Optimal. Leaf size=63 \[ \frac{x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{2 d \sec ^{\frac{3}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0137245, 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 = {17, 2635, 8} \[ \frac{x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{b \sec (c+d x)}}{2 d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 17
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \frac{\sqrt{b \sec (c+d x)}}{\sec ^{\frac{5}{2}}(c+d x)} \, dx &=\frac{\sqrt{b \sec (c+d x)} \int \cos ^2(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sqrt{b \sec (c+d x)} \int 1 \, dx}{2 \sqrt{\sec (c+d x)}}\\ &=\frac{x \sqrt{b \sec (c+d x)}}{2 \sqrt{\sec (c+d x)}}+\frac{\sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x)}\\ \end{align*}
Mathematica [A] time = 0.0639679, size = 45, normalized size = 0.71 \[ \frac{(2 (c+d x)+\sin (2 (c+d x))) \sqrt{b \sec (c+d x)}}{4 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.14, 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}}\sqrt{{\frac{b}{\cos \left ( dx+c \right ) }}} \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.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 2.01943, size = 34, normalized size = 0.54 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt{b}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.6486, size = 428, normalized size = 6.79 \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 \, 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 \, d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (d x + c\right )}}{\sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]