Optimal. Leaf size=33 \[ \frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.0113825, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {17, 3767, 8} \[ \frac{b \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (b \sec (c+d x))^{3/2} \, dx &=\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=-\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt{\sec (c+d x)}}\\ &=\frac{b \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.0203905, size = 32, normalized size = 0.97 \[ \frac{\sin (c+d x) (b \sec (c+d x))^{3/2}}{d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 39, normalized size = 1.2 \begin{align*}{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{d}\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}} \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 = 2.03621, size = 73, normalized size = 2.21 \begin{align*} \frac{2 \, b^{\frac{3}{2}} \sin \left (2 \, d x + 2 \, c\right )}{{\left (\cos \left (2 \, d x + 2 \, c\right )^{2} + \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.41589, size = 81, normalized size = 2.45 \begin{align*} \frac{b \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{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 \left (b \sec \left (d x + c\right )\right )^{\frac{3}{2}} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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