Optimal. Leaf size=118 \[ \frac{4 a^2 (3 A+2 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d \sqrt{\cos (c+d x)}}+\frac{4 a^2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
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Rubi [A] time = 0.26874, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {2975, 2968, 3023, 2748, 2641, 2639} \[ \frac{4 a^2 (3 A+2 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (3 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) \left (a^2 \cos (c+d x)+a^2\right )}{d \sqrt{\cos (c+d x)}}+\frac{4 a^2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2975
Rule 2968
Rule 3023
Rule 2748
Rule 2641
Rule 2639
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^2 (A+B \cos (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\frac{2 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+2 \int \frac{(a+a \cos (c+d x)) \left (\frac{1}{2} a (3 A+B)-\frac{1}{2} a (3 A-B) \cos (c+d x)\right )}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{2 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+2 \int \frac{\frac{1}{2} a^2 (3 A+B)+\left (-\frac{1}{2} a^2 (3 A-B)+\frac{1}{2} a^2 (3 A+B)\right ) \cos (c+d x)-\frac{1}{2} a^2 (3 A-B) \cos ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 a^2 (3 A-B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{4}{3} \int \frac{\frac{1}{2} a^2 (3 A+2 B)+\frac{3}{2} a^2 B \cos (c+d x)}{\sqrt{\cos (c+d x)}} \, dx\\ &=-\frac{2 a^2 (3 A-B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\left (2 a^2 B\right ) \int \sqrt{\cos (c+d x)} \, dx+\frac{1}{3} \left (2 a^2 (3 A+2 B)\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{4 a^2 B E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{4 a^2 (3 A+2 B) F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{2 a^2 (3 A-B) \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{2 A \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.33418, size = 623, normalized size = 5.28 \[ -\frac{A \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{d \sqrt{\cot ^2(c)+1}}-\frac{B \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{2 d}-\frac{2 B \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (-\frac{\csc (c) \sec (c) (A \cos (2 c)-A+2 B \cos (2 c)+2 B)}{4 d}+\frac{A \sec (c) \sin (d x) \sec (c+d x)}{2 d}+\frac{B \sin (c) \cos (d x)}{6 d}+\frac{B \cos (c) \sin (d x)}{6 d}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 3.598, size = 388, normalized size = 3.3 \begin{align*} -{\frac{4\,{a}^{2}}{3\,d} \left ( 2\,B\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}-\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 3\,A+B \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +3\,A\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}+2\,B\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}-3\,B\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}} \right ){\frac{1}{\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}}} \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}}}} \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 (B \cos \left (d x + c\right ) + 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{B a^{2} \cos \left (d x + c\right )^{3} +{\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} +{\left (2 \, A + B\right )} 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 (B \cos \left (d x + c\right ) + 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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