Optimal. Leaf size=134 \[ -\frac {a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {a A \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f} \]
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Rubi [A]
time = 0.23, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {3052, 2819,
2817} \begin {gather*} -\frac {a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a \sin (e+f x)+a}}-\frac {a A \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 2819
Rule 3052
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}+A \int (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac {a A \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}+\frac {1}{3} (2 a A) \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx\\ &=-\frac {a^2 A \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a+a \sin (e+f x)}}-\frac {a A \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}{3 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{4 f}\\ \end {align*}
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Mathematica [A]
time = 0.51, size = 96, normalized size = 0.72 \begin {gather*} -\frac {c \sec ^3(e+f x) (-1+\sin (e+f x)) (a (1+\sin (e+f x)))^{3/2} \sqrt {c-c \sin (e+f x)} (-12 B \cos (2 (e+f x))-3 B \cos (4 (e+f x))+8 A (9 \sin (e+f x)+\sin (3 (e+f x))))}{96 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 86, normalized size = 0.64
method | result | size |
default | \(\frac {\left (3 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+4 A \left (\cos ^{2}\left (f x +e \right )\right )+3 B \sin \left (f x +e \right )+8 A \right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}}}{12 f \cos \left (f x +e \right )^{3}}\) | \(86\) |
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 = 89, normalized size = 0.66 \begin {gather*} -\frac {{\left (3 \, B a c \cos \left (f x + e\right )^{4} - 3 \, B a c - 4 \, {\left (A a c \cos \left (f x + e\right )^{2} + 2 \, A a 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 [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs.
\(2 (125) = 250\).
time = 0.53, size = 252, normalized size = 1.88 \begin {gather*} \frac {4 \, {\left (3 \, B a 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 ) + 2 \, A a 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 ) - 6 \, B a 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 ) - 3 \, A a c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \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 ) + 3 \, B a c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \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 = 1.84, size = 103, normalized size = 0.77 \begin {gather*} -\frac {a\,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\,B\,\cos \left (e+f\,x\right )+15\,B\,\cos \left (3\,e+3\,f\,x\right )+3\,B\,\cos \left (5\,e+5\,f\,x\right )-80\,A\,\sin \left (2\,e+2\,f\,x\right )-8\,A\,\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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