Optimal. Leaf size=121 \[ \frac{52 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{28 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d} \]
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Rubi [A] time = 0.128461, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2757, 2639, 2635, 2641} \[ \frac{52 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{28 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^3 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 a^3 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{52 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2639
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{\cos (c+d x)} (a+a \cos (c+d x))^3 \, dx &=\int \left (a^3 \sqrt{\cos (c+d x)}+3 a^3 \cos ^{\frac{3}{2}}(c+d x)+3 a^3 \cos ^{\frac{5}{2}}(c+d x)+a^3 \cos ^{\frac{7}{2}}(c+d x)\right ) \, dx\\ &=a^3 \int \sqrt{\cos (c+d x)} \, dx+a^3 \int \cos ^{\frac{7}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^{\frac{5}{2}}(c+d x) \, dx\\ &=\frac{2 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^3 \sqrt{\cos (c+d x)} \sin (c+d x)}{d}+\frac{6 a^3 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{7} \left (5 a^3\right ) \int \cos ^{\frac{3}{2}}(c+d x) \, dx+a^3 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\frac{1}{5} \left (9 a^3\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{28 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{2 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{6 a^3 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac{1}{21} \left (5 a^3\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx\\ &=\frac{28 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{5 d}+\frac{52 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{21 d}+\frac{52 a^3 \sqrt{\cos (c+d x)} \sin (c+d x)}{21 d}+\frac{6 a^3 \cos ^{\frac{3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac{2 a^3 \cos ^{\frac{5}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end{align*}
Mathematica [C] time = 6.11681, size = 500, normalized size = 4.13 \[ -\frac{7 \csc (c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3 \left (\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt{\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac{\frac{\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{20 d}-\frac{13 \csc (c) \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3 \sqrt{1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin (c) \left (-\sqrt{\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt{\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{42 d \sqrt{\cot ^2(c)+1}}+\sqrt{\cos (c+d x)} \sec ^6\left (\frac{c}{2}+\frac{d x}{2}\right ) (a \cos (c+d x)+a)^3 \left (\frac{107 \sin (c) \cos (d x)}{336 d}+\frac{3 \sin (2 c) \cos (2 d x)}{40 d}+\frac{\sin (3 c) \cos (3 d x)}{112 d}+\frac{107 \cos (c) \sin (d x)}{336 d}+\frac{3 \cos (2 c) \sin (2 d x)}{40 d}+\frac{\cos (3 c) \sin (3 d x)}{112 d}-\frac{7 \cot (c)}{10 d}\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.362, size = 272, normalized size = 2.3 \begin{align*} -{\frac{4\,{a}^{3}}{105\,d}\sqrt{ \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1 \right ) \left ( \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}} \left ( 120\,\cos \left ( 1/2\,dx+c/2 \right ) \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{8}-432\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{6}\cos \left ( 1/2\,dx+c/2 \right ) +602\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +65\,\sqrt{2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}\sqrt{ \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}}{\it EllipticF} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -147\,\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 EllipticE} \left ( \cos \left ( 1/2\,dx+c/2 \right ) ,\sqrt{2} \right ) -208\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \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{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \sqrt{\cos \left (d x + c\right )}\,{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 ({\left (a^{3} \cos \left (d x + c\right )^{3} + 3 \, a^{3} \cos \left (d x + c\right )^{2} + 3 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )} \sqrt{\cos \left (d x + c\right )}, 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(-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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