Optimal. Leaf size=94 \[ -\frac {a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt {a+a \sin (e+f x)}}+\frac {a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.21, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3050, 2817}
\begin {gather*} \frac {a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt {a \sin (e+f x)+a}}-\frac {a (A+B) \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
Rule 3050
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{7/2} \, dx &=(A+B) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2} \, dx-\frac {B \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2} \, dx}{c}\\ &=-\frac {a (A+B) \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 f \sqrt {a+a \sin (e+f x)}}+\frac {a B \cos (e+f x) (c-c \sin (e+f x))^{9/2}}{5 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.71, size = 118, normalized size = 1.26 \begin {gather*} -\frac {c^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (4 (-60 A+23 B) \sin (e+f x)+4 \cos (2 (e+f x)) (-35 A+25 B+4 (5 A-6 B) \sin (e+f x))+\cos (4 (e+f x)) (5 A-15 B+4 B \sin (e+f x)))}{160 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. \(173\) vs.
\(2(82)=164\).
time = 22.54, size = 174, normalized size = 1.85
method | result | size |
default | \(\frac {\left (-4 B \left (\cos ^{4}\left (f x +e \right )\right )+5 A \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-15 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-20 A \left (\cos ^{2}\left (f x +e \right )\right )+28 B \left (\cos ^{2}\left (f x +e \right )\right )-35 A \sin \left (f x +e \right )+25 B \sin \left (f x +e \right )+40 A -24 B \right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {7}{2}} \sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{20 f \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-3 \left (\cos ^{2}\left (f x +e \right )\right )-4 \sin \left (f x +e \right )+4\right ) \cos \left (f x +e \right )}\) | \(174\) |
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 = 0.38, size = 148, normalized size = 1.57 \begin {gather*} -\frac {{\left (5 \, {\left (A - 3 \, B\right )} c^{3} \cos \left (f x + e\right )^{4} - 40 \, {\left (A - B\right )} c^{3} \cos \left (f x + e\right )^{2} + 5 \, {\left (7 \, A - 5 \, B\right )} c^{3} + 4 \, {\left (B c^{3} \cos \left (f x + e\right )^{4} + {\left (5 \, A - 7 \, B\right )} c^{3} \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, A - 3 \, B\right )} 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}}{20 \, 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 = 0.51, size = 159, normalized size = 1.69 \begin {gather*} -\frac {4 \, {\left (8 \, B c^{3} \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 )^{10} - 5 \, A c^{3} \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} - 5 \, B c^{3} \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}\right )} \sqrt {a} \sqrt {c}}{5 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 16.17, size = 173, normalized size = 1.84 \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 (100\,B\,\cos \left (e+f\,x\right )-140\,A\,\cos \left (e+f\,x\right )-135\,A\,\cos \left (3\,e+3\,f\,x\right )+5\,A\,\cos \left (5\,e+5\,f\,x\right )+85\,B\,\cos \left (3\,e+3\,f\,x\right )-15\,B\,\cos \left (5\,e+5\,f\,x\right )-240\,A\,\sin \left (2\,e+2\,f\,x\right )+40\,A\,\sin \left (4\,e+4\,f\,x\right )+90\,B\,\sin \left (2\,e+2\,f\,x\right )-48\,B\,\sin \left (4\,e+4\,f\,x\right )+2\,B\,\sin \left (6\,e+6\,f\,x\right )\right )}{160\,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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