Optimal. Leaf size=69 \[ \frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{c f}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{c f} \]
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Rubi [A] time = 0.31, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {2934, 2773, 206, 2736, 2673} \[ \frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{c f}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{c f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2673
Rule 2736
Rule 2773
Rule 2934
Rubi steps
\begin {align*} \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx &=\frac {\int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{c}+\int \frac {\sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx\\ &=\frac {\int \sec ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx}{a c}-\frac {(2 a) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f}\\ &=-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{c f}\\ \end {align*}
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Mathematica [B] time = 0.35, size = 157, normalized size = 2.28 \[ \frac {\sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right ) \left (\log \left (\sin \left (\frac {1}{2} (e+f x)\right )-\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-\log \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )+1\right )\right )+\sin \left (\frac {1}{2} (e+f x)\right ) \left (\log \left (-\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )+1\right )-\log \left (\sin \left (\frac {1}{2} (e+f x)\right )-\cos \left (\frac {1}{2} (e+f x)\right )+1\right )\right )+2\right )}{c f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 202, normalized size = 2.93 \[ \frac {\sqrt {a} \cos \left (f x + e\right ) \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, c f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.37, size = 79, normalized size = 1.14 \[ -\frac {2 \left (1+\sin \left (f x +e \right )\right ) \left (\arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right ) a \sqrt {a -a \sin \left (f x +e \right )}-a^{\frac {3}{2}}\right )}{\sqrt {a}\, c \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ -\int \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) - c\right )} \sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (c-c\,\sin \left (e+f\,x\right )\right )} \,d x \]
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 \[ - \frac {\int \frac {\sqrt {a \sin {\left (e + f x \right )} + a}}{\sin ^{2}{\left (e + f x \right )} - \sin {\left (e + f x \right )}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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