Optimal. Leaf size=145 \[ \frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d} \]
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Rubi [A] time = 0.118375, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3880, 80, 50, 63, 207} \[ \frac{2 (a \sec (c+d x)+a)^{9/2}}{9 a^2 d}-\frac{2 a^2 \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 d}-\frac{2 a (a \sec (c+d x)+a)^{3/2}}{3 d} \]
Antiderivative was successfully verified.
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Rule 3880
Rule 80
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int (a+a \sec (c+d x))^{5/2} \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x) (a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{7/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{a \operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{x^2}{a}} \, dx,x,\sqrt{a+a \sec (c+d x)}\right )}{d}\\ &=\frac{2 a^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 a^2 \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 a (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 d}-\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a d}+\frac{2 (a+a \sec (c+d x))^{9/2}}{9 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.583184, size = 102, normalized size = 0.7 \[ \frac{2 (a (\sec (c+d x)+1))^{5/2} \left (\sqrt{\sec (c+d x)+1} \left (35 \sec ^4(c+d x)+95 \sec ^3(c+d x)+12 \sec ^2(c+d x)-226 \sec (c+d x)-493\right )+315 \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )\right )}{315 d (\sec (c+d x)+1)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.225, size = 362, normalized size = 2.5 \begin{align*} -{\frac{{a}^{2}}{5040\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 315\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1260\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1890\,\sqrt{2} \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +1260\,\sqrt{2}\cos \left ( dx+c \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{9/2}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}} \right ) +315\,\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 ) ^{9/2}+15776\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+7232\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-384\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-3040\,\cos \left ( dx+c \right ) -1120 \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 = 2.01509, size = 873, normalized size = 6.02 \begin{align*} \left [\frac{315 \, a^{\frac{5}{2}} \cos \left (d x + c\right )^{4} \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 (493 \, a^{2} \cos \left (d x + c\right )^{4} + 226 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 95 \, a^{2} \cos \left (d x + c\right ) - 35 \, a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{630 \, d \cos \left (d x + c\right )^{4}}, -\frac{315 \, \sqrt{-a} a^{2} \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 )^{4} + 2 \,{\left (493 \, a^{2} \cos \left (d x + c\right )^{4} + 226 \, a^{2} \cos \left (d x + c\right )^{3} - 12 \, a^{2} \cos \left (d x + c\right )^{2} - 95 \, a^{2} \cos \left (d x + c\right ) - 35 \, a^{2}\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{315 \, d \cos \left (d x + c\right )^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.8191, size = 266, normalized size = 1.83 \begin{align*} -\frac{\sqrt{2}{\left (\frac{315 \, \sqrt{2} a^{3} \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}} + \frac{2 \,{\left (315 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} a^{3} - 210 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} a^{4} + 252 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} a^{5} - 360 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a^{6} - 560 \, a^{7}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{4} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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