Optimal. Leaf size=128 \[ \frac {2 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.24, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {2803, 2663, 2661, 2807, 2805} \[ \frac {2 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2661
Rule 2663
Rule 2803
Rule 2805
Rule 2807
Rubi steps
\begin {align*} \int \csc (e+f x) \sqrt {a+b \sin (e+f x)} \, dx &=a \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx+b \int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx\\ &=\frac {\left (a \sqrt {\frac {a+b \sin (e+f x)}{a+b}}\right ) \int \frac {\csc (e+f x)}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}} \, dx}{\sqrt {a+b \sin (e+f x)}}+\frac {\left (b \sqrt {\frac {a+b \sin (e+f x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}} \, dx}{\sqrt {a+b \sin (e+f x)}}\\ &=\frac {2 b F\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}+\frac {2 a \Pi \left (2;\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}{f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 16.18, size = 89, normalized size = 0.70 \[ -\frac {2 \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \left (b F\left (\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 b}{a+b}\right )+a \Pi \left (2;\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 b}{a+b}\right )\right )}{f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \sqrt {b \sin \left (f x + e\right ) + a} \csc \left (f x + e\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.21, size = 169, normalized size = 1.32 \[ \frac {2 \left (a -b \right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) b}{a +b}}\, \sqrt {-\frac {\left (1+\sin \left (f x +e \right )\right ) b}{a -b}}\, \left (\EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )-\EllipticPi \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \frac {a -b}{a}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{\cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f} \]
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 \[ \int \sqrt {b \sin \left (f x + e\right ) + a} \csc \left (f x + e\right )\,{d x} \]
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 \[ \int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )} \,d x \]
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 \[ \int \sqrt {a + b \sin {\left (e + f x \right )}} \csc {\left (e + f x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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