Optimal. Leaf size=33 \[ \frac{2 \log (\cos (c+d x)+1)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.0448387, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3879, 72} \[ \frac{2 \log (\cos (c+d x)+1)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 72
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x (a+a x)} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{x}-\frac{2}{1+x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}+\frac{2 \log (1+\cos (c+d x))}{a^2 d}\\ \end{align*}
Mathematica [A] time = 0.060438, size = 30, normalized size = 0.91 \[ \frac{4 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )-\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 34, normalized size = 1. \begin{align*} 2\,{\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 = 1.12223, size = 42, normalized size = 1.27 \begin{align*} \frac{\frac{2 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{2}} - \frac{\log \left (\cos \left (d x + c\right )\right )}{a^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19847, size = 85, normalized size = 2.58 \begin{align*} -\frac{\log \left (-\cos \left (d x + c\right )\right ) - 2 \, \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right )}{a^{2} d} \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*} \frac{\int \frac{\tan ^{3}{\left (c + d x \right )}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec{\left (c + d x \right )} + 1}\, dx}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.91302, size = 45, normalized size = 1.36 \begin{align*} -\frac{\log \left ({\left | \frac{{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - 1 \right |}\right )}{a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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