Optimal. Leaf size=24 \[ \frac{x \sqrt{\sec (c+d x)}}{\sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0022741, antiderivative size = 24, 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, 8} \[ \frac{x \sqrt{\sec (c+d x)}}{\sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{b \sec (c+d x)}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int 1 \, dx}{\sqrt{b \sec (c+d x)}}\\ &=\frac{x \sqrt{\sec (c+d x)}}{\sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0147899, size = 24, normalized size = 1. \[ \frac{x \sqrt{\sec (c+d x)}}{\sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 32, normalized size = 1.3 \begin{align*}{\frac{dx+c}{d}\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.57971, size = 35, normalized size = 1.46 \begin{align*} \frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{\sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97238, size = 274, normalized size = 11.42 \begin{align*} \left [-\frac{\sqrt{-b} \log \left (2 \, \sqrt{-b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{2 \, b d}, \frac{\arctan \left (\frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{b} \sqrt{\cos \left (d x + c\right )}}\right )}{\sqrt{b} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 19.0768, size = 5, normalized size = 0.21 \begin{align*} \frac{x}{\sqrt{b}} \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{\sec \left (d x + c\right )}}{\sqrt{b \sec \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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