Optimal. Leaf size=61 \[ -\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \]
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Rubi [A] time = 0.19, antiderivative size = 61, 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 = {2943, 206} \[ -\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2943
Rubi steps
\begin {align*} \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx &=-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{1-a c x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{f}\\ &=-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{\sqrt {c} f}\\ \end {align*}
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Mathematica [C] time = 1.97, size = 367, normalized size = 6.02 \[ -\frac {\sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-i \sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\log \left (-\frac {(1+i) e^{\frac {i e}{2}} f \left (-2 i \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt {2} c \left (-1+e^{i (e+f x)}\right )+i \sqrt {2} d \left (1+e^{i (e+f x)}\right )\right )}{\sqrt {c} \left (1+e^{i (e+f x)}\right )}\right )+\log \left (\frac {(1+i) e^{\frac {i e}{2}} f \left (2 \sqrt {c} \sqrt {2 c e^{i (e+f x)}-i d \left (-1+e^{2 i (e+f x)}\right )}+\sqrt {2} c \left (1+e^{i (e+f x)}\right )-i \sqrt {2} d \left (-1+e^{i (e+f x)}\right )\right )}{\sqrt {c} \left (-1+e^{i (e+f x)}\right )}\right )\right ) \sqrt {(\cos (e+f x)+i \sin (e+f x)) (c+d \sin (e+f x))}}{\sqrt {c} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 1044, normalized size = 17.11 \[ \text {result too large to display} \]
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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.54, size = 231, normalized size = 3.79 \[ \frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {c +d \sin \left (f x +e \right )}\, \sqrt {2}\, \left (\ln \left (\frac {\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-c \cos \left (f x +e \right )+d \sin \left (f x +e \right )+c}{\sin \left (f x +e \right ) \sqrt {c}}\right )-\ln \left (-\frac {2 \left (\sqrt {c}\, \sqrt {2}\, \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-d \cos \left (f x +e \right )+c \sin \left (f x +e \right )+d \right )}{-1+\cos \left (f x +e \right )}\right )\right ) \left (-1+\cos \left (f x +e \right )\right )}{f \sin \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )\right ) \sqrt {\frac {c +d \sin \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {c}} \]
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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{\sqrt {d \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\sqrt {c + d \sin {\left (e + f x \right )}} \sin {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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