Optimal. Leaf size=205 \[ \frac{(5 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A-3 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d} \]
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Rubi [A] time = 0.348067, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14, Rules used = {4221, 3041, 2748, 2636, 2641, 2639} \[ \frac{(5 A-3 B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(3 A-3 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d}+\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a d} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3041
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)} \, 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))} \, dx\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} a (5 A-3 B+3 C)-\frac{1}{2} a (3 A-3 B+C) \cos (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac{\left ((3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{2 a}+\frac{\left ((5 A-3 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}{2 a}\\ &=-\frac{(3 A-3 B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{a d}+\frac{(5 A-3 B+3 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}+\frac{\left ((3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a}+\frac{\left ((5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a}\\ &=\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a d}+\frac{(5 A-3 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)}}{3 a d}-\frac{(3 A-3 B+C) \sqrt{\sec (c+d x)} \sin (c+d x)}{a d}+\frac{(5 A-3 B+3 C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}-\frac{(A-B+C) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 2.50083, size = 162, normalized size = 0.79 \[ \frac{\cos ^2\left (\frac{1}{2} (c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x) \left (2 (5 A-3 B+3 C) \cos ^{\frac{3}{2}}(c+d x) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+6 (3 A-3 B+C) \cos ^{\frac{3}{2}}(c+d x) E\left (\left .\frac{1}{2} (c+d x)\right |2\right )-\tan \left (\frac{1}{2} (c+d x)\right ) (3 (3 A-3 B+C) \cos (2 (c+d x))+4 (2 A-3 B) \cos (c+d x)+5 A-9 B+3 C)\right )}{3 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 3.162, size = 494, normalized size = 2.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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}}}{a \cos \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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 \cos \left (d x + c\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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}}}{a \cos \left (d x + c\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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