Optimal. Leaf size=201 \[ \frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.332743, antiderivative size = 201, 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, 4015, 3805, 3804} \[ \frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1168 a^3 \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3813
Rule 4015
Rule 3805
Rule 3804
Rubi steps
\begin{align*} \int \frac{(a+a \sec (c+d x))^{5/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{9} (2 a) \int \frac{\sqrt{a+a \sec (c+d x)} \left (\frac{19 a}{2}+\frac{15}{2} a \sec (c+d x)\right )}{\sec ^{\frac{7}{2}}(c+d x)} \, dx\\ &=\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{21} \left (73 a^2\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{105} \left (292 a^2\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sec ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{1}{315} \left (584 a^2\right ) \int \frac{\sqrt{a+a \sec (c+d x)}}{\sqrt{\sec (c+d x)}} \, dx\\ &=\frac{38 a^3 \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{146 a^3 \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}}+\frac{584 a^3 \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}}+\frac{1168 a^3 \sqrt{\sec (c+d x)} \sin (c+d x)}{315 d \sqrt{a+a \sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \sec (c+d x)} \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.564219, size = 80, normalized size = 0.4 \[ \frac{2 a^3 \sin (c+d x) \left (584 \sec ^4(c+d x)+292 \sec ^3(c+d x)+219 \sec ^2(c+d x)+130 \sec (c+d x)+35\right )}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 105, normalized size = 0.5 \begin{align*} -{\frac{2\,{a}^{2} \left ( 35\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+95\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+89\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+73\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+292\,\cos \left ( dx+c \right ) -584 \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 = 2.18681, size = 570, normalized size = 2.84 \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 = 1.89736, size = 302, normalized size = 1.5 \begin{align*} \frac{2 \,{\left (35 \, a^{2} \cos \left (d x + c\right )^{5} + 130 \, a^{2} \cos \left (d x + c\right )^{4} + 219 \, a^{2} \cos \left (d x + c\right )^{3} + 292 \, a^{2} \cos \left (d x + c\right )^{2} + 584 \, a^{2} \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{5}{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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