Optimal. Leaf size=138 \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac{b^2 \left (3 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^2}+\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)^2}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.14238, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3885, 894} \[ \frac{b^2}{a d \left (a^2-b^2\right ) (a+b \sec (c+d x))}-\frac{b^2 \left (3 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 d \left (a^2-b^2\right )^2}+\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 d (a+b)^2}+\frac{\log (\sec (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 894
Rubi steps
\begin{align*} \int \frac{\cot (c+d x)}{(a+b \sec (c+d x))^2} \, dx &=-\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{x (a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{b^2 \operatorname{Subst}\left (\int \left (\frac{1}{2 b^2 (a+b)^2 (b-x)}+\frac{1}{a^2 b^2 x}+\frac{1}{a (a-b) (a+b) (a+x)^2}+\frac{3 a^2-b^2}{a^2 (a-b)^2 (a+b)^2 (a+x)}-\frac{1}{2 (a-b)^2 b^2 (b+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{\log (\cos (c+d x))}{a^2 d}+\frac{\log (1-\sec (c+d x))}{2 (a+b)^2 d}+\frac{\log (1+\sec (c+d x))}{2 (a-b)^2 d}-\frac{b^2 \left (3 a^2-b^2\right ) \log (a+b \sec (c+d x))}{a^2 \left (a^2-b^2\right )^2 d}+\frac{b^2}{a \left (a^2-b^2\right ) d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.325961, size = 189, normalized size = 1.37 \[ \frac{a \cos (c+d x) \left (\left (b^4-3 a^2 b^2\right ) \log (a \cos (c+d x)+b)+a^2 (a-b)^2 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+a^2 (a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+b \left ((a-b) \left (a^2 (a-b) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-b^2 (a+b)\right )+\left (b^4-3 a^2 b^2\right ) \log (a \cos (c+d x)+b)+a^2 (a+b)^2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{a^2 d (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 141, normalized size = 1. \begin{align*} -{\frac{{b}^{3}}{d{a}^{2} \left ( a+b \right ) \left ( a-b \right ) \left ( b+a\cos \left ( dx+c \right ) \right ) }}-3\,{\frac{{b}^{2}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}+{\frac{{b}^{4}\ln \left ( b+a\cos \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}{a}^{2}}}+{\frac{\ln \left ( \cos \left ( dx+c \right ) +1 \right ) }{2\,d \left ( a-b \right ) ^{2}}}+{\frac{\ln \left ( -1+\cos \left ( dx+c \right ) \right ) }{2\,d \left ( a+b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994884, size = 192, normalized size = 1.39 \begin{align*} -\frac{\frac{2 \, b^{3}}{a^{4} b - a^{2} b^{3} +{\left (a^{5} - a^{3} b^{2}\right )} \cos \left (d x + c\right )} + \frac{2 \,{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}} - \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{\log \left (\cos \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26706, size = 525, normalized size = 3.8 \begin{align*} -\frac{2 \, a^{2} b^{3} - 2 \, b^{5} + 2 \,{\left (3 \, a^{2} b^{3} - b^{5} +{\left (3 \, a^{3} b^{2} - a b^{4}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) -{\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3} +{\left (a^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) -{\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3} +{\left (a^{5} - 2 \, a^{4} b + a^{3} b^{2}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{2 \,{\left ({\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} d \cos \left (d x + c\right ) +{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (c + d x \right )}}{\left (a + b \sec{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.32971, size = 409, normalized size = 2.96 \begin{align*} -\frac{\frac{2 \,{\left (3 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | -a - b - \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}} - \frac{\log \left (\frac{{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (3 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} + \frac{3 \, a^{2} b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b^{4}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (a^{5} + a^{4} b - a^{3} b^{2} - a^{2} b^{3}\right )}{\left (a + b + \frac{a{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}} + \frac{2 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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