Optimal. Leaf size=161 \[ -\frac {4 a (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {8 (5 c-d) d (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{35 f}-\frac {12 d^2 (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.18, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2849, 2840,
2830, 2725} \begin {gather*} -\frac {4 a (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{35 f \sqrt {a \sin (e+f x)+a}}-\frac {12 d^2 (c+d) \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{35 a f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a \sin (e+f x)+a}}-\frac {8 d (5 c-d) (c+d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{35 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2725
Rule 2830
Rule 2840
Rule 2849
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^3 \, dx &=-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {1}{7} (6 (c+d)) \int \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2 \, dx\\ &=-\frac {12 d^2 (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {(12 (c+d)) \int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a \left (5 c^2+3 d^2\right )+a (5 c-d) d \sin (e+f x)\right ) \, dx}{35 a}\\ &=-\frac {8 (5 c-d) d (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{35 f}-\frac {12 d^2 (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}+\frac {1}{35} \left (2 (c+d) \left (15 c^2+10 c d+7 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \, dx\\ &=-\frac {4 a (c+d) \left (15 c^2+10 c d+7 d^2\right ) \cos (e+f x)}{35 f \sqrt {a+a \sin (e+f x)}}-\frac {8 (5 c-d) d (c+d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{35 f}-\frac {12 d^2 (c+d) \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{35 a f}-\frac {2 a \cos (e+f x) (c+d \sin (e+f x))^3}{7 f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 146, normalized size = 0.91 \begin {gather*} -\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {a (1+\sin (e+f x))} \left (140 c^3+280 c^2 d+266 c d^2+76 d^3-6 d^2 (7 c+2 d) \cos (2 (e+f x))+d \left (140 c^2+112 c d+47 d^2\right ) \sin (e+f x)-5 d^3 \sin (3 (e+f x))\right )}{70 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.20, size = 141, normalized size = 0.88
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) a \left (\sin \left (f x +e \right )-1\right ) \left (5 d^{3} \left (\sin ^{3}\left (f x +e \right )\right )+21 c \,d^{2} \left (\sin ^{2}\left (f x +e \right )\right )+6 d^{3} \left (\sin ^{2}\left (f x +e \right )\right )+35 c^{2} d \sin \left (f x +e \right )+28 c \,d^{2} \sin \left (f x +e \right )+8 d^{3} \sin \left (f x +e \right )+35 c^{3}+70 c^{2} d +56 c \,d^{2}+16 d^{3}\right )}{35 \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(141\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 251, normalized size = 1.56 \begin {gather*} \frac {2 \, {\left (5 \, d^{3} \cos \left (f x + e\right )^{4} + 3 \, {\left (7 \, c d^{2} + 2 \, d^{3}\right )} \cos \left (f x + e\right )^{3} - 35 \, c^{3} - 35 \, c^{2} d - 49 \, c d^{2} - 9 \, d^{3} - {\left (35 \, c^{2} d + 7 \, c d^{2} + 12 \, d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, c^{3} + 70 \, c^{2} d + 77 \, c d^{2} + 22 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (5 \, d^{3} \cos \left (f x + e\right )^{3} + 35 \, c^{3} + 35 \, c^{2} d + 49 \, c d^{2} + 9 \, d^{3} - {\left (21 \, c d^{2} + d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (35 \, c^{2} d + 28 \, c d^{2} + 13 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{35 \, {\left (f \cos \left (f x + e\right ) + f \sin \left (f x + e\right ) + f\right )}} \end {gather*}
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 \begin {gather*} \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \left (c + d \sin {\left (e + f x \right )}\right )^{3}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.60, size = 266, normalized size = 1.65 \begin {gather*} \frac {\sqrt {2} {\left (5 \, d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) + 35 \, {\left (8 \, c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 12 \, c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 12 \, c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 3 \, d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 35 \, {\left (4 \, c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 2 \, c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) + 7 \, {\left (6 \, c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sin \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right )\right )} \sqrt {a}}{140 \, f} \end {gather*}
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 \begin {gather*} \int \sqrt {a+a\,\sin \left (e+f\,x\right )}\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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