Optimal. Leaf size=238 \[ \frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}-\frac {(a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}} \]
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Rubi [A] time = 0.50, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2935, 2807, 2805, 2768, 2752, 2663, 2661, 2655, 2653} \[ \frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}-\frac {(a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\sqrt {a+b \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}+\frac {2 a \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{c f \sqrt {a+b \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2752
Rule 2768
Rule 2805
Rule 2807
Rule 2935
Rubi steps
\begin {align*} \int \frac {\csc (e+f x) \sqrt {a+b \sin (e+f x)}}{c+c \sin (e+f x)} \, dx &=(-a+b) \int \frac {1}{\sqrt {a+b \sin (e+f x)} (c+c \sin (e+f x))} \, dx+\frac {a \int \frac {\csc (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx}{c}\\ &=\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}-\frac {b \int \frac {-\frac {c}{2}-\frac {1}{2} c \sin (e+f x)}{\sqrt {a+b \sin (e+f x)}} \, dx}{c^2}+\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}{c \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}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}+\frac {\int \sqrt {a+b \sin (e+f x)} \, dx}{2 c}-\frac {(a-b) \int \frac {1}{\sqrt {a+b \sin (e+f x)}} \, dx}{2 c}\\ &=\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}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}+\frac {\sqrt {a+b \sin (e+f x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin (e+f x)}{a+b}} \, dx}{2 c \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {\left ((a-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}{2 c \sqrt {a+b \sin (e+f x)}}\\ &=\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)}}{c f \sqrt {\frac {a+b \sin (e+f x)}{a+b}}}-\frac {(a-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}}}{c 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}}}{c f \sqrt {a+b \sin (e+f x)}}+\frac {\cos (e+f x) \sqrt {a+b \sin (e+f x)}}{f (c+c \sin (e+f x))}\\ \end {align*}
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Mathematica [C] time = 6.64, size = 611, normalized size = 2.57 \[ -\frac {2 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a+b \sin (e+f x)}}{f (c \sin (e+f x)+c)}+\frac {\left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\frac {2 i b \cos (e+f x) \cos (2 (e+f x)) \sqrt {\frac {b-b \sin (e+f x)}{a+b}} \sqrt {-\frac {b \sin (e+f x)+b}{a-b}} \left (2 a (a-b) E\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )+b \left (2 a F\left (i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )-b \Pi \left (\frac {a+b}{a};i \sinh ^{-1}\left (\sqrt {-\frac {1}{a+b}} \sqrt {a+b \sin (e+f x)}\right )|\frac {a+b}{a-b}\right )\right )\right )}{a \sqrt {-\frac {1}{a+b}} \sqrt {1-\sin ^2(e+f x)} \left (-2 a^2+4 a (a+b \sin (e+f x))-2 (a+b \sin (e+f x))^2+b^2\right ) \sqrt {-\frac {a^2-2 a (a+b \sin (e+f x))+(a+b \sin (e+f x))^2-b^2}{b^2}}}+\frac {2 \sin (2 (e+f x)) \cot (e+f x) \sqrt {a+b \sin (e+f x)}}{1-\sin ^2(e+f x)}-\frac {4 b \sqrt {\frac {a+b \sin (e+f x)}{a+b}} F\left (\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (e+f x)}}+\frac {2 (-4 a-b) \sqrt {\frac {a+b \sin (e+f x)}{a+b}} \Pi \left (2;\frac {1}{2} \left (-e-f x+\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{\sqrt {a+b \sin (e+f x)}}\right )}{4 f (c \sin (e+f x)+c)} \]
Warning: Unable to verify antiderivative.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 4.93, size = 593, normalized size = 2.49 \[ \frac {\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (-\frac {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}}\, 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 )}{\sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}+\left (-a +b \right ) \left (-\frac {-b \left (\sin ^{2}\left (f x +e \right )\right )-a \sin \left (f x +e \right )+b \sin \left (f x +e \right )+a}{\left (a -b \right ) \sqrt {\left (-b \sin \left (f x +e \right )-a \right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}}-\frac {2 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}}\, \EllipticF \left (\sqrt {\frac {a +b \sin \left (f x +e \right )}{a -b}}, \sqrt {\frac {a -b}{a +b}}\right )}{\left (2 a -2 b \right ) \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\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 )}{\left (a -b \right ) \sqrt {-\left (-b \sin \left (f x +e \right )-a \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )\right )}{c \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 \[ \int \frac {\sqrt {b \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) + c\right )} \sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {a+b\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (c+c\,\sin \left (e+f\,x\right )\right )} \,d x \]
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 \[ \frac {\int \frac {\sqrt {a + b \sin {\left (e + f x \right )}}}{\sin ^{2}{\left (e + f x \right )} + \sin {\left (e + f x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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