Optimal. Leaf size=44 \[ \frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
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Rubi [A] time = 0.0804822, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2757, 2636, 2639, 2641} \[ \frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2757
Rule 2636
Rule 2639
Rule 2641
Rubi steps
\begin{align*} \int \frac{(a+a \cos (c+d x))^2}{\cos ^{\frac{3}{2}}(c+d x)} \, dx &=\int \left (\frac{a^2}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2}{\sqrt{\cos (c+d x)}}+a^2 \sqrt{\cos (c+d x)}\right ) \, dx\\ &=a^2 \int \frac{1}{\cos ^{\frac{3}{2}}(c+d x)} \, dx+a^2 \int \sqrt{\cos (c+d x)} \, dx+\left (2 a^2\right ) \int \frac{1}{\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 )}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}-a^2 \int \sqrt{\cos (c+d x)} \, dx\\ &=\frac{4 a^2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d}+\frac{2 a^2 \sin (c+d x)}{d \sqrt{\cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.16049, size = 39, normalized size = 0.89 \[ \frac{2 a^2 \left (2 F\left (\left .\frac{1}{2} (c+d x)\right |2\right )+\frac{\sin (c+d x)}{\sqrt{\cos (c+d x)}}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 2.259, size = 104, normalized size = 2.4 \begin{align*} -4\,{\frac{{a}^{2} \left ( \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 ) - \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}\cos \left ( 1/2\,dx+c/2 \right ) \right ) }{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}-1}d}} \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{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^{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{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 )}^{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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