Optimal. Leaf size=86 \[ \frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{14 a \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.152013, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3800, 4001, 3792} \[ \frac{2 \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{14 a \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3800
Rule 4001
Rule 3792
Rubi steps
\begin{align*} \int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \, dx &=\frac{2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac{2 \int \sec (c+d x) \left (\frac{3 a}{2}-a \sec (c+d x)\right ) \sqrt{a+a \sec (c+d x)} \, dx}{5 a}\\ &=-\frac{4 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}+\frac{7}{15} \int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \, dx\\ &=\frac{14 a \tan (c+d x)}{15 d \sqrt{a+a \sec (c+d x)}}-\frac{4 \sqrt{a+a \sec (c+d x)} \tan (c+d x)}{15 d}+\frac{2 (a+a \sec (c+d x))^{3/2} \tan (c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 0.102349, size = 48, normalized size = 0.56 \[ \frac{2 a \tan (c+d x) \left (3 \sec ^2(c+d x)+4 \sec (c+d x)+8\right )}{15 d \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.169, size = 72, normalized size = 0.8 \begin{align*} -{\frac{16\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\cos \left ( dx+c \right ) -6}{15\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}\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 = 1.67688, size = 185, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (8 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 3\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \,{\left (d \cos \left (d x + c\right )^{3} + 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 \sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )} \sec ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 4.85842, size = 136, normalized size = 1.58 \begin{align*} \frac{2 \, \sqrt{2}{\left (15 \, a^{3} +{\left (7 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 10 \, a^{3}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}\right )} \mathrm{sgn}\left (\cos \left (d x + c\right )\right ) \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{15 \,{\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} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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