Optimal. Leaf size=156 \[ \frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.46271, antiderivative size = 156, 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 = {4221, 3044, 2984, 12, 2782, 205} \[ \frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a \cos (c+d x)+a}}\right )}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d \sqrt{a \cos (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4221
Rule 3044
Rule 2984
Rule 12
Rule 2782
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x)}{\sqrt{a+a \cos (c+d x)}} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx\\ &=\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{a A}{2}+\frac{1}{2} a (2 A+3 C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{3 a^2 (A+C)}{4 \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx}{3 a^2}\\ &=-\frac{2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\left ((A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}} \, dx\\ &=-\frac{2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}-\frac{\left (2 a (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{2 a^2+a x^2} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{\sqrt{2} (A+C) \tan ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{2} \sqrt{\cos (c+d x)} \sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{\sqrt{a} d}-\frac{2 A \sqrt{\sec (c+d x)} \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.79832, size = 576, normalized size = 3.69 \[ \frac{2 \sqrt{\frac{1}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}} \sqrt{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )} \cos \left (\frac{c}{2}+\frac{d x}{2}\right ) \left (\frac{(A+C) \csc ^5\left (\frac{c}{2}+\frac{d x}{2}\right ) \left (-12 \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \cos ^4\left (\frac{1}{2} (c+d x)\right ) \, _3F_2\left (2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}\right )-12 \left (3 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )-7 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )+4\right ) \sin ^8\left (\frac{c}{2}+\frac{d x}{2}\right ) \, _2F_1\left (2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}\right )+7 \sqrt{-\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}} \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )^3 \left (8 \sin ^4\left (\frac{c}{2}+\frac{d x}{2}\right )-20 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )+15\right ) \left (\left (3-7 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \sqrt{-\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}}-3 \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right ) \tanh ^{-1}\left (\sqrt{-\frac{\sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}}\right )\right )\right )}{63 \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )^{7/2}}-\frac{4 C \sin ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 \left (1-2 \sin ^2\left (\frac{c}{2}+\frac{d x}{2}\right )\right )^{3/2}}\right )}{d \sqrt{a (\cos (c+d x)+1)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.187, size = 366, normalized size = 2.4 \begin{align*}{\frac{\sqrt{2}\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{3\,ad \left ( -1+\cos \left ( dx+c \right ) \right ) \left ( 1+\cos \left ( dx+c \right ) \right ) ^{2}} \left ( 3\,A \left ( \cos \left ( dx+c \right ) \right ) ^{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+3\,C \left ( \cos \left ( dx+c \right ) \right ) ^{2}\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+6\,A\cos \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+6\,C\cos \left ( dx+c \right ) \arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+3\,A\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+3\,C\arcsin \left ({\frac{-1+\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \left ({\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }} \right ) ^{3/2}+A\sqrt{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -A\sqrt{2}\sin \left ( dx+c \right ) \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{5}{2}}}\sqrt{a \left ( 1+\cos \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.95146, size = 382, normalized size = 2.45 \begin{align*} -\frac{\frac{3 \, \sqrt{2}{\left ({\left (A + C\right )} a \cos \left (d x + c\right )^{2} +{\left (A + C\right )} a \cos \left (d x + c\right )\right )} \arctan \left (\frac{\sqrt{2} \sqrt{a \cos \left (d x + c\right ) + a} \sqrt{\cos \left (d x + c\right )}}{\sqrt{a} \sin \left (d x + c\right )}\right )}{\sqrt{a}} + \frac{2 \,{\left (A \cos \left (d x + c\right ) - A\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{3 \,{\left (a d \cos \left (d x + c\right )^{2} + a 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 \frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{\sqrt{a \cos \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]