Optimal. Leaf size=54 \[ -\frac{\log (a+b \sec (c+d x))}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))} \]
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Rubi [A] time = 0.0428775, antiderivative size = 54, 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, 44} \[ -\frac{\log (a+b \sec (c+d x))}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 44
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{(a+b \sec (c+d x))^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x (a+x)^2} \, dx,x,b \sec (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{1}{a (a+x)^2}-\frac{1}{a^2 (a+x)}\right ) \, dx,x,b \sec (c+d x)\right )}{d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{\log (a+b \sec (c+d x))}{a^2 d}+\frac{1}{a d (a+b \sec (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.038492, size = 54, normalized size = 1. \[ -\frac{b \log (a \cos (c+d x)+b)+a \cos (c+d x) \log (a \cos (c+d x)+b)+b}{a^2 d (a \cos (c+d x)+b)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 54, normalized size = 1. \begin{align*} -{\frac{\ln \left ( a+b\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\frac{1}{ad \left ( a+b\sec \left ( dx+c \right ) \right ) }}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{d{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96915, size = 55, normalized size = 1.02 \begin{align*} -\frac{\frac{b}{a^{3} \cos \left (d x + c\right ) + a^{2} b} + \frac{\log \left (a \cos \left (d x + c\right ) + b\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.882972, size = 113, normalized size = 2.09 \begin{align*} -\frac{{\left (a \cos \left (d x + c\right ) + b\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) + b}{a^{3} d \cos \left (d x + c\right ) + a^{2} b d} \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 [B] time = 1.31203, size = 321, normalized size = 5.94 \begin{align*} -\frac{\frac{{\left (a - b\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^{3} - a^{2} b} - \frac{a^{2} - 2 \, a b - b^{2} + \frac{a^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{2 \, a b{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac{b^{2}{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}}{{\left (a^{3} - a^{2} b\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{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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