Optimal. Leaf size=69 \[ \frac{x \sqrt{\sec (c+d x)}}{2 b \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 b d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
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Rubi [A] time = 0.0141337, antiderivative size = 69, 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 = {18, 2635, 8} \[ \frac{x \sqrt{\sec (c+d x)}}{2 b \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 b d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{\sec (c+d x)} (b \sec (c+d x))^{3/2}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos ^2(c+d x) \, dx}{b \sqrt{b \sec (c+d x)}}\\ &=\frac{\sin (c+d x)}{2 b d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \int 1 \, dx}{2 b \sqrt{b \sec (c+d x)}}\\ &=\frac{x \sqrt{\sec (c+d x)}}{2 b \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 b d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0659757, size = 45, normalized size = 0.65 \[ \frac{(2 (c+d x)+\sin (2 (c+d x))) \sec ^{\frac{3}{2}}(c+d x)}{4 d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 54, normalized size = 0.8 \begin{align*}{\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +dx+c}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{\frac{1}{\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 = 2.04361, size = 34, normalized size = 0.49 \begin{align*} \frac{2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )}{4 \, b^{\frac{3}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97708, size = 443, normalized size = 6.42 \begin{align*} \left [\frac{2 \, \sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) - \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 )}{4 \, b^{2} d}, \frac{\sqrt{\frac{b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac{3}{2}} \sin \left (d x + c\right ) + \sqrt{b} \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 )}{2 \, b^{2} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 49.1731, size = 82, normalized size = 1.19 \begin{align*} \begin{cases} \frac{x \tan ^{2}{\left (c + d x \right )}}{2 b^{\frac{3}{2}} \sec ^{2}{\left (c + d x \right )}} + \frac{x}{2 b^{\frac{3}{2}} \sec ^{2}{\left (c + d x \right )}} + \frac{\tan{\left (c + d x \right )}}{2 b^{\frac{3}{2}} d \sec ^{2}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\left (b \sec{\left (c \right )}\right )^{\frac{3}{2}} \sqrt{\sec{\left (c \right )}}} & \text{otherwise} \end{cases} \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}} \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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