Optimal. Leaf size=28 \[ \frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d} \]
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Rubi [A] time = 0.0481519, antiderivative size = 28, 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, 43} \[ \frac{\sec (c+d x)}{a d}+\frac{\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan ^3(c+d x)}{a+a \sec (c+d x)} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{a-a x}{x^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{a}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^2 d}\\ &=\frac{\log (\cos (c+d x))}{a d}+\frac{\sec (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.065714, size = 21, normalized size = 0.75 \[ \frac{\sec (c+d x)+\log (\cos (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 30, normalized size = 1.1 \begin{align*}{\frac{\sec \left ( dx+c \right ) }{da}}-{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14619, size = 38, normalized size = 1.36 \begin{align*} \frac{\frac{\log \left (\cos \left (d x + c\right )\right )}{a} + \frac{1}{a \cos \left (d x + c\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.15497, size = 78, normalized size = 2.79 \begin{align*} \frac{\cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) + 1}{a d \cos \left (d x + c\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*} \frac{\int \frac{\tan ^{3}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.97942, size = 150, normalized size = 5.36 \begin{align*} -\frac{\frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right )}{a} - \frac{\log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a} + \frac{\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1}{a{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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