Optimal. Leaf size=310 \[ \frac{(33 A-13 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.710594, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3041, 2978, 2748, 2636, 2641, 2639} \[ \frac{(33 A-13 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{30 d \left (a^3 \cos (c+d x)+a^3\right )}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{6 a^3 d}+\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4221
Rule 3041
Rule 2978
Rule 2748
Rule 2636
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{\left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^{\frac{5}{2}}(c+d x)}{(a+a \cos (c+d x))^3} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a (13 A-3 B+3 C)-\frac{1}{2} a (7 A-7 B-3 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(2 A-B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{3}{2} a^2 (23 A-8 B+3 C)-\frac{25}{2} a^2 (2 A-B) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{15 a^4}\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(2 A-B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{(119 A-49 B+9 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{15}{4} a^3 (33 A-13 B+3 C)-\frac{3}{4} a^3 (119 A-49 B+9 C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(2 A-B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{(119 A-49 B+9 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\left ((33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{4 a^3}-\frac{\left ((119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{20 a^3}\\ &=-\frac{(119 A-49 B+9 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac{(33 A-13 B+3 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(2 A-B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{(119 A-49 B+9 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}+\frac{\left ((33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{12 a^3}+\frac{\left ((119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sqrt{\sec (c+d x)} \sin (c+d x)}{10 a^3 d}+\frac{(33 A-13 B+3 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{6 a^3 d}-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}-\frac{(2 A-B) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d (a+a \cos (c+d x))^2}-\frac{(119 A-49 B+9 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{30 d \left (a^3+a^3 \cos (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 5.84773, size = 249, normalized size = 0.8 \[ \frac{2 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x) \left (10 (33 A-13 B+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 (119 A-49 B+9 C) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\frac{1}{16} \tan \left (\frac{1}{2} (c+d x)\right ) \sec ^4\left (\frac{1}{2} (c+d x)\right ) (12 (533 A-228 B+33 C) \cos (c+d x)+8 (526 A-221 B+36 C) \cos (2 (c+d x))+1812 A \cos (3 (c+d x))+357 A \cos (4 (c+d x))+3691 A-752 B \cos (3 (c+d x))-147 B \cos (4 (c+d x))-1621 B+132 C \cos (3 (c+d x))+27 C \cos (4 (c+d x))+261 C)\right )}{15 a^3 d (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 5.271, size = 1040, normalized size = 3.4 \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}}, x\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} + B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]