Optimal. Leaf size=33 \[ \frac{b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}} \]
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Rubi [A] time = 0.0074591, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {17, 2637} \[ \frac{b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 2637
Rubi steps
\begin{align*} \int \frac{(b \sec (c+d x))^{3/2}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{\left (b \sqrt{b \sec (c+d x)}\right ) \int \cos (c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{b \sqrt{b \sec (c+d x)} \sin (c+d x)}{d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0440589, size = 32, normalized size = 0.97 \[ \frac{\sin (c+d x) (b \sec (c+d x))^{3/2}}{d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 41, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( dx+c \right ) }{d\cos \left ( dx+c \right ) } \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{3}{2}}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.00161, size = 18, normalized size = 0.55 \begin{align*} \frac{b^{\frac{3}{2}} \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57804, size = 78, normalized size = 2.36 \begin{align*} \frac{b \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d} \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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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