Optimal. Leaf size=43 \[ -\frac {a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817}
\begin {gather*} -\frac {a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx &=-\frac {a \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 83, normalized size = 1.93 \begin {gather*} -\frac {c^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (-28 \cos (2 (e+f x))+\cos (4 (e+f x))+8 (-7 \sin (e+f x)+\sin (3 (e+f x))))}{32 f} \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. \(102\) vs.
\(2(37)=74\).
time = 16.28, size = 103, normalized size = 2.40
method | result | size |
default | \(\frac {\left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \left (\cos ^{6}\left (f x +e \right )+\sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right )+4 \sin \left (f x +e \right )+4\right )}{4 f \cos \left (f x +e \right )^{7}}\) | \(103\) |
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. 102 vs.
\(2 (40) = 80\).
time = 0.32, size = 102, normalized size = 2.37 \begin {gather*} -\frac {{\left (c^{3} \cos \left (f x + e\right )^{4} - 8 \, c^{3} \cos \left (f x + e\right )^{2} + 7 \, c^{3} + 4 \, {\left (c^{3} \cos \left (f x + e\right )^{2} - 2 \, c^{3}\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}}{4 \, 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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 54, normalized size = 1.26 \begin {gather*} \frac {4 \, \sqrt {a} c^{\frac {7}{2}} \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 ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.25, size = 99, normalized size = 2.30 \begin {gather*} \frac {c^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (28\,\cos \left (e+f\,x\right )+27\,\cos \left (3\,e+3\,f\,x\right )-\cos \left (5\,e+5\,f\,x\right )+48\,\sin \left (2\,e+2\,f\,x\right )-8\,\sin \left (4\,e+4\,f\,x\right )\right )}{32\,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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