Optimal. Leaf size=91 \[ \frac{20 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.110365, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {2757, 2636, 2639, 2641, 2635} \[ \frac{20 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{4 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^3 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2636
Rule 2639
Rule 2641
Rule 2635
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^3}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\int \left (\frac{a^3}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{3 a^3}{\sqrt{\cos (c+d x)}}+3 a^3 \sqrt{\cos (c+d x)}+a^3 \cos ^{\frac{3}{2}}(c+d x)\right ) \, dx\\ &=a^3 \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+a^3 \int \cos ^{\frac{3}{2}}(c+d x) \, dx+\left (3 a^3\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx+\left (3 a^3\right ) \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{6 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{6 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a^3 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} a^3 \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx-a^3 \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^3 E\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{20 a^3 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{3 d}+\frac{2 a^3 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a^3 \sqrt{\cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [C] time = 4.6832, size = 240, normalized size = 2.64 \[ \frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left (\frac{1}{2} (c+d x)\right ) \left (-6 \cos (c) \sqrt{\sec ^2(c)} \sqrt{\sin ^2\left (\tan ^{-1}(\tan (c))+d x\right )} \csc \left (\tan ^{-1}(\tan (c))+d x\right ) \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},-\frac{1}{4}\right \},\left \{\frac{3}{4}\right \},\cos ^2\left (\tan ^{-1}(\tan (c))+d x\right )\right )-20 \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left (d x-\tan ^{-1}(\cot (c))\right )} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \text{HypergeometricPFQ}\left (\left \{\frac{1}{4},\frac{1}{2}\right \},\left \{\frac{5}{4}\right \},\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )+\sin (2 (c+d x))-3 \csc (c) \cos (d x)-9 \csc (c) \cos (2 c+d x)+9 \cot (c) \sqrt{\sec ^2(c)} \cos \left (c-\tan ^{-1}(\tan (c))-d x\right )+3 \cot (c) \sqrt{\sec ^2(c)} \cos \left (c+\tan ^{-1}(\tan (c))+d x\right )\right )}{24 d \sqrt{\cos (c+d x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 2.25, size = 172, normalized size = 1.9 \begin{align*} -{\frac{4\,{a}^{3}}{3\,d} \left ( 2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}\cos \left ( 1/2\,dx+c/2 \right ) +5\,\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 ) -3\,\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 ) -4\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) \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 (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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{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}}{\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 (a \cos \left (d x + c\right ) + a\right )}^{3}}{\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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