{2} b - 2 \, A a b^{2} - B b^{3} +{\left (B a^{2} b - 2 \, A a b^{2} - B b^{3}\right )} \cos \left (2 \, d x + 2 \, c\right ) +{\left (B a^{3} - 2 \, A a^{2} b - B a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \log \left (a b \sin \left (2 \, d x + 2 \, c\right ) + \frac{1}{2} \, a^{2} + \frac{1}{2} \, b^{2} - \frac{1}{2} \,{\left (a^{2} - b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )\right ) - 2 \,{\left (B a b^{2} - A b^{3} -{\left (A a^{3} + 2 \, B a^{2} b - A a b^{2}\right )} d x\right )} \sin \left (2 \, d x + 2 \, c\right )}{2 \,{\left ({\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d \cos \left (2 \, d x + 2 \, c\right ) +{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} d \sin \left (2 \, d x + 2 \, c\right ) +{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.38256, size = 325, normalized size = 2.93 \begin{align*} \frac{\frac{2 \,{\left (A a^{2} + 2 \, B a b - A b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (B a^{2} - 2 \, A a b - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (B a^{3} - 2 \, A a^{2} b - B a b^{2}\right )} \log \left ({\left | a \tan \left (d x + c\right ) + b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} - \frac{2 \,{\left (B a^{4} \tan \left (d x + c\right ) - 2 \, A a^{3} b \tan \left (d x + c\right ) - B a^{2} b^{2} \tan \left (d x + c\right ) - A a^{2} b^{2} - 2 \, B a b^{3} + A b^{4}\right )}}{{\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )}{\left (a \tan \left (d x + c\right ) + b\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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