Optimal. Leaf size=330 \[ -\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left (a^2-b^2\right )}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a b d \left (a^2-b^2\right )}+\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d \left (a^2-b^2\right )}+\frac{\left (-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 b d (a-b) (a+b)^2} \]
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Rubi [A] time = 1.14817, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3056, 3055, 3059, 2639, 3002, 2641, 2805} \[ -\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d \left (a^2-b^2\right )}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left (a^2-b^2\right ) (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a b d \left (a^2-b^2\right )}+\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d \left (a^2-b^2\right )}+\frac{\left (-a^2 b^2 (5 A+C)+a^4 (-C)+3 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 b d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3056
Rule 3055
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{\left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac{3}{2}}(c+d x)}{(a+b \cos (c+d x))^2} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx\\ &=\frac{\left (A b^2+a^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} \left (-3 A b^2+2 a^2 \left (A-\frac{C}{2}\right )\right )-a b (A+C) \cos (c+d x)+\frac{1}{2} \left (A b^2+a^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2+a^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} b \left (3 A b^2-a^2 (4 A+C)\right )+\frac{1}{2} a \left (2 A b^2-a^2 (A-C)\right ) \cos (c+d x)+\frac{1}{4} b \left (3 A b^2-a^2 (2 A-C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2+a^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{1}{4} b^2 \left (3 A b^2-a^2 (4 A+C)\right )-\frac{1}{4} a b \left (A b^2+a^2 C\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{a^2 b \left (a^2-b^2\right )}+\frac{\left (\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{2 a^2 \left (a^2-b^2\right )}\\ &=\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}-\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2+a^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}+\frac{\left (\left (A b^2+a^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{2 a b \left (a^2-b^2\right )}+\frac{\left (\left (3 A b^4-a^4 C-a^2 b^2 (5 A+C)\right ) \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}{2 a^2 b \left (a^2-b^2\right )}\\ &=\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2+a^2 C\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{a b \left (a^2-b^2\right ) d}+\frac{\left (3 A b^4-a^4 C-a^2 b^2 (5 A+C)\right ) \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) b (a+b)^2 d}-\frac{\left (3 A b^2-a^2 (2 A-C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a^2 \left (a^2-b^2\right ) d}+\frac{\left (A b^2+a^2 C\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{a \left (a^2-b^2\right ) d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [B] time = 7.00557, size = 682, normalized size = 2.07 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{\left (2 a^2 A-a^2 C-3 A b^2\right ) \sin (c+d x)}{a^2 \left (a^2-b^2\right )}+\frac{a^2 C \sin (c+d x)+A b^2 \sin (c+d x)}{a \left (a^2-b^2\right ) (a+b \cos (c+d x))}\right )}{d}-\frac{-\frac{2 \left (4 a^3 A-4 a^3 C-8 a A b^2\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{2 \left (10 a^2 A b+a^2 b C-9 A b^3\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{\left (2 a^2 A b-a^2 b C-3 A b^3\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}}{4 a^2 d (a-b) (a+b)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 4.196, size = 899, normalized size = 2.7 \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(-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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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 (C \cos \left (d x + c\right )^{2} + A\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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