Optimal. Leaf size=554 \[ \frac{\left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left (a^2-b^2\right )^2}-\frac{\left (-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{\left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac{b \left (-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3} \]
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Rubi [A] time = 2.23811, antiderivative size = 554, normalized size of antiderivative = 1., number of steps used = 10, 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} \[ \frac{\left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 a^3 d \left (a^2-b^2\right )^2}-\frac{b \left (-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right ) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^4 d \left (a^2-b^2\right )^2}-\frac{\left (-a^2 b^2 (13 A+C)-5 a^4 C+7 A b^4\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{\left (a^2 C+A b^2\right ) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{\left (-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+35 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{12 a^3 d \left (a^2-b^2\right )^2}+\frac{b \left (-a^2 b^2 (65 A-3 C)+3 a^4 (8 A-3 C)+35 A b^4\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d \left (a^2-b^2\right )^2}+\frac{\left (3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+15 a^6 C+35 A b^6\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{4 a^4 d (a-b)^2 (a+b)^3} \]
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{5}{2}}(c+d x)}{(a+b \cos (c+d x))^3} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^3} \, dx\\ &=\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{2} \left (-7 A b^2+a^2 (4 A-3 C)\right )-2 a b (A+C) \cos (c+d x)+\frac{5}{2} \left (A b^2+a^2 C\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))^2} \, dx}{2 a \left (a^2-b^2\right )}\\ &=\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{1}{4} \left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right )+a b \left (A b^2-a^2 (4 A+3 C)\right ) \cos (c+d x)-\frac{3}{4} \left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{2 a^2 \left (a^2-b^2\right )^2}\\ &=\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{3}{8} b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right )-\frac{1}{2} a \left (7 A b^4-2 a^4 (A+3 C)-a^2 b^2 (14 A+3 C)\right ) \cos (c+d x)+\frac{1}{8} b \left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \cos (c+d x))} \, dx}{3 a^3 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 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}{16} \left (105 A b^6+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right )+\frac{1}{4} a b \left (35 A b^4+4 a^4 (5 A-3 C)-a^2 b^2 (64 A-3 C)\right ) \cos (c+d x)+\frac{3}{16} b^2 \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 \left (a^2-b^2\right )^2}\\ &=-\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 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}{16} b \left (105 A b^6+a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)\right )-\frac{1}{16} a b^2 \left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{3 a^4 b \left (a^2-b^2\right )^2}+\frac{\left (b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 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)}}{4 a^4 \left (a^2-b^2\right )^2 d}-\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}+\frac{\left (\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{24 a^3 \left (a^2-b^2\right )^2}+\frac{\left (\left (35 A b^6-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 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}{8 a^4 \left (a^2-b^2\right )^2}\\ &=\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 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)}}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \sqrt{\sec (c+d x)}}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^6-a^2 b^4 (86 A-3 C)+3 a^4 b^2 (21 A-2 C)+15 a^6 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)}}{4 a^4 (a-b)^2 (a+b)^3 d}-\frac{b \left (35 A b^4+3 a^4 (8 A-3 C)-a^2 b^2 (65 A-3 C)\right ) \sqrt{\sec (c+d x)} \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2 d}+\frac{\left (35 A b^4+a^4 (8 A-21 C)-a^2 b^2 (61 A-3 C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{12 a^3 \left (a^2-b^2\right )^2 d}+\frac{\left (A b^2+a^2 C\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{2 a \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac{\left (7 A b^4-5 a^4 C-a^2 b^2 (13 A+C)\right ) \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{4 a^2 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 7.2569, size = 886, normalized size = 1.6 \[ \frac{-\frac{2 \left (160 A b a^5-96 b C a^5-512 A b^3 a^3+24 b^3 C a^3+280 A b^5 a\right ) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{2 \left (16 A a^6+48 C a^6+328 A b^2 a^4-57 b^2 C a^4-641 A b^4 a^2+27 b^4 C a^2+315 A b^6\right ) \left (F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right ) (b+a \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right )}+\frac{\left (105 A b^6-195 a^2 A b^4+9 a^2 C b^4+72 a^4 A b^2-27 a^4 C b^2\right ) \cos (2 (c+d x)) (b+a \sec (c+d x)) \left (4 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a+2 (2 a-b) b F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}-2 b^2 \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right ) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right ) \sin (c+d x)}{a b^2 (a+b \cos (c+d x)) \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right )}}{48 a^4 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left (-\frac{b \left (24 A a^4-9 C a^4-65 A b^2 a^2+3 b^2 C a^2+35 A b^4\right ) \sin (c+d x)}{4 a^4 \left (a^2-b^2\right )^2}+\frac{-A \sin (c+d x) b^3-a^2 C \sin (c+d x) b}{2 a^2 \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}+\frac{9 A \sin (c+d x) b^5-15 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3-7 a^4 C \sin (c+d x) b}{4 a^3 \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}+\frac{2 A \tan (c+d x)}{3 a^3}\right )}{d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 9.041, size = 2140, normalized size = 3.9 \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{5}{2}}}{{\left (b \cos \left (d x + c\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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