Optimal. Leaf size=62 \[ \frac {2 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \]
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Rubi [A]
time = 0.03, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2734, 2732}
\begin {gather*} \frac {2 \sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rubi steps
\begin {align*} \int \sqrt {a+b \sin (e+f x)} \, dx &=\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}} \, dx}{\sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\\ &=\frac {2 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 61, normalized size = 0.98 \begin {gather*} -\frac {2 E\left (\frac {1}{4} (-2 e+\pi -2 f x)|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (e+f x)}}{f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}} \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. \(238\) vs.
\(2(91)=182\).
time = 3.32, size = 239, normalized size = 3.85
method | result | size |
default | \(-\frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \left (\EllipticE \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) a +\EllipticE \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b -a \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )-\EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right ) b \right )}{b \cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f}\) | \(239\) |
risch | \(\text {Expression too large to display}\) | \(1514\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.11, size = 371, normalized size = 5.98 \begin {gather*} \frac {\sqrt {2} a \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right ) + \sqrt {2} a \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right ) - 3 i \, \sqrt {2} \sqrt {i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) - 3 i \, b \sin \left (f x + e\right ) - 2 i \, a}{3 \, b}\right )\right ) + 3 i \, \sqrt {2} \sqrt {-i \, b} b {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (f x + e\right ) + 3 i \, b \sin \left (f x + e\right ) + 2 i \, a}{3 \, b}\right )\right )}{3 \, b f} \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 \sqrt {a + b \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 [B]
time = 6.87, size = 55, normalized size = 0.89 \begin {gather*} \frac {2\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |\frac {2\,b}{a+b}\right )\,\sqrt {a+b\,\sin \left (e+f\,x\right )}}{f\,\sqrt {\frac {a+b\,\sin \left (e+f\,x\right )}{a+b}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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