Optimal. Leaf size=210 \[ -\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 b (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d} \]
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Rubi [A] time = 0.807839, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.303, Rules used = {2960, 4033, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641} \[ -\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{2 b (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d (a+b)}+\frac{2 A \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2960
Rule 4033
Rule 4102
Rule 4106
Rule 3849
Rule 2805
Rule 3787
Rule 3771
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(A+B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}{a+b \cos (c+d x)} \, dx &=\int \frac{\sec ^{\frac{5}{2}}(c+d x) (B+A \sec (c+d x))}{b+a \sec (c+d x)} \, dx\\ &=\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{2 \int \frac{\sqrt{\sec (c+d x)} \left (\frac{A b}{2}+\frac{1}{2} a A \sec (c+d x)-\frac{3}{2} (A b-a B) \sec ^2(c+d x)\right )}{b+a \sec (c+d x)} \, dx}{3 a}\\ &=-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{4 \int \frac{\frac{3}{4} b (A b-a B)+\frac{1}{4} a (4 A b-3 a B) \sec (c+d x)+\frac{1}{4} \left (a^2 A+3 A b^2-3 a b B\right ) \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (b+a \sec (c+d x))} \, dx}{3 a^2}\\ &=-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{4 \int \frac{\frac{3}{4} b^2 (A b-a B)-\left (-\frac{1}{4} a b (4 A b-3 a B)+\frac{3}{4} a b (A b-a B)\right ) \sec (c+d x)}{\sqrt{\sec (c+d x)}} \, dx}{3 a^2 b^2}+\frac{(b (A b-a B)) \int \frac{\sec ^{\frac{3}{2}}(c+d x)}{b+a \sec (c+d x)} \, dx}{a^2}\\ &=-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{A \int \sqrt{\sec (c+d x)} \, dx}{3 a}+\frac{(A b-a B) \int \frac{1}{\sqrt{\sec (c+d x)}} \, dx}{a^2}+\frac{\left (b (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^2}\\ &=\frac{2 b (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 (a+b) d}-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}+\frac{\left (A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 a}+\frac{\left ((A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{a^2}\\ &=\frac{2 (A b-a B) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 A \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{2 b (A b-a B) \sqrt{\cos (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 (a+b) d}-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 d}+\frac{2 A \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 3.40825, size = 229, normalized size = 1.09 \[ -\frac{\cot (c+d x) \left (-2 \left (a^2 (A-3 B)+3 a b (A-B)+3 A b^2\right ) \sqrt{-\tan ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-a^2 A \sec ^{\frac{5}{2}}(c+d x)+a^2 A \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)-6 A b^2 \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-6 a (a B-A b) \sqrt{-\tan ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+6 a b B \sqrt{-\tan ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{3 a^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 9.955, size = 468, normalized size = 2.2 \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 (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{b \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(-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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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 (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac{5}{2}}}{b \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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