Optimal. Leaf size=430 \[ \frac{\left (7 a^2 C+5 A b^2-2 b^2 C\right ) \sin (c+d x)}{5 b^2 d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (7 a^2 C+3 A b^2-4 b^2 C\right ) \sin (c+d x)}{3 b^3 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left (a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^5 d \left (a^2-b^2\right )}+\frac{\left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^4 d \left (a^2-b^2\right )}-\frac{a^2 \left (-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^5 d (a-b) (a+b)^2} \]
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Rubi [A] time = 1.54768, antiderivative size = 430, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.229, Rules used = {4221, 3048, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac{\left (7 a^2 C+5 A b^2-2 b^2 C\right ) \sin (c+d x)}{5 b^2 d \left (a^2-b^2\right ) \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (7 a^2 C+3 A b^2-4 b^2 C\right ) \sin (c+d x)}{3 b^3 d \left (a^2-b^2\right ) \sqrt{\sec (c+d x)}}-\frac{\left (a^2 C+A b^2\right ) \sin (c+d x)}{b d \left (a^2-b^2\right ) \sec ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))}-\frac{a \left (a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 b^5 d \left (a^2-b^2\right )}+\frac{\left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 b^4 d \left (a^2-b^2\right )}-\frac{a^2 \left (-3 a^2 b^2 (A-3 C)-7 a^4 C+5 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^5 d (a-b) (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3048
Rule 3049
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{(a+b \cos (c+d x))^2 \sec ^{\frac{5}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{5}{2}}(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}-\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (\frac{5}{2} \left (A b^2+a^2 C\right )-a b (A+C) \cos (c+d x)-\frac{1}{2} \left (5 A b^2+7 a^2 C-2 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (5 A b^2+7 a^2 C-2 b^2 C\right ) \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}-\frac{\left (2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\sqrt{\cos (c+d x)} \left (-\frac{3}{4} a \left (5 A b^2+7 a^2 C-2 b^2 C\right )+\frac{1}{2} b \left (5 A b^2+2 a^2 C+3 b^2 C\right ) \cos (c+d x)+\frac{5}{4} a \left (3 A b^2+7 a^2 C-4 b^2 C\right ) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b^2 \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (5 A b^2+7 a^2 C-2 b^2 C\right ) \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (3 A b^2+7 a^2 C-4 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}-\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{8} a^2 \left (3 A b^2+7 a^2 C-4 b^2 C\right )-\frac{1}{4} a b \left (15 A b^2+\left (14 a^2+b^2\right ) C\right ) \cos (c+d x)-\frac{3}{8} \left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^3 \left (a^2-b^2\right )}\\ &=-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (5 A b^2+7 a^2 C-2 b^2 C\right ) \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (3 A b^2+7 a^2 C-4 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{5}{8} a^2 b \left (3 A b^2+7 a^2 C-4 b^2 C\right )-\frac{5}{8} a \left (a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^4 \left (a^2-b^2\right )}+\frac{\left (\left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{10 b^4 \left (a^2-b^2\right )}\\ &=\frac{\left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 b^4 \left (a^2-b^2\right ) d}-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (5 A b^2+7 a^2 C-2 b^2 C\right ) \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (3 A b^2+7 a^2 C-4 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}-\frac{\left (a^2 \left (5 A b^4-3 a^2 b^2 (A-3 C)-7 a^4 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 b^5 \left (a^2-b^2\right )}-\frac{\left (a \left (a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 b^5 \left (a^2-b^2\right )}\\ &=\frac{\left (3 a^2 b^2 (5 A-8 C)+35 a^4 C-2 b^4 (5 A+3 C)\right ) \sqrt{\cos (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{5 b^4 \left (a^2-b^2\right ) d}-\frac{a \left (a^2 b^2 (9 A-20 C)+21 a^4 C-4 b^4 (3 A+C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{3 b^5 \left (a^2-b^2\right ) d}-\frac{a^2 \left (5 A b^4-3 a^2 b^2 (A-3 C)-7 a^4 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-b) b^5 (a+b)^2 d}-\frac{\left (A b^2+a^2 C\right ) \sin (c+d x)}{b \left (a^2-b^2\right ) d (a+b \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}+\frac{\left (5 A b^2+7 a^2 C-2 b^2 C\right ) \sin (c+d x)}{5 b^2 \left (a^2-b^2\right ) d \sec ^{\frac{3}{2}}(c+d x)}-\frac{a \left (3 A b^2+7 a^2 C-4 b^2 C\right ) \sin (c+d x)}{3 b^3 \left (a^2-b^2\right ) d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [A] time = 7.14896, size = 768, normalized size = 1.79 \[ \frac{\sqrt{\sec (c+d x)} \left (-\frac{\left (10 a^2 A b^2-a^2 b^2 C+10 a^4 C+b^4 C\right ) \sin (c+d x)}{10 b^4 \left (a^2-b^2\right )}-\frac{a^3 A b^2 \sin (c+d x)+a^5 C \sin (c+d x)}{b^4 \left (b^2-a^2\right ) (a+b \cos (c+d x))}-\frac{2 a C \sin (2 (c+d x))}{3 b^3}+\frac{C \sin (3 (c+d x))}{10 b^2}\right )}{d}+\frac{-\frac{2 \left (56 a^3 b C+60 a A b^3+4 a b^3 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{2 \left (15 a^2 A b^2-32 a^2 b^2 C+35 a^4 C-30 A b^4-18 b^4 C\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \left (\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{\left (45 a^2 A b^2-72 a^2 b^2 C+105 a^4 C-30 A b^4-18 b^4 C\right ) \sin (c+d x) \cos (2 (c+d x)) (a \sec (c+d x)+b) \left (4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}}{60 b^3 d (a-b) (a+b)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 4.954, size = 1337, normalized size = 3.1 \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{C \cos \left (d x + c\right )^{2} + A}{{\left (b \cos \left (d x + c\right ) + a\right )}^{2} \sec \left (d x + c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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