Optimal. Leaf size=133 \[ \frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.367609, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.162, Rules used = {4089, 4023, 3808, 206, 3801, 215} \[ \frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4089
Rule 4023
Rule 3808
Rule 206
Rule 3801
Rule 215
Rubi steps
\begin{align*} \int \frac{\sqrt{\sec (c+d x)} \left (A+C \sec ^2(c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{\int \frac{\sqrt{\sec (c+d x)} \left (\frac{1}{2} a (2 A+C)-\frac{1}{2} a C \sec (c+d x)\right )}{\sqrt{a+a \sec (c+d x)}} \, dx}{a}\\ &=\frac{C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}-\frac{C \int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \, dx}{2 a}+(A+C) \int \frac{\sqrt{\sec (c+d x)}}{\sqrt{a+a \sec (c+d x)}} \, dx\\ &=\frac{C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}+\frac{C \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a}}} \, dx,x,-\frac{a \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a d}-\frac{(2 (A+C)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,-\frac{a \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{C \sinh ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{\sec (c+d x)} \sin (c+d x)}{\sqrt{2} \sqrt{a+a \sec (c+d x)}}\right )}{\sqrt{a} d}+\frac{C \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 6.6647, size = 717, normalized size = 5.39 \[ \frac{(2 A+C) \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left (\log \left (-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right )-\log \left (-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right )\right ) \left (A+C \sec ^2(c+d x)\right )}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}-\frac{C \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left (8 \log \left (\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right )+\sqrt{2} \left (\log \left (-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right )-\log \left (-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right )\right )-8 \log (\sec (c+d x)+1)\right ) \left (A+C \sec ^2(c+d x)\right )}{4 d (\cos (c+d x)+1) \left (1-\cos ^2(c+d x)\right ) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left (A+C \sec ^2(c+d x)\right ) \left (\frac{2 C \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{2 C \tan \left (\frac{c}{2}\right )}{d}\right )}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.375, size = 252, normalized size = 1.9 \begin{align*}{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}-1}{4\,ad \left ( \sin \left ( dx+c \right ) \right ) ^{2}}\sqrt{ \left ( \cos \left ( dx+c \right ) \right ) ^{-1}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( -C\cos \left ( dx+c \right ) \arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( dx+c \right ) +1+\sin \left ( dx+c \right ) \right ) }{4}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \sqrt{2}+C\cos \left ( dx+c \right ) \arctan \left ({\frac{\sqrt{2} \left ( \cos \left ( dx+c \right ) +1-\sin \left ( dx+c \right ) \right ) }{4}\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \right ) \sqrt{2}+4\,A\cos \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) +4\,C\cos \left ( dx+c \right ) \arctan \left ( 1/2\,\sin \left ( dx+c \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}} \right ) +2\,C\sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}\sin \left ( dx+c \right ) \right ) \sqrt{-2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 2.12321, size = 1307, normalized size = 9.83 \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 = 0.659541, size = 1364, normalized size = 10.26 \begin{align*} \left [\frac{{\left (C \cos \left (d x + c\right ) + C\right )} \sqrt{a} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} + \frac{4 \,{\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt{a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac{2 \, \sqrt{2}{\left ({\left (A + C\right )} a \cos \left (d x + c\right ) +{\left (A + C\right )} a\right )} \log \left (-\frac{\cos \left (d x + c\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt{a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt{a}} + \frac{4 \, C \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{4 \,{\left (a d \cos \left (d x + c\right ) + a d\right )}}, -\frac{2 \, \sqrt{2}{\left ({\left (A + C\right )} a \cos \left (d x + c\right ) +{\left (A + C\right )} a\right )} \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{-\frac{1}{a}} \sqrt{\cos \left (d x + c\right )}}{\sin \left (d x + c\right )}\right ) +{\left (C \cos \left (d x + c\right ) + C\right )} \sqrt{-a} \arctan \left (\frac{2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt{\cos \left (d x + c\right )} \sin \left (d x + c\right )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) - \frac{2 \, C \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{2 \,{\left (a d \cos \left (d x + c\right ) + a d\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 \frac{{\left (C \sec \left (d x + c\right )^{2} + A\right )} \sqrt{\sec \left (d x + c\right )}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]