Optimal. Leaf size=34 \[ \frac {2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2815, 2752}
\begin {gather*} \frac {2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2752
Rule 2815
Rubi steps
\begin {align*} \int (a+a \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx &=(a c) \int \frac {\cos ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(71\) vs. \(2(34)=68\).
time = 0.09, size = 71, normalized size = 2.09 \begin {gather*} \frac {2 a \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {c-c \sin (e+f x)}}{3 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 = 1.33, size = 47, normalized size = 1.38
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right )^{2} a}{3 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(47\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 86 vs.
\(2 (32) = 64\).
time = 0.35, size = 86, normalized size = 2.53 \begin {gather*} -\frac {2 \, {\left (a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - {\left (a \cos \left (f x + e\right ) + 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, a\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\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*} a \left (\int \sqrt {- c \sin {\left (e + f x \right )} + c} \sin {\left (e + f x \right )}\, dx + \int \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 71 vs.
\(2 (32) = 64\).
time = 0.49, size = 71, normalized size = 2.09 \begin {gather*} -\frac {\sqrt {2} {\left (3 \, a \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + a \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{3 \, 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 = -0.03 \begin {gather*} \int \left (a+a\,\sin \left (e+f\,x\right )\right )\,\sqrt {c-c\,\sin \left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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