Optimal. Leaf size=121 \[ \frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d} \]
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Rubi [A] time = 0.10358, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{2 (a \sec (c+d x)+a)^{7/2}}{7 a^2 d}+\frac{2 a^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a \sec (c+d x)+a}}{\sqrt{a}}\right )}{d}-\frac{2 (a \sec (c+d x)+a)^{5/2}}{5 a d}-\frac{2 (a \sec (c+d x)+a)^{3/2}}{3 d}-\frac{2 a \sqrt{a \sec (c+d x)+a}}{d} \]
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))^{3/2} \tan ^3(c+d x) \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(-a+a x) (a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a^2 d}\\ &=\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{5/2}}{x} \, dx,x,\sec (c+d x)\right )}{a d}\\ &=-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{a \operatorname{Subst}\left (\int \frac{\sqrt{a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+a x}} \, dx,x,\sec (c+d x)\right )}{d}\\ &=-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}-\frac{(2 a) \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^{3/2} \tanh ^{-1}\left (\frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 a \sqrt{a+a \sec (c+d x)}}{d}-\frac{2 (a+a \sec (c+d x))^{3/2}}{3 d}-\frac{2 (a+a \sec (c+d x))^{5/2}}{5 a d}+\frac{2 (a+a \sec (c+d x))^{7/2}}{7 a^2 d}\\ \end{align*}
Mathematica [A] time = 0.247848, size = 92, normalized size = 0.76 \[ \frac{2 (a (\sec (c+d x)+1))^{3/2} \left (\sqrt{\sec (c+d x)+1} \left (15 \sec ^3(c+d x)+24 \sec ^2(c+d x)-32 \sec (c+d x)-146\right )+105 \tanh ^{-1}\left (\sqrt{\sec (c+d x)+1}\right )\right )}{105 d (\sec (c+d x)+1)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.204, size = 291, normalized size = 2.4 \begin{align*}{\frac{a}{840\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 105\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\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 ) ^{7/2}+315\, \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 ) ^{7/2}+315\,\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 ) ^{7/2}+105\,\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 ) ^{7/2}-2336\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-512\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+384\,\cos \left ( dx+c \right ) +240 \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 = 2.01883, size = 782, normalized size = 6.46 \begin{align*} \left [\frac{105 \, a^{\frac{3}{2}} \cos \left (d x + c\right )^{3} \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 (146 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} - 24 \, a \cos \left (d x + c\right ) - 15 \, a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{210 \, d \cos \left (d x + c\right )^{3}}, -\frac{105 \, \sqrt{-a} 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 )^{3} + 2 \,{\left (146 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} - 24 \, a \cos \left (d x + c\right ) - 15 \, a\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{105 \, d \cos \left (d x + c\right )^{3}}\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 = 4.98603, size = 238, normalized size = 1.97 \begin{align*} -\frac{\sqrt{2}{\left (\frac{105 \, \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 (105 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} a^{3} - 70 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} a^{4} + 84 \,{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )} a^{5} + 120 \, a^{6}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{3} \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 )}{105 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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