Optimal. Leaf size=76 \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{b \sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{\sec (c+d x)}}{3 b d \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0168956, antiderivative size = 76, 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 = {18, 2633} \[ \frac{\sin (c+d x) \sqrt{\sec (c+d x)}}{b d \sqrt{b \sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{\sec (c+d x)}}{3 b d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 2633
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos ^3(c+d x) \, dx}{b \sqrt{b \sec (c+d x)}}\\ &=-\frac{\sqrt{\sec (c+d x)} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{b d \sqrt{b \sec (c+d x)}}\\ &=\frac{\sqrt{\sec (c+d x)} \sin (c+d x)}{b d \sqrt{b \sec (c+d x)}}-\frac{\sqrt{\sec (c+d x)} \sin ^3(c+d x)}{3 b d \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0848518, size = 45, normalized size = 0.59 \[ \frac{\sin (c+d x) (\cos (2 (c+d x))+5) \sec ^{\frac{3}{2}}(c+d x)}{6 d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 52, normalized size = 0.7 \begin{align*}{\frac{\sin \left ( dx+c \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}+2 \right ) }{3\,d \left ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.04891, size = 57, normalized size = 0.75 \begin{align*} \frac{\sin \left (3 \, d x + 3 \, c\right ) + 9 \, \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 )}{12 \, b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71806, size = 135, normalized size = 1.78 \begin{align*} \frac{{\left (\cos \left (d x + c\right )^{3} + 2 \, \cos \left (d x + c\right )\right )} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, b^{2} 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{1}{\left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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