Optimal. Leaf size=118 \[ -\frac {2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]
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Rubi [A] time = 0.25, antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2968, 3023, 2751, 2646} \[ -\frac {2 (5 A d+5 B c-2 B d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{15 f}-\frac {2 a (15 A c+5 A d+5 B c+7 B d) \cos (e+f x)}{15 f \sqrt {a \sin (e+f x)+a}}-\frac {2 B d \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{5 a f} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2751
Rule 2968
Rule 3023
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\int \sqrt {a+a \sin (e+f x)} \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx\\ &=-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac {2 \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (5 A c+3 B d)+\frac {1}{2} a (5 B c+5 A d-2 B d) \sin (e+f x)\right ) \, dx}{5 a}\\ &=-\frac {2 (5 B c+5 A d-2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}+\frac {1}{15} (15 A c+5 B c+5 A d+7 B d) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {2 a (15 A c+5 B c+5 A d+7 B d) \cos (e+f x)}{15 f \sqrt {a+a \sin (e+f x)}}-\frac {2 (5 B c+5 A d-2 B d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{15 f}-\frac {2 B d \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{5 a f}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 117, normalized size = 0.99 \[ -\frac {\sqrt {a (\sin (e+f x)+1)} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (2 (5 A d+5 B c+4 B d) \sin (e+f x)+30 A c+20 A d+20 B c-3 B d \cos (2 (e+f x))+19 B d)}{15 f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 175, normalized size = 1.48 \[ \frac {2 \, {\left (3 \, B d \cos \left (f x + e\right )^{3} - {\left (5 \, B c + {\left (5 \, A + B\right )} d\right )} \cos \left (f x + e\right )^{2} - 5 \, {\left (3 \, A + B\right )} c - {\left (5 \, A + 7 \, B\right )} d - {\left (5 \, {\left (3 \, A + 2 \, B\right )} c + {\left (10 \, A + 11 \, B\right )} d\right )} \cos \left (f x + e\right ) - {\left (3 \, B d \cos \left (f x + e\right )^{2} - 5 \, {\left (3 \, A + B\right )} c - {\left (5 \, A + 7 \, B\right )} d + {\left (5 \, B c + {\left (5 \, A + 4 \, B\right )} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{15 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \]
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 = 1.36, size = 102, normalized size = 0.86 \[ \frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (3 B \left (\sin ^{2}\left (f x +e \right )\right ) d +5 A \sin \left (f x +e \right ) d +5 B \sin \left (f x +e \right ) c +4 B \sin \left (f x +e \right ) d +15 A c +10 A d +10 B c +8 B d \right )}{15 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\sin \left (e+f\,x\right )\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\left (c+d\,\sin \left (e+f\,x\right )\right ) \,d x \]
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 \[ \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (A + B \sin {\left (e + f x \right )}\right ) \left (c + d \sin {\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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