Optimal. Leaf size=270 \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right )}{3 b^4 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right )}{5 b^3 d}+\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)} \]
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Rubi [A] time = 1.05872, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.163, Rules used = {4221, 3049, 3059, 2639, 3002, 2641, 2805} \[ \frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (3 a^2 b B-3 a^3 C-a b^2 (3 A+C)+b^3 B\right )}{3 b^4 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left (\left .\frac{1}{2} (c+d x)\right |2\right ) \left (5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right )}{5 b^3 d}+\frac{2 a^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left (A b^2-a (b B-a C)\right ) \Pi \left (\frac{2 b}{a+b};\left .\frac{1}{2} (c+d x)\right |2\right )}{b^4 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 4221
Rule 3049
Rule 3059
Rule 2639
Rule 3002
Rule 2641
Rule 2805
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)+C \cos ^2(c+d x)}{(a+b \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x)} \, dx &=\left (\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\cos ^{\frac{3}{2}}(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx\\ &=\frac{2 C \sin (c+d x)}{5 b 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 a C}{2}+\frac{1}{2} b (5 A+3 C) \cos (c+d x)+\frac{5}{2} (b B-a C) \cos ^2(c+d x)\right )}{a+b \cos (c+d x)} \, dx}{5 b}\\ &=\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{\frac{5}{4} a (b B-a C)+\frac{1}{4} b (5 b B+4 a C) \cos (c+d x)+\frac{3}{4} \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^2}\\ &=\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}-\frac{\left (4 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{-\frac{5}{4} a b (b B-a C)-\frac{5}{4} \left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (3 A+C)\right ) \cos (c+d x)}{\sqrt{\cos (c+d x)} (a+b \cos (c+d x))} \, dx}{15 b^3}+\frac{\left (\left (5 A b^2-5 a b B+5 a^2 C+3 b^2 C\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \sqrt{\cos (c+d x)} \, dx}{5 b^3}\\ &=\frac{2 \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 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^3 d}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}+\frac{\left (\left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (3 A+C)\right ) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{3 b^4}+\frac{\left (a^2 \left (A b^2-a (b B-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}{b^4}\\ &=\frac{2 \left (5 A b^2-5 a b B+5 a^2 C+3 b^2 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^3 d}+\frac{2 \left (3 a^2 b B+b^3 B-3 a^3 C-a b^2 (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^4 d}+\frac{2 a^2 \left (A b^2-a (b B-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)}}{b^4 (a+b) d}+\frac{2 C \sin (c+d x)}{5 b d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \sqrt{\sec (c+d x)}}\\ \end{align*}
Mathematica [B] time = 7.01389, size = 632, normalized size = 2.34 \[ \frac{\sqrt{\sec (c+d x)} \left (\frac{(b B-a C) \sin (2 (c+d x))}{3 b^2}+\frac{C \sin (c+d x)}{10 b}+\frac{C \sin (3 (c+d x))}{10 b}\right )}{d}-\frac{\frac{2 \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} \left (-5 a^2 C+5 a b B-15 A b^2-9 b^2 C\right ) (a \sec (c+d x)+b) \left (\Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )\right )}{a \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}+\frac{\sin (c+d x) \cos (2 (c+d x)) \left (-15 a^2 C+15 a b B-15 A b^2-9 b^2 C\right ) (a \sec (c+d x)+b) \left (4 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-2 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )+4 a b \sec ^2(c+d x)+2 b (2 a-b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left (\left .\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )-4 a b\right )}{a b^2 \left (1-\cos ^2(c+d x)\right ) \sqrt{\sec (c+d x)} \left (2-\sec ^2(c+d x)\right ) (a+b \cos (c+d x))}-\frac{2 \left (-8 a b C-10 b^2 B\right ) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a \sec (c+d x)+b) \Pi \left (-\frac{a}{b};\left .-\sin ^{-1}\left (\sqrt{\sec (c+d x)}\right )\right |-1\right )}{b \left (1-\cos ^2(c+d x)\right ) (a+b \cos (c+d x))}}{30 b^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 3.438, size = 803, normalized size = 3. \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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{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{C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{{\left (b \cos \left (d x + c\right ) + a\right )} \sec \left (d x + c\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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