Optimal. Leaf size=155 \[ -\frac{(5 A-C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2} \]
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Rubi [A] time = 0.323075, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.171, Rules used = {3042, 2978, 2748, 2636, 2639, 2641} \[ -\frac{(5 A-C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 3042
Rule 2978
Rule 2748
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{A+C \cos ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx &=-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}+\frac{\int \frac{\frac{1}{2} a (7 A+C)-\frac{3}{2} a (A-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{3 a^2}\\ &=-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}+\frac{\int \frac{6 a^2 A-\frac{1}{2} a^2 (5 A-C) \cos (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{3 a^4}\\ &=-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}+\frac{(2 A) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx}{a^2}-\frac{(5 A-C) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{6 a^2}\\ &=-\frac{(5 A-C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}-\frac{(2 A) \int \sqrt{\cos (c+d x)} \, dx}{a^2}\\ &=-\frac{4 A E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{a^2 d}-\frac{(5 A-C) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac{4 A \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(5 A-C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (1+\cos (c+d x))}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^2}\\ \end{align*}
Mathematica [C] time = 6.63269, size = 834, normalized size = 5.38 \[ -\frac{2 i A \csc \left (\frac{c}{2}\right ) \sec \left (\frac{c}{2}\right ) \left (\frac{2 e^{2 i d x} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) \sqrt{e^{-i d x} \left (2 \left (1+e^{2 i d x}\right ) \cos (c)+2 i \left (-1+e^{2 i d x}\right ) \sin (c)\right )} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left (1+e^{2 i d x}\right ) \cos (c)-3 d \left (-1+e^{2 i d x}\right ) \sin (c)}-\frac{2 \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right ) \sqrt{e^{-i d x} \left (2 \left (1+e^{2 i d x}\right ) \cos (c)+2 i \left (-1+e^{2 i d x}\right ) \sin (c)\right )} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left (-1+e^{2 i d x}\right ) \sin (c)-i d \left (1+e^{2 i d x}\right ) \cos (c)}\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left (\frac{2 \sec \left (\frac{c}{2}\right ) \left (A \sin \left (\frac{d x}{2}\right )+C \sin \left (\frac{d x}{2}\right )\right ) \sec ^3\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{2 (A+C) \tan \left (\frac{c}{2}\right ) \sec ^2\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d}+\frac{8 A \sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right ) \sec \left (\frac{c}{2}+\frac{d x}{2}\right )}{d}+\frac{8 A \cot \left (\frac{c}{2}\right ) \sec (c)}{d}+\frac{8 A \sec (c) \sec (c+d x) \sin (d x)}{d}\right ) \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{(\cos (c+d x) a+a)^2}+\frac{10 A \csc \left (\frac{c}{2}\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (\frac{c}{2}\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc \left (\frac{c}{2}\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right ) \sec \left (\frac{c}{2}\right ) \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \cos ^4\left (\frac{c}{2}+\frac{d x}{2}\right )}{3 d (\cos (c+d x) a+a)^2 \sqrt{\cot ^2(c)+1}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.152, size = 452, normalized size = 2.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt{\cos \left (d x + c\right )}}{a^{2} \cos \left (d x + c\right )^{4} + 2 \, a^{2} \cos \left (d x + c\right )^{3} + a^{2} \cos \left (d x + c\right )^{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{C \cos \left (d x + c\right )^{2} + A}{{\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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