Optimal. Leaf size=160 \[ \frac{4 a^2 (3 A+C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (15 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )}{5 d}+\frac{16 a^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.414087, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3044, 2976, 2968, 3023, 2748, 2641, 2639} \[ \frac{4 a^2 (3 A+C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (15 A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{2 (5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )}{5 d}+\frac{16 a^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 A \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3044
Rule 2976
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^2 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 \int \frac{(a+a \cos (c+d x))^2 \left (2 a A-\frac{1}{2} a (5 A-C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{a}\\ &=\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (5 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{4 \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{4} a^2 (15 A+C)-\frac{1}{4} a^2 (15 A-7 C) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx}{5 a}\\ &=\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (5 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{4 \int \frac{\frac{1}{4} a^3 (15 A+C)+\left (-\frac{1}{4} a^3 (15 A-7 C)+\frac{1}{4} a^3 (15 A+C)\right ) \cos (c+d x)-\frac{1}{4} a^3 (15 A-7 C) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{5 a}\\ &=-\frac{2 a^2 (15 A-7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (5 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{8 \int \frac{\frac{5}{4} a^3 (3 A+C)+3 a^3 C \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx}{15 a}\\ &=-\frac{2 a^2 (15 A-7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (5 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}+\frac{1}{5} \left (8 a^2 C\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (2 a^2 (3 A+C)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{16 a^2 C E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^2 (3 A+C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (15 A-7 C) \sqrt{\cos (c+d x)} \sin (c+d x)}{15 d}+\frac{2 A (a+a \cos (c+d x))^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-\frac{2 (5 A-C) \sqrt{\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{5 d}\\ \end{align*}
Mathematica [C] time = 6.37171, size = 658, normalized size = 4.11 \[ -\frac{A \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{d \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{5 d}-\frac{C \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (-\frac{\csc (c) \sec (c) (5 A \cos (2 c)-5 A+8 C \cos (2 c)+8 C)}{20 d}+\frac{A \sec (c) \sin (d x) \sec (c+d x)}{2 d}+\frac{C \sin (c) \cos (d x)}{3 d}+\frac{C \sin (2 c) \cos (2 d x)}{20 d}+\frac{C \cos (c) \sin (d x)}{3 d}+\frac{C \cos (2 c) \sin (2 d x)}{20 d}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.091, size = 440, normalized size = 2.8 \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 (C \cos \left (d x + c\right )^{2} + A\right )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{C a^{2} \cos \left (d x + c\right )^{4} + 2 \, C a^{2} \cos \left (d x + c\right )^{3} +{\left (A + C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, A a^{2} \cos \left (d x + c\right ) + A a^{2}}{\cos \left (d x + c\right )^{\frac{3}{2}}}, x\right ) \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 )}{\left (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \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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