Optimal. Leaf size=251 \[ \frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.954069, antiderivative size = 251, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4221, 3043, 2976, 2981, 2774, 216} \[ \frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sin ^{-1}\left (\frac{\sqrt{a} \sin (c+d x)}{\sqrt{a \cos (c+d x)+a}}\right )}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+a}}-\frac{a^2 (8 A-2 B-3 C) \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) \sin (c+d x) (a \cos (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)} (a \cos (c+d x)+a)^{5/2}}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4221
Rule 3043
Rule 2976
Rule 2981
Rule 2774
Rule 216
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^{5/2} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \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} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx\\ &=\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{5/2} \left (\frac{1}{2} a (5 A+B)-\frac{1}{2} a (6 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{(a+a \cos (c+d x))^{3/2} \left (\frac{1}{4} a^2 (24 A+6 B+C)-\frac{3}{4} a^2 (8 A-2 B-3 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^2 (8 A-2 B-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{a+a \cos (c+d x)} \left (\frac{1}{8} a^3 (72 A+30 B+13 C)-\frac{1}{8} a^3 (24 A-54 B-49 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{3 a}\\ &=-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-2 B-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}+\frac{1}{16} \left (a^2 (40 A+38 B+25 C) \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 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-2 B-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}-\frac{\left (a^2 (40 A+38 B+25 C) \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 )}{8 d}\\ &=\frac{a^{5/2} (40 A+38 B+25 C) \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)}}{8 d}-\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{a+a \cos (c+d x)} \sqrt{\sec (c+d x)}}-\frac{a^2 (8 A-2 B-3 C) \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{4 d \sqrt{\sec (c+d x)}}-\frac{a (6 A-C) (a+a \cos (c+d x))^{3/2} \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A (a+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \sin (c+d x)}{d}\\ \end{align*}
Mathematica [A] time = 0.995426, size = 156, normalized size = 0.62 \[ \frac{a^2 \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{\sec (c+d x)} \sqrt{a (\cos (c+d x)+1)} \left (3 \sqrt{2} (40 A+38 B+25 C) \sin ^{-1}\left (\sqrt{2} \sin \left (\frac{1}{2} (c+d x)\right )\right ) \sqrt{\cos (c+d x)}+2 \sin \left (\frac{1}{2} (c+d x)\right ) (3 (8 A+22 B+27 C) \cos (c+d x)+48 A+(6 B+17 C) \cos (2 (c+d x))+6 B+2 C \cos (3 (c+d x))+17 C)\right )}{48 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.185, size = 513, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [A] time = 3.26127, size = 478, normalized size = 1.9 \begin{align*} -\frac{3 \,{\left ({\left (40 \, A + 38 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right ) +{\left (40 \, A + 38 \, B + 25 \, 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 (8 \, C a^{2} \cos \left (d x + c\right )^{3} + 2 \,{\left (6 \, B + 17 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (8 \, A + 22 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, A 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 )}}}{24 \,{\left (d \cos \left (d x + c\right ) + 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(-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]