Optimal. Leaf size=161 \[ \frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.232306, antiderivative size = 161, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3813, 21, 3805, 3804} \[ \frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{208 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3813
Rule 21
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{3/2}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{7} (2 a) \int \frac{\frac{13 a}{2}+\frac{13}{2} a \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{7} (13 a) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{35} (52 a) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{1}{105} (104 a) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{26 a^2 \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{104 a^2 \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{208 a^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{105 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.337498, size = 72, normalized size = 0.45 \[ \frac{a (253 \cos (c+d x)+78 \cos (2 (c+d x))+15 \cos (3 (c+d x))+494) \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\sec (c+d x)+1)}}{210 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.205, size = 93, normalized size = 0.6 \begin{align*} -{\frac{2\,a \left ( 15\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+24\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+13\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+52\,\cos \left ( dx+c \right ) -104 \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{105\,d\sin \left ( dx+c \right ) }\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{7}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.82408, size = 409, normalized size = 2.54 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.65348, size = 255, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (15 \, a \cos \left (d x + c\right )^{4} + 39 \, a \cos \left (d x + c\right )^{3} + 52 \, a \cos \left (d x + c\right )^{2} + 104 \, a \cos \left (d x + c\right )\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 ) + d\right )} \sqrt{\cos \left (d x + c\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}{\sec \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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