Optimal. Leaf size=48 \[ \frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d} \]
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Rubi [A] time = 0.0497635, antiderivative size = 48, 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} \[ \frac{\sec ^2(c+d x)}{2 a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}-\frac{\log (\cos (c+d x))}{a^2 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 43
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^2}{x^3} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a^2}{x^3}-\frac{2 a^2}{x^2}+\frac{a^2}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\log (\cos (c+d x))}{a^2 d}-\frac{2 \sec (c+d x)}{a^2 d}+\frac{\sec ^2(c+d x)}{2 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.114018, size = 51, normalized size = 1.06 \[ -\frac{\sec ^2(c+d x) (4 \cos (c+d x)+\cos (2 (c+d x)) \log (\cos (c+d x))+\log (\cos (c+d x))-1)}{2 a^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 46, normalized size = 1. \begin{align*}{\frac{ \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{2\,d{a}^{2}}}-2\,{\frac{\sec \left ( dx+c \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 = 1.17518, size = 54, normalized size = 1.12 \begin{align*} -\frac{\frac{2 \, \log \left (\cos \left (d x + c\right )\right )}{a^{2}} + \frac{4 \, \cos \left (d x + c\right ) - 1}{a^{2} \cos \left (d x + c\right )^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.20419, size = 119, normalized size = 2.48 \begin{align*} -\frac{2 \, \cos \left (d x + c\right )^{2} \log \left (-\cos \left (d x + c\right )\right ) + 4 \, \cos \left (d x + c\right ) - 1}{2 \, a^{2} d \cos \left (d x + c\right )^{2}} \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 ^{5}{\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 [B] time = 3.51986, size = 184, normalized size = 3.83 \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{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{\frac{6 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 5}{a^{2}{\left (\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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