Optimal. Leaf size=45 \[ -\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.21, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2920, 2817}
\begin {gather*} -\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2920
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx &=\frac {\int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx}{a c}\\ &=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(120\) vs. \(2(45)=90\).
time = 0.35, size = 120, normalized size = 2.67 \begin {gather*} -\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (-1+\sin (e+f x)) \sqrt {c-c \sin (e+f x)} (6 \cos (2 (e+f x))+15 \sin (e+f x)-\sin (3 (e+f x)))}{12 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \sqrt {a (1+\sin (e+f x))}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(140\) vs.
\(2(39)=78\).
time = 0.15, size = 141, normalized size = 3.13
method | result | size |
default | \(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )-\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+2 \left (\cos ^{2}\left (f x +e \right )\right )+3 \cos \left (f x +e \right ) \sin \left (f x +e \right )-4 \cos \left (f x +e \right )+\sin \left (f x +e \right )+1\right )}{3 f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\cos ^{2}\left (f x +e \right )-\cos \left (f x +e \right ) \sin \left (f x +e \right )+\cos \left (f x +e \right )+2 \sin \left (f x +e \right )-2\right )}\) | \(141\) |
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 = 0.36, size = 84, normalized size = 1.87 \begin {gather*} \frac {{\left (3 \, c \cos \left (f x + e\right )^{2} - {\left (c \cos \left (f x + e\right )^{2} - 4 \, c\right )} \sin \left (f x + e\right ) - 3 \, c\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\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 \frac {\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}} \cos ^{2}{\left (e + f x \right )}}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 56, normalized size = 1.24 \begin {gather*} \frac {8 \, c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}{3 \, \sqrt {a} f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.96, size = 83, normalized size = 1.84 \begin {gather*} -\frac {c\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (6\,\cos \left (e+f\,x\right )+6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{24\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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