Optimal. Leaf size=134 \[ \frac{5 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}-\frac{a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.282289, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4222, 2762, 2981, 2774, 216} \[ \frac{5 a^{5/2} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{d}-\frac{a^3 \sin (c+d x)}{d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}+\frac{2 a^2 \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4222
Rule 2762
Rule 2981
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \sec ^{\frac{3}{2}}(c+d x) \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{5/2}}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\left (2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\left (-\frac{3 a}{2}+\frac{1}{2} a \cos (c+d x)\right ) \sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{a^3 \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{2} \left (5 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)}}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{a^3 \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\frac{\left (5 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,-\frac{a \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right )}{d}\\ &=\frac{5 a^{5/2} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a+a \cos (c+d x)}}\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}{d}-\frac{a^3 \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 a^2 \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [C] time = 3.07871, size = 202, normalized size = 1.51 \[ \frac{\sqrt{\cos (c+d x)} \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} (a (\cos (c+d x)+1))^{5/2} \left (6 \sin ^4(c+d x) \csc ^2\left (\frac{1}{2} (c+d x)\right ) \, _3F_2\left (\frac{3}{2},2,\frac{5}{2};1,\frac{9}{2};2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )+24 \sin ^2(c+d x) (\cos (c+d x)+3) \, _2F_1\left (\frac{3}{2},\frac{5}{2};\frac{9}{2};2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )+7 (28 \cos (c+d x)+3 \cos (2 (c+d x))+89) \, _2F_1\left (\frac{1}{2},\frac{3}{2};\frac{7}{2};2 \sin ^2\left (\frac{1}{2} (c+d x)\right )\right )\right )}{420 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.414, size = 186, normalized size = 1.4 \begin{align*}{\frac{{a}^{2}\cos \left ( dx+c \right ) }{d \left ( 1+\cos \left ( dx+c \right ) \right ) } \left ( 5\,\cos \left ( dx+c \right ) \arctan \left ({\frac{\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +5\,\arctan \left ({\frac{\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}+2\,\sin \left ( dx+c \right ) \right ) \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{-1} \right ) ^{{\frac{3}{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 [B] time = 2.05394, size = 1314, normalized size = 9.81 \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 = 1.75789, size = 298, normalized size = 2.22 \begin{align*} -\frac{5 \,{\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt{a} \arctan \left (\frac{\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 ) - \frac{{\left (a^{2} \cos \left (d x + c\right ) + 2 \, a^{2}\right )} \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt{\cos \left (d x + c\right )}}}{d \cos \left (d x + c\right ) + d} \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(-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]