Optimal. Leaf size=222 \[ -\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}+\frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.50, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {2802, 3060, 2655, 2653, 3002, 2663, 2661, 2807, 2805} \[ -\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}+\frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{f \sqrt {a+b \sin (e+f x)}}-\frac {\sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{a f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2802
Rule 2805
Rule 2807
Rule 3002
Rule 3060
Rubi steps
\begin {align*} \int \frac {\csc ^2(e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx &=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}+\frac {\int \frac {\csc (e+f x) \left (-\frac {b}{2}-\frac {1}{2} b \sin ^2(e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{a}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}-\frac {\int \sqrt {a+b \sin (e+f x)} \, dx}{2 a}-\frac {\int \frac {\csc (e+f x) \left (\frac {b^2}{2}-\frac {1}{2} a b \sin (e+f x)\right )}{\sqrt {a+b \sin (e+f x)}} \, dx}{a b}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}+\frac {1}{2} \int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx-\frac {b \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx}{2 a}-\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}} \, dx}{2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}-\frac {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)}}{a f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {\sqrt {\frac {a+b \sin (e+f x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}}} \, dx}{2 \sqrt {a+b \sin (e+f x)}}-\frac {\left (b \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}{2 a \sqrt {a+b \sin (e+f x)}}\\ &=-\frac {\cot (e+f x) \sqrt {a+b \sin (e+f x)}}{a f}-\frac {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)}}{a f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {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 {b \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}}}{a f \sqrt {a+b \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 10.05, size = 315, normalized size = 1.42 \[ \frac {-4 \cot (e+f x) \sqrt {a+b \sin (e+f x)}+\frac {6 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{4} (-2 e-2 f x+\pi )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (e+f x)}}+\frac {2 i \sec (e+f x) \sqrt {-\frac {b (\sin (e+f x)-1)}{a+b}} \sqrt {-\frac {b (\sin (e+f x)+1)}{a-b}} \left (b \left (b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )-2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )\right )-2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )\right )}{a b \sqrt {-\frac {1}{a+b}}}}{4 a f} \]
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 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.92, size = 412, normalized size = 1.86 \[ \frac {\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}{a \sin \left (f x +e \right )}-\frac {b \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {b \left (1-\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) b}{a -b}}\, \left (\left (-\frac {a}{b}-1\right ) \EllipticE \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )+\EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )\right )}{a \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\frac {b^{2} \left (\frac {a}{b}-1\right ) \sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}\, \sqrt {\frac {b \left (1-\sin \left (f x +e \right )\right )}{a +b}}\, \sqrt {\frac {\left (-\sin \left (f x +e \right )-1\right ) b}{a -b}}\, \EllipticPi \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, -\frac {\left (-\frac {a}{b}+1\right ) b}{a}, \sqrt {\frac {a -b}{a +b}}\right )}{a^{2} \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )}{\cos \left (f x +e \right ) \sqrt {a +b \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {\csc \left (f x + e\right )^{2}}{\sqrt {b \sin \left (f x + e\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\sin \left (e+f\,x\right )}^2\,\sqrt {a+b\,\sin \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\csc ^{2}{\left (e + f x \right )}}{\sqrt {a + b \sin {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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