Optimal. Leaf size=116 \[ \frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d} \]
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Rubi [A] time = 0.193941, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3800, 4001, 3793, 3792} \[ \frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{38 a \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d} \]
Antiderivative was successfully verified.
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Rule 3800
Rule 4001
Rule 3793
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{2 \int \sec (c+d x) \left (\frac{5 a}{2}-a \sec (c+d x)\right ) (a+a \sec (c+d x))^{3/2} \, dx}{7 a}\\ &=-\frac{4 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{19}{35} \int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \, dx\\ &=\frac{38 a \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}-\frac{4 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}+\frac{1}{105} (76 a) \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{152 a^2 \tan (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}+\frac{38 a \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{105 d}-\frac{4 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{35 d}+\frac{2 (a+a \sec (c+d x))^{5/2} \tan (c+d x)}{7 a d}\\ \end{align*}
Mathematica [A] time = 0.178844, size = 60, normalized size = 0.52 \[ \frac{2 a^2 \tan (c+d x) \left (15 \sec ^3(c+d x)+39 \sec ^2(c+d x)+52 \sec (c+d x)+104\right )}{105 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.148, size = 83, normalized size = 0.7 \begin{align*} -{\frac{2\,a \left ( 104\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-52\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-13\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-24\,\cos \left ( dx+c \right ) -15 \right ) }{105\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 1.69004, size = 230, normalized size = 1.98 \begin{align*} \frac{2 \,{\left (104 \, a \cos \left (d x + c\right )^{3} + 52 \, a \cos \left (d x + c\right )^{2} + 39 \, 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 )}} \sin \left (d x + c\right )}{105 \,{\left (d \cos \left (d x + c\right )^{4} + 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 = 4.83551, size = 204, normalized size = 1.76 \begin{align*} -\frac{4 \,{\left (105 \, \sqrt{2} a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) -{\left (140 \, \sqrt{2} a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) + 19 \,{\left (2 \, \sqrt{2} a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7 \, \sqrt{2} a^{5} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{105 \,{\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} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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