Optimal. Leaf size=51 \[ \frac {E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2665, 2652,
2719} \begin {gather*} \frac {\sqrt {\sin (e+f x)} E\left (\left .e+f x-\frac {\pi }{4}\right |2\right )}{f \sqrt {\sin (2 e+2 f x)} \sqrt {b \sec (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2652
Rule 2665
Rule 2719
Rubi steps
\begin {align*} \int \frac {\sqrt {\sin (e+f x)}}{\sqrt {b \sec (e+f x)}} \, dx &=\frac {\int \sqrt {b \cos (e+f x)} \sqrt {\sin (e+f x)} \, dx}{\sqrt {b \cos (e+f x)} \sqrt {b \sec (e+f x)}}\\ &=\frac {\sqrt {\sin (e+f x)} \int \sqrt {\sin (2 e+2 f x)} \, dx}{\sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}\\ &=\frac {E\left (\left .e-\frac {\pi }{4}+f x\right |2\right ) \sqrt {\sin (e+f x)}}{f \sqrt {b \sec (e+f x)} \sqrt {\sin (2 e+2 f x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.81, size = 60, normalized size = 1.18 \begin {gather*} -\frac {b \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f (b \sec (e+f x))^{3/2} \sqrt {\sin (e+f x)}} \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. \(496\) vs.
\(2(71)=142\).
time = 0.29, size = 497, normalized size = 9.75
method | result | size |
default | \(-\frac {\left (2 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )-\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right ) \cos \left (f x +e \right )+2 \sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticE \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {\sin \left (f x +e \right )-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \sqrt {\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {1-\cos \left (f x +e \right )+\sin \left (f x +e \right )}{\sin \left (f x +e \right )}}, \frac {\sqrt {2}}{2}\right )+\left (\cos ^{2}\left (f x +e \right )\right ) \sqrt {2}-\cos \left (f x +e \right ) \sqrt {2}\right ) \sqrt {2}}{2 f \cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, \sqrt {\frac {b}{\cos \left (f x +e \right )}}}\) | \(497\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sin {\left (e + f x \right )}}}{\sqrt {b \sec {\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 = -0.02 \begin {gather*} \int \frac {\sqrt {\sin \left (e+f\,x\right )}}{\sqrt {\frac {b}{\cos \left (e+f\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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