Optimal. Leaf size=36 \[ \frac {2 a^2 c^3 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752}
\begin {gather*} \frac {2 a^2 c^3 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}} \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))^2 \sqrt {c-c \sin (e+f x)} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {2 a^2 c^3 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(73\) vs. \(2(36)=72\).
time = 0.16, size = 73, normalized size = 2.03 \begin {gather*} \frac {2 a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \sqrt {c-c \sin (e+f x)}}{5 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 = 1.32, size = 49, normalized size = 1.36
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c \left (1+\sin \left (f x +e \right )\right )^{3} a^{2}}{5 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(49\) |
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. 123 vs.
\(2 (34) = 68\).
time = 0.34, size = 123, normalized size = 3.42 \begin {gather*} -\frac {2 \, {\left (a^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) - 4 \, a^{2} + {\left (a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \cos \left (f x + e\right ) - 4 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{5 \, {\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^{2} \left (\int 2 \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} \sin ^{2}{\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. 107 vs.
\(2 (34) = 68\).
time = 0.51, size = 107, normalized size = 2.97 \begin {gather*} -\frac {\sqrt {2} {\left (10 \, a^{2} \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 ) + 5 \, a^{2} \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 ) + a^{2} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{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}}{10 \, 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 )}^2\,\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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