Optimal. Leaf size=78 \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{2 d}+\frac{b^2 \sqrt{b \sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\sec (c+d x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.020254, antiderivative size = 78, 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, 3768, 3770} \[ \frac{b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{b \sec (c+d x)}}{2 d}+\frac{b^2 \sqrt{b \sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt{\sec (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 17
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sqrt{\sec (c+d x)} (b \sec (c+d x))^{5/2} \, dx &=\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{\sqrt{\sec (c+d x)}}\\ &=\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d}+\frac{\left (b^2 \sqrt{b \sec (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 \sqrt{\sec (c+d x)}}\\ &=\frac{b^2 \tanh ^{-1}(\sin (c+d x)) \sqrt{b \sec (c+d x)}}{2 d \sqrt{\sec (c+d x)}}+\frac{b^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{b \sec (c+d x)} \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0767913, size = 50, normalized size = 0.64 \[ \frac{(b \sec (c+d x))^{5/2} \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{2 d \sec ^{\frac{5}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.132, size = 112, normalized size = 1.4 \begin{align*}{\frac{\cos \left ( dx+c \right ) }{2\,d} \left ( \ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) -\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}-\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+\sin \left ( dx+c \right ) \right ) \sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}} \left ({\frac{b}{\cos \left ( dx+c \right ) }} \right ) ^{{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.19142, size = 1008, normalized size = 12.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.97983, size = 544, normalized size = 6.97 \begin{align*} \left [\frac{b^{\frac{5}{2}} \cos \left (d x + c\right ) \log \left (-\frac{b \cos \left (d x + c\right )^{2} - 2 \, \sqrt{b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b}{\cos \left (d x + c\right )^{2}}\right ) + \frac{2 \, b^{2} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}, -\frac{\sqrt{-b} b^{2} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b}\right ) \cos \left (d x + c\right ) - \frac{b^{2} \sqrt{\frac{b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{2 \, d \cos \left (d x + c\right )}\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 \left (b \sec \left (d x + c\right )\right )^{\frac{5}{2}} \sqrt{\sec \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]