Optimal. Leaf size=201 \[ \frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{544 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.291753, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3813, 21, 3805, 3804} \[ \frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{544 a^2 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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{9}{2}}(c+d x)} \, dx &=\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{9} (2 a) \int \frac{\frac{17 a}{2}+\frac{17}{2} a \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{9} (17 a) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{21} (34 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)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{1}{105} (136 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)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{1}{315} (272 a) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{34 a^2 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{68 a^2 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{272 a^2 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{544 a^2 \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.520719, size = 80, normalized size = 0.4 \[ \frac{2 a^2 \sin (c+d x) \left (272 \sec ^4(c+d x)+136 \sec ^3(c+d x)+102 \sec ^2(c+d x)+85 \sec (c+d x)+35\right )}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.196, size = 103, normalized size = 0.5 \begin{align*} -{\frac{2\,a \left ( 35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+50\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+17\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+34\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+136\,\cos \left ( dx+c \right ) -272 \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315\,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{9}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 3.37442, size = 535, normalized size = 2.66 \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 = 1.65599, size = 288, normalized size = 1.43 \begin{align*} \frac{2 \,{\left (35 \, a \cos \left (d x + c\right )^{5} + 85 \, a \cos \left (d x + c\right )^{4} + 102 \, a \cos \left (d x + c\right )^{3} + 136 \, a \cos \left (d x + c\right )^{2} + 272 \, 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 )}{315 \,{\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{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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