Optimal. Leaf size=37 \[ -\frac {2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f} \]
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Rubi [A]
time = 0.04, 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} \cos (e+f x)}{\sqrt {a \sin (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 \sin (e+f x)}}{\sqrt {\sin (e+f x)}} \, dx &=-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a}}} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac {2 \sqrt {a} \sin ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{f}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.36, size = 164, normalized size = 4.43 \begin {gather*} \frac {(1+i) e^{\frac {1}{2} i (e+f x)} \sqrt {-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \left (\tan ^{-1}\left (\sqrt {-1+e^{2 i (e+f x)}}\right )-i \tanh ^{-1}\left (\frac {e^{i (e+f x)}}{\sqrt {-1+e^{2 i (e+f x)}}}\right )\right ) \sqrt {a (1+\sin (e+f x))}}{\sqrt {2} \sqrt {-1+e^{2 i (e+f x)}} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \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. \(319\) vs.
\(2(31)=62\).
time = 16.06, size = 320, normalized size = 8.65
method | result | size |
default | \(-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\sqrt {\sin }\left (f x +e \right )\right ) \left (\ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )+1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )-1}\right )+4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}+1\right )+4 \arctan \left (\sqrt {2}\, \sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}-1\right )+\ln \left (-\frac {\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-\sin \left (f x +e \right )-1}{\sqrt {-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {2}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+\sin \left (f x +e \right )+1}\right )\right ) \sqrt {2}}{2 f \left (-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right )}\) | \(320\) |
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. 224 vs.
\(2 (33) = 66\).
time = 0.55, size = 224, normalized size = 6.05 \begin {gather*} -\frac {2 \, \sqrt {2} \sqrt {a} \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac {3}{2}} - 3 \, \sqrt {2} {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right ) + \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right )\right )} \sqrt {a} + 6 \, \sqrt {2} \sqrt {a} \sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac {2 \, {\left (\frac {3 \, \sqrt {2} \sqrt {a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {\sqrt {2} \sqrt {a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{\sqrt {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs.
\(2 (33) = 66\).
time = 0.42, size = 360, normalized size = 9.73 \begin {gather*} \left [\frac {\sqrt {-a} \log \left (\frac {128 \, a \cos \left (f x + e\right )^{5} - 128 \, a \cos \left (f x + e\right )^{4} - 416 \, a \cos \left (f x + e\right )^{3} + 128 \, a \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} - 66 \, \cos \left (f x + e\right )^{2} + {\left (16 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 51\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 51\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-a} \sqrt {\sin \left (f x + e\right )} + 289 \, a \cos \left (f x + e\right ) + {\left (128 \, a \cos \left (f x + e\right )^{4} + 256 \, a \cos \left (f x + e\right )^{3} - 160 \, a \cos \left (f x + e\right )^{2} - 288 \, a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right )}{4 \, f}, \frac {\sqrt {a} \arctan \left (\frac {{\left (8 \, \cos \left (f x + e\right )^{2} + 8 \, \sin \left (f x + e\right ) - 9\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} \sqrt {\sin \left (f x + e\right )}}{4 \, {\left (2 \, a \cos \left (f x + e\right )^{3} + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a \cos \left (f x + e\right )\right )}}\right )}{2 \, 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 (\sin {\left (e + f x \right )} + 1\right )}}{\sqrt {\sin {\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\,\sin \left (e+f\,x\right )}}{\sqrt {\sin \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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