Optimal. Leaf size=158 \[ \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f} \]
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Rubi [A]
time = 0.17, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {2967, 2895}
\begin {gather*} \frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \tan (e+f x) \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (\csc (e+f x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right )}{\sqrt {d} f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2895
Rule 2967
Rubi steps
\begin {align*} \int \frac {\sec ^2(e+f x) \sqrt {a+b \sin (e+f x)}}{\sqrt {d \sin (e+f x)}} \, dx &=\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}+\frac {1}{2} a \int \frac {1}{\sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}} \, dx\\ &=\frac {\sec (e+f x) \sqrt {d \sin (e+f x)} \sqrt {a+b \sin (e+f x)}}{d f}-\frac {\sqrt {a+b} \sqrt {\frac {a (1-\csc (e+f x))}{a+b}} \sqrt {\frac {a (1+\csc (e+f x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \sin (e+f x)}}{\sqrt {a+b} \sqrt {d \sin (e+f x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (e+f x)}{\sqrt {d} f}\\ \end {align*}
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Mathematica [A]
time = 23.75, size = 198, normalized size = 1.25 \begin {gather*} \frac {4 a^2 \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{a-b}} F\left (\sin ^{-1}\left (\sqrt {-\frac {a+b \sin (e+f x)}{a (-1+\sin (e+f x))}}\right )|\frac {2 a}{a-b}\right ) \sec (e+f x) \sqrt {-\frac {(a+b) \sin (e+f x) (a+b \sin (e+f x))}{a^2 (-1+\sin (e+f x))^2}} \sin ^4\left (\frac {1}{4} (2 e-\pi +2 f x)\right )+(a+b) (a+b \sin (e+f x)) \tan (e+f x)}{(a+b) f \sqrt {d \sin (e+f x)} \sqrt {a+b \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. \(649\) vs.
\(2(142)=284\).
time = 3.54, size = 650, normalized size = 4.11
method | result | size |
default | \(-\frac {\left (\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {\frac {\cos \left (f x +e \right ) a +\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )}}\, \sqrt {\frac {a \left (-1+\cos \left (f x +e \right )\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \sqrt {-a^{2}+b^{2}}+\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}\, \sqrt {\frac {\cos \left (f x +e \right ) a +\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )-b \sin \left (f x +e \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )}}\, \sqrt {\frac {a \left (-1+\cos \left (f x +e \right )\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (f x +e \right )}}\, \EllipticF \left (\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (f x +e \right )+b \sin \left (f x +e \right )-\cos \left (f x +e \right ) a +a}{\sin \left (f x +e \right ) \left (b +\sqrt {-a^{2}+b^{2}}\right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) b -\sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {2}\, b +\sin \left (f x +e \right ) \sqrt {2}\, b -\cos \left (f x +e \right ) \sqrt {2}\, a +\sqrt {2}\, a \right ) \sin \left (f x +e \right ) \sqrt {2}}{2 f \left (-1+\cos \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {d \sin \left (f x +e \right )}\, \sqrt {a +b \sin \left (f x +e \right )}}\) | \(650\) |
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 {a + b \sin {\left (e + f x \right )}} \sec ^{2}{\left (e + f x \right )}}{\sqrt {d \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 = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{{\cos \left (e+f\,x\right )}^2\,\sqrt {d\,\sin \left (e+f\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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