Optimal. Leaf size=70 \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0163579, antiderivative size = 70, 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} \[ \frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}}-\frac{\sin ^3(c+d x) \sqrt{b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2633
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{7}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos ^3(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\sqrt{b \sec (c+d x)} \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{d \sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{d \sqrt{\sec (c+d x)}}-\frac{\sqrt{b \sec (c+d x)} \sin ^3(c+d x)}{3 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.108713, size = 45, normalized size = 0.64 \[ \frac{\sin (c+d x) (\cos (2 (c+d x))+5) \sqrt{b \sec (c+d x)}}{6 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, 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}}\sqrt{{\frac{b}{\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 [A] time = 2.07164, size = 57, normalized size = 0.81 \begin{align*} \frac{\sqrt{b}{\left (\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 )\right )}}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38295, size = 130, normalized size = 1.86 \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 \, 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{\sqrt{b \sec \left (d x + c\right )}}{\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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