Optimal. Leaf size=72 \[ \frac{b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{b \sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0170607, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2633} \[ \frac{b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{b \sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2633
Rubi steps
\begin{align*} \int \frac{(b \sec (c+d x))^{3/2}}{\sec ^{\frac{9}{2}}(c+d x)} \, dx &=\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \int \cos ^3(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\sec (c+d x)}}\\ &=\frac{b \sqrt{b \sec (c+d x)} \sin (c+d x)}{d \sqrt{\sec (c+d x)}}-\frac{b \sqrt{b \sec (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.138329, size = 45, normalized size = 0.62 \[ \frac{\sin (c+d x) (\cos (2 (c+d x))+5) (b \sec (c+d x))^{3/2}}{6 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 52, normalized size = 0.7 \begin{align*}{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) \sin \left ( dx+c \right ) }{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \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 [A] time = 2.08281, size = 61, normalized size = 0.85 \begin{align*} \frac{{\left (b \sin \left (3 \, d x + 3 \, c\right ) + 9 \, b \sin \left (\frac{1}{3} \, \arctan \left (\sin \left (3 \, d x + 3 \, c\right ), \cos \left (3 \, d x + 3 \, c\right )\right )\right )\right )} \sqrt{b}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66819, size = 135, normalized size = 1.88 \begin{align*} \frac{{\left (b \cos \left (d x + c\right )^{3} + 2 \, b \cos \left (d x + c\right )\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, d \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 (b \sec \left (d x + c\right )\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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