Optimal. Leaf size=61 \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)}}{d e} \]
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Rubi [A] time = 0.0463531, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2669, 2642, 2641} \[ \frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}-\frac{2 a \sqrt{e \cos (c+d x)}}{d e} \]
Antiderivative was successfully verified.
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Rule 2669
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{a+a \sin (c+d x)}{\sqrt{e \cos (c+d x)}} \, dx &=-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}+a \int \frac{1}{\sqrt{e \cos (c+d x)}} \, dx\\ &=-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}+\frac{\left (a \sqrt{\cos (c+d x)}\right ) \int \frac{1}{\sqrt{\cos (c+d x)}} \, dx}{\sqrt{e \cos (c+d x)}}\\ &=-\frac{2 a \sqrt{e \cos (c+d x)}}{d e}+\frac{2 a \sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )}{d \sqrt{e \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 20.9496, size = 48, normalized size = 0.79 \[ -\frac{2 a \left (\cos (c+d x)-\sqrt{\cos (c+d x)} F\left (\left .\frac{1}{2} (c+d x)\right |2\right )\right )}{d \sqrt{e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.22, size = 103, normalized size = 1.7 \begin{align*} -2\,{\frac{a \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 ) -2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+\sin \left ( 1/2\,dx+c/2 \right ) \right ) }{\sin \left ( 1/2\,dx+c/2 \right ) \sqrt{-2\, \left ( \sin \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}e+e}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{a \sin \left (d x + c\right ) + a}{\sqrt{e \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 (\frac{\sqrt{e \cos \left (d x + c\right )}{\left (a \sin \left (d x + c\right ) + a\right )}}{e \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a \sin \left (d x + c\right ) + a}{\sqrt{e \cos \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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