Optimal. Leaf size=36 \[ -\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d} \]
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Rubi [A] time = 0.0347561, antiderivative size = 36, 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 = {3879, 43} \[ -\frac{1}{a^2 d (\cos (c+d x)+1)}-\frac{\log (\cos (c+d x)+1)}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan (c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{x}{(a+a x)^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2 (1+x)^2}+\frac{1}{a^2 (1+x)}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{1}{a^2 d (1+\cos (c+d x))}-\frac{\log (1+\cos (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.132192, size = 56, normalized size = 1.56 \[ -\frac{\sec ^2\left (\frac{1}{2} (c+d x)\right ) \left (2 \cos (c+d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+1\right )}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 50, normalized size = 1.4 \begin{align*}{\frac{1}{d{a}^{2} \left ( 1+\sec \left ( dx+c \right ) \right ) }}-{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{d{a}^{2}}}+{\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 = 1.13091, size = 47, normalized size = 1.31 \begin{align*} -\frac{\frac{1}{a^{2} \cos \left (d x + c\right ) + a^{2}} + \frac{\log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12163, size = 113, normalized size = 3.14 \begin{align*} -\frac{{\left (\cos \left (d x + c\right ) + 1\right )} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1}{a^{2} d \cos \left (d x + c\right ) + a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.8006, size = 177, normalized size = 4.92 \begin{align*} \begin{cases} \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} + \frac{\log{\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )} \sec{\left (c + d x \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} - \frac{2 \log{\left (\sec{\left (c + d x \right )} + 1 \right )}}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} + \frac{2}{2 a^{2} d \sec{\left (c + d x \right )} + 2 a^{2} d} & \text{for}\: d \neq 0 \\\frac{x \tan{\left (c \right )}}{\left (a \sec{\left (c \right )} + a\right )^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26897, size = 77, normalized size = 2.14 \begin{align*} \frac{\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}} + \frac{\cos \left (d x + c\right ) - 1}{a^{2}{\left (\cos \left (d x + c\right ) + 1\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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