Optimal. Leaf size=121 \[ \frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{16 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.118515, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2757, 2636, 2639, 2641} \[ \frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}-\frac{16 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{7}{2}}(c+d x)} \, dx &=\int \left (\frac{a^2}{\cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a^2}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{a^2}{\cos ^{\frac{3}{2}}(c+d x)}\right ) \, dx\\ &=a^2 \int \frac{1}{\cos ^{\frac{7}{2}}(c+d x)} \, dx+a^2 \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\left (2 a^2\right ) \int \frac{1}{\cos ^{\frac{5}{2}}(c+d x)} \, dx\\ &=\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{1}{5} \left (3 a^2\right ) \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+\frac{1}{3} \left (2 a^2\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-a^2 \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{2 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}-\frac{1}{5} \left (3 a^2\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=-\frac{16 a^2 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^2 \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{16 a^2 \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [C] time = 6.19474, size = 487, normalized size = 4.02 \[ \frac{2 \csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{5 d}-\frac{\csc (c) \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{3 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^4\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^2 \left (\frac{\sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 \sin (c)+10 \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (5 \sin (c)+12 \sin (d x)) \sec (c+d x)}{15 d}+\frac{4 \csc (c) \sec (c)}{5 d}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 3.846, size = 386, normalized size = 3.2 \begin{align*} -8\,{\frac{\sqrt{- \left ( -2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{a}^{2}}{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d} \left ( -1/12\,{\frac{\cos \left ( 1/2\,dx+c/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}}}{ \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{2}}}+{\frac{17\,\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) }{30\,\sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}+ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}}-4/5\,{\frac{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) }{\sqrt{- \left ( -2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}}}-2/5\,{\frac{\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1} \left ({\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -{\it EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) \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}}}}-{\frac{\cos \left ( 1/2\,dx+c/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}}}{80\, \left ( \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1/2 \right ) ^{3}}} \right ) } \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 (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac{7}{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{a^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right ) + a^{2}}{\cos \left (d x + c\right )^{\frac{7}{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 (a \cos \left (d x + c\right ) + a\right )}^{2}}{\cos \left (d x + c\right )^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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