Optimal. Leaf size=89 \[ \frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 f} \]
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Rubi [A]
time = 0.12, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2819, 2817}
\begin {gather*} \frac {c^2 \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{6 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2819
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx &=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 f}+\frac {1}{2} c \int (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {c^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{6 f \sqrt {c-c \sin (e+f x)}}+\frac {c \cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{4 f}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 133, normalized size = 1.49 \begin {gather*} -\frac {c (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{5/2} \sqrt {c-c \sin (e+f x)} (-12 \cos (2 (e+f x))-3 \cos (4 (e+f x))+8 (9 \sin (e+f x)+\sin (3 (e+f x))))}{96 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 17.41, size = 91, normalized size = 1.02
method | result | size |
default | \(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \left (3 \left (\cos ^{4}\left (f x +e \right )\right )-\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )+4 \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )+5\right )}{12 f \cos \left (f x +e \right )^{5}}\) | \(91\) |
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 = 93, normalized size = 1.04 \begin {gather*} -\frac {{\left (3 \, a^{2} c \cos \left (f x + e\right )^{4} - 3 \, a^{2} c - 4 \, {\left (a^{2} c \cos \left (f x + e\right )^{2} + 2 \, a^{2} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{12 \, 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 110, normalized size = 1.24 \begin {gather*} \frac {4 \, {\left (3 \, a^{2} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 4 \, a^{2} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.27, size = 100, normalized size = 1.12 \begin {gather*} -\frac {a^2\,c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (12\,\cos \left (e+f\,x\right )+15\,\cos \left (3\,e+3\,f\,x\right )+3\,\cos \left (5\,e+5\,f\,x\right )-80\,\sin \left (2\,e+2\,f\,x\right )-8\,\sin \left (4\,e+4\,f\,x\right )\right )}{96\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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