Optimal. Leaf size=100 \[ \frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{a^3 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{a^{3/2} d} \]
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Rubi [A] time = 0.0972202, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3880, 88, 50, 63, 207} \[ \frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a^4 d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{a^3 d}+\frac{2 \sqrt{a \sec (c+d x)+a}}{a^2 d}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 88
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\tan ^5(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x)^2 \sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-3 a^2 \sqrt{a+a x}+\frac{a^2 \sqrt{a+a x}}{x}+a (a+a x)^{3/2}\right ) \, dx,x,\sec (c+d x)\right )}{a^4 d}\\ &=-\frac{2 (a+a \sec (c+d x))^{3/2}}{a^3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 \sqrt{a+a \sec (c+d x)}}{a^2 d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{a^3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^4 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=\frac{2 \sqrt{a+a \sec (c+d x)}}{a^2 d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{a^3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^4 d}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{a^2 d}\\ &=-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{a^{3/2} d}+\frac{2 \sqrt{a+a \sec (c+d x)}}{a^2 d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{a^3 d}+\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a^4 d}\\ \end{align*}
Mathematica [A] time = 0.14375, size = 79, normalized size = 0.79 \[ \frac{2 \left (\sec ^3(c+d x)-2 \sec ^2(c+d x)-2 \sec (c+d x)-5 \sqrt{\sec (c+d x)+1} \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )+1\right )}{5 a d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.202, size = 224, normalized size = 2.2 \begin{align*}{\frac{1}{20\,d{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}+10\,\cos \left ( dx+c \right ) \sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}+5\,\sqrt{2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}+8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-24\,\cos \left ( dx+c \right ) +8 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.94421, size = 687, normalized size = 6.87 \begin{align*} \left [\frac{5 \, \sqrt{a} \cos \left (d x + c\right )^{2} \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \,{\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \,{\left (\cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{10 \, a^{2} d \cos \left (d x + c\right )^{2}}, \frac{5 \, \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right )^{2} + 2 \,{\left (\cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{5 \, a^{2} d \cos \left (d x + c\right )^{2}}\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*} \int \frac{\tan ^{5}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 12.0769, size = 227, normalized size = 2.27 \begin{align*} -\frac{2 \,{\left (\frac{5 \, \arctan \left (\frac{\sqrt{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} + \frac{\sqrt{2}{\left (5 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} + 10 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a + 4 \, a^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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