Optimal. Leaf size=46 \[ \frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d} \]
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Rubi [A] time = 0.0541435, antiderivative size = 46, 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, 88} \[ \frac{\sec (c+d x)}{a^3 d}+\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (\cos (c+d x)+1)}{a^3 d} \]
Antiderivative was successfully verified.
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Rule 3879
Rule 88
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{(a-a x)^2}{x^2 (a+a x)} \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{a}{x^2}-\frac{3 a}{x}+\frac{4 a}{1+x}\right ) \, dx,x,\cos (c+d x)\right )}{a^4 d}\\ &=\frac{3 \log (\cos (c+d x))}{a^3 d}-\frac{4 \log (1+\cos (c+d x))}{a^3 d}+\frac{\sec (c+d x)}{a^3 d}\\ \end{align*}
Mathematica [A] time = 0.113691, size = 36, normalized size = 0.78 \[ \frac{\sec (c+d x)-8 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+3 \log (\cos (c+d x))}{a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.071, size = 46, normalized size = 1. \begin{align*}{\frac{\sec \left ( dx+c \right ) }{{a}^{3}d}}-4\,{\frac{\ln \left ( 1+\sec \left ( dx+c \right ) \right ) }{{a}^{3}d}}+{\frac{\ln \left ( \sec \left ( dx+c \right ) \right ) }{{a}^{3}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.16601, size = 61, normalized size = 1.33 \begin{align*} -\frac{\frac{4 \, \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{3}} - \frac{3 \, \log \left (\cos \left (d x + c\right )\right )}{a^{3}} - \frac{1}{a^{3} \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 = 1.22972, size = 144, normalized size = 3.13 \begin{align*} \frac{3 \, \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right ) - 4 \, \cos \left (d x + c\right ) \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) + 1}{a^{3} 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 ^{5}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec{\left (c + d x \right )} + 1}\, dx}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.67744, size = 151, normalized size = 3.28 \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^{3}} + \frac{3 \, \log \left ({\left | -\frac{\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right )}{a^{3}} - \frac{\frac{3 \,{\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + 1}{a^{3}{\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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