Optimal. Leaf size=109 \[ -\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d} \]
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Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2895}
\begin {gather*} -\frac {2 \sqrt {a+b} \tan (c+d x) \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (\csc (c+d x)+1)}{a-b}} F\left (\text {ArcSin}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2895
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d x)}} \, dx &=-\frac {2 \sqrt {a+b} \sqrt {\frac {a (1-\csc (c+d x))}{a+b}} \sqrt {\frac {a (1+\csc (c+d x))}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \sin (c+d x)}}{\sqrt {a+b} \sqrt {\sin (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \tan (c+d x)}{a d}\\ \end {align*}
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Mathematica [A]
time = 4.21, size = 172, normalized size = 1.58 \begin {gather*} \frac {8 a \sqrt {-\frac {(a+b) \cot ^2\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{a-b}} F\left (\sin ^{-1}\left (\sqrt {-\frac {a+b \sin (c+d x)}{a (-1+\sin (c+d x))}}\right )|\frac {2 a}{a-b}\right ) \sec (c+d x) \sqrt {-\frac {(a+b) \sin (c+d x) (a+b \sin (c+d x))}{a^2 (-1+\sin (c+d x))^2}} \sin ^4\left (\frac {1}{4} (2 c-\pi +2 d x)\right )}{(a+b) d \sqrt {\sin (c+d x)} \sqrt {a+b \sin (c+d 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. \(309\) vs.
\(2(101)=202\).
time = 13.29, size = 310, normalized size = 2.84
method | result | size |
default | \(-\frac {\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )-a \cos \left (d x +c \right )+b \sin \left (d x +c \right )+a}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}\, \sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )+a \cos \left (d x +c \right )-b \sin \left (d x +c \right )-a}{\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )}}\, \sqrt {\frac {a \left (-1+\cos \left (d x +c \right )\right )}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {\sqrt {-a^{2}+b^{2}}\, \sin \left (d x +c \right )-a \cos \left (d x +c \right )+b \sin \left (d x +c \right )+a}{\left (b +\sqrt {-a^{2}+b^{2}}\right ) \sin \left (d x +c \right )}}, \frac {\sqrt {2}\, \sqrt {\frac {b +\sqrt {-a^{2}+b^{2}}}{\sqrt {-a^{2}+b^{2}}}}}{2}\right ) \left (\sin ^{\frac {3}{2}}\left (d x +c \right )\right ) \sqrt {2}\, \left (b +\sqrt {-a^{2}+b^{2}}\right )}{d \sqrt {a +b \sin \left (d x +c \right )}\, \left (-1+\cos \left (d x +c \right )\right ) a}\) | \(310\) |
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 {1}{\sqrt {a + b \sin {\left (c + d x \right )}} \sqrt {\sin {\left (c + d 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 {1}{\sqrt {\sin \left (c+d\,x\right )}\,\sqrt {a+b\,\sin \left (c+d\,x\right )}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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