Optimal. Leaf size=112 \[ -\frac{B \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{b^4 B \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}+\frac{b B x}{a^2+b^2}+\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \cot ^2(c+d x)}{2 a d} \]
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Rubi [A] time = 0.324771, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {21, 3569, 3649, 3652, 3530, 3475} \[ -\frac{B \left (a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac{b^4 B \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 d \left (a^2+b^2\right )}+\frac{b B x}{a^2+b^2}+\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \cot ^2(c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 3569
Rule 3649
Rule 3652
Rule 3530
Rule 3475
Rubi steps
\begin{align*} \int \frac{\cot ^3(c+d x) (a B+b B \tan (c+d x))}{(a+b \tan (c+d x))^2} \, dx &=B \int \frac{\cot ^3(c+d x)}{a+b \tan (c+d x)} \, dx\\ &=-\frac{B \cot ^2(c+d x)}{2 a d}-\frac{B \int \frac{\cot ^2(c+d x) \left (2 b+2 a \tan (c+d x)+2 b \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a}\\ &=\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \cot ^2(c+d x)}{2 a d}+\frac{B \int \frac{\cot (c+d x) \left (-2 \left (a^2-b^2\right )+2 b^2 \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^2}\\ &=\frac{b B x}{a^2+b^2}+\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \cot ^2(c+d x)}{2 a d}-\frac{\left (\left (a^2-b^2\right ) B\right ) \int \cot (c+d x) \, dx}{a^3}-\frac{\left (b^4 B\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^3 \left (a^2+b^2\right )}\\ &=\frac{b B x}{a^2+b^2}+\frac{b B \cot (c+d x)}{a^2 d}-\frac{B \cot ^2(c+d x)}{2 a d}-\frac{\left (a^2-b^2\right ) B \log (\sin (c+d x))}{a^3 d}-\frac{b^4 B \log (a \cos (c+d x)+b \sin (c+d x))}{a^3 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.621862, size = 107, normalized size = 0.96 \[ -\frac{B \left (\frac{2 b^4 \log (a \cot (c+d x)+b)}{a^3 \left (a^2+b^2\right )}-\frac{2 b \cot (c+d x)}{a^2}-\frac{\log (-\cot (c+d x)+i)}{a-i b}-\frac{\log (\cot (c+d x)+i)}{a+i b}+\frac{\cot ^2(c+d x)}{a}\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.095, size = 151, normalized size = 1.4 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) aB}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) b}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B}{2\,ad \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ) }{ad}}+{\frac{B\ln \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{{a}^{3}d}}+{\frac{Bb}{{a}^{2}d\tan \left ( dx+c \right ) }}-{\frac{{b}^{4}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{{a}^{3}d \left ({a}^{2}+{b}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.59219, size = 176, normalized size = 1.57 \begin{align*} -\frac{\frac{2 \, B b^{4} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{5} + a^{3} b^{2}} - \frac{2 \,{\left (d x + c\right )} B b}{a^{2} + b^{2}} - \frac{B a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (B a^{2} - B b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{3}} - \frac{2 \, B b \tan \left (d x + c\right ) - B a}{a^{2} \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 = 1.90301, size = 435, normalized size = 3.88 \begin{align*} -\frac{B b^{4} \log \left (\frac{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) \tan \left (d x + c\right )^{2} + B a^{4} + B a^{2} b^{2} +{\left (B a^{4} - B b^{4}\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} -{\left (2 \, B a^{3} b d x - B a^{4} - B a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} - 2 \,{\left (B a^{3} b + B a b^{3}\right )} \tan \left (d x + c\right )}{2 \,{\left (a^{5} + a^{3} b^{2}\right )} 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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32287, size = 223, normalized size = 1.99 \begin{align*} -\frac{\frac{2 \, B b^{5} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{5} b + a^{3} b^{3}} - \frac{2 \,{\left (d x + c\right )} B b}{a^{2} + b^{2}} - \frac{B a \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (B a^{2} - B b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac{3 \, B a^{2} \tan \left (d x + c\right )^{2} - 3 \, B b^{2} \tan \left (d x + c\right )^{2} + 2 \, B a b \tan \left (d x + c\right ) - B a^{2}}{a^{3} \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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