Optimal. Leaf size=62 \[ -\frac {2 a (3 c+d) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 d \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f} \]
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Rubi [A] time = 0.06, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2751, 2646} \[ -\frac {2 a (3 c+d) \cos (e+f x)}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {2 d \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{3 f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x)) \, dx &=-\frac {2 d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}+\frac {1}{3} (3 c+d) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {2 a (3 c+d) \cos (e+f x)}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {2 d \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 82, normalized size = 1.32 \[ -\frac {2 \sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (3 c+d \sin (e+f x)+2 d)}{3 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 85, normalized size = 1.37 \[ -\frac {2 \, {\left (d \cos \left (f x + e\right )^{2} + {\left (3 \, c + 2 \, d\right )} \cos \left (f x + e\right ) + {\left (d \cos \left (f x + e\right ) - 3 \, c - d\right )} \sin \left (f x + e\right ) + 3 \, c + d\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{3 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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 = 0.64, size = 58, normalized size = 0.94 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (d \sin \left (f x +e \right )+3 c +2 d \right )}{3 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \]
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 \[ \int \sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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