Optimal. Leaf size=37 \[ \frac {2 \sqrt {a} \text {ArcSin}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+a \cos (e+f x)}}\right )}{f} \]
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Rubi [A]
time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2853, 222}
\begin {gather*} \frac {2 \sqrt {a} \text {ArcSin}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a \cos (e+f x)+a}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 2853
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \cos (e+f x)}}{\sqrt {\cos (e+f x)}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,-\frac {a \sin (e+f x)}{\sqrt {a+a \cos (e+f x)}}\right )}{f}\\ &=\frac {2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a} \sin (e+f x)}{\sqrt {a+a \cos (e+f x)}}\right )}{f}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 50, normalized size = 1.35 \begin {gather*} \frac {\sqrt {2} \text {ArcSin}\left (\sqrt {2} \sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\cos (e+f x))} \sec \left (\frac {1}{2} (e+f x)\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(79\) vs.
\(2(31)=62\).
time = 1.12, size = 80, normalized size = 2.16
method | result | size |
default | \(\frac {2 \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {a \left (\cos \left (f x +e \right )+1\right )}\, \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}{\cos \left (f x +e \right )}\right )}{f \sqrt {\cos \left (f x +e \right )}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 158 vs.
\(2 (33) = 66\).
time = 0.56, size = 158, normalized size = 4.27 \begin {gather*} \frac {\sqrt {a} \arctan \left ({\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{4}} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + \sin \left (f x + e\right ), {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}^{\frac {1}{4}} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right )\right ) + \cos \left (f x + e\right )\right )}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.44, size = 128, normalized size = 3.46 \begin {gather*} \left [\frac {\sqrt {-a} \log \left (\frac {2 \, a \cos \left (f x + e\right )^{2} - 2 \, \sqrt {a \cos \left (f x + e\right ) + a} \sqrt {-a} \sqrt {\cos \left (f x + e\right )} \sin \left (f x + e\right ) + a \cos \left (f x + e\right ) - a}{\cos \left (f x + e\right ) + 1}\right )}{f}, -\frac {2 \, \sqrt {a} \arctan \left (\frac {\sqrt {a \cos \left (f x + e\right ) + a} \sqrt {\cos \left (f x + e\right )}}{\sqrt {a} \sin \left (f x + e\right )}\right )}{f}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a \left (\cos {\left (e + f x \right )} + 1\right )}}{\sqrt {\cos {\left (e + f x \right )}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {a+a\,\cos \left (e+f\,x\right )}}{\sqrt {\cos \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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