Optimal. Leaf size=63 \[ \frac{x \sqrt{\sec (c+d x)}}{2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0135085, antiderivative size = 63, 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} \[ \frac{x \sqrt{\sec (c+d x)}}{2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 18
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{1}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{b \sec (c+d x)}} \, dx &=\frac{\sqrt{\sec (c+d x)} \int \cos ^2(c+d x) \, dx}{\sqrt{b \sec (c+d x)}}\\ &=\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \int 1 \, dx}{2 \sqrt{b \sec (c+d x)}}\\ &=\frac{x \sqrt{\sec (c+d x)}}{2 \sqrt{b \sec (c+d x)}}+\frac{\sin (c+d x)}{2 d \sqrt{\sec (c+d x)} \sqrt{b \sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0769065, size = 45, normalized size = 0.71 \[ \frac{(2 (c+d x)+\sin (2 (c+d x))) \sqrt{\sec (c+d x)}}{4 d \sqrt{b \sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.135, size = 54, normalized size = 0.9 \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}} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.05379, size = 34, normalized size = 0.54 \begin{align*} \frac{2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )}{4 \, \sqrt{b} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.00488, size = 437, normalized size = 6.94 \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 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 d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 44.2609, size = 82, normalized size = 1.3 \begin{align*} \begin{cases} \frac{x \tan ^{2}{\left (c + d x \right )}}{2 \sqrt{b} \sec ^{2}{\left (c + d x \right )}} + \frac{x}{2 \sqrt{b} \sec ^{2}{\left (c + d x \right )}} + \frac{\tan{\left (c + d x \right )}}{2 \sqrt{b} d \sec ^{2}{\left (c + d x \right )}} & \text{for}\: d \neq 0 \\\frac{x}{\sqrt{b \sec{\left (c \right )}} \sec ^{\frac{3}{2}}{\left (c \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (d x + c\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]