Optimal. Leaf size=98 \[ \frac {1}{8} a^2 (4 A+B) c x-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f} \]
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Rubi [A]
time = 0.11, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3046, 2939,
2748, 2715, 8} \begin {gather*} -\frac {a^2 c (4 A+B) \cos ^3(e+f x)}{12 f}+\frac {a^2 c (4 A+B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^2 c x (4 A+B)-\frac {B c \cos ^3(e+f x) \left (a^2 \sin (e+f x)+a^2\right )}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
Rule 2748
Rule 2939
Rule 3046
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx &=(a c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) (A+B \sin (e+f x)) \, dx\\ &=-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} (a (4 A+B) c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx\\ &=-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{4} \left (a^2 (4 A+B) c\right ) \int \cos ^2(e+f x) \, dx\\ &=-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}+\frac {1}{8} \left (a^2 (4 A+B) c\right ) \int 1 \, dx\\ &=\frac {1}{8} a^2 (4 A+B) c x-\frac {a^2 (4 A+B) c \cos ^3(e+f x)}{12 f}+\frac {a^2 (4 A+B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {B c \cos ^3(e+f x) \left (a^2+a^2 \sin (e+f x)\right )}{4 f}\\ \end {align*}
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Mathematica [A]
time = 0.58, size = 67, normalized size = 0.68 \begin {gather*} -\frac {a^2 c (-12 (4 A+B) f x+24 (A+B) \cos (e+f x)+8 (A+B) \cos (3 (e+f x))-24 A \sin (2 (e+f x))+3 B \sin (4 (e+f x)))}{96 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. \(185\) vs.
\(2(90)=180\).
time = 0.13, size = 186, normalized size = 1.90
method | result | size |
risch | \(\frac {a^{2} c x A}{2}+\frac {a^{2} c x B}{8}-\frac {a^{2} c \cos \left (f x +e \right ) A}{4 f}-\frac {a^{2} c \cos \left (f x +e \right ) B}{4 f}-\frac {B \,a^{2} c \sin \left (4 f x +4 e \right )}{32 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) A}{12 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right ) B}{12 f}+\frac {a^{2} A c \sin \left (2 f x +2 e \right )}{4 f}\) | \(126\) |
derivativedivides | \(\frac {-a^{2} A c \cos \left (f x +e \right )+a^{2} A c \left (f x +e \right )+B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \cos \left (f x +e \right )-a^{2} A c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {a^{2} A c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(186\) |
default | \(\frac {-a^{2} A c \cos \left (f x +e \right )+a^{2} A c \left (f x +e \right )+B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c \cos \left (f x +e \right )-a^{2} A c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {a^{2} A c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-B \,a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\) | \(186\) |
norman | \(\frac {\left (\frac {1}{2} a^{2} A c +\frac {1}{8} B \,a^{2} c \right ) x +\left (2 a^{2} A c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (2 a^{2} A c +\frac {1}{2} B \,a^{2} c \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 a^{2} A c +\frac {3}{4} B \,a^{2} c \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {1}{2} a^{2} A c +\frac {1}{8} B \,a^{2} c \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {2 a^{2} A c +2 B \,a^{2} c}{3 f}-\frac {2 \left (a^{2} A c +B \,a^{2} c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {2 \left (a^{2} A c +B \,a^{2} c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (a^{2} A c +B \,a^{2} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {a^{2} c \left (4 A -B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{2} c \left (4 A -B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} c \left (4 A +7 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 193 vs.
\(2 (95) = 190\).
time = 0.27, size = 193, normalized size = 1.97 \begin {gather*} -\frac {32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c - 96 \, {\left (f x + e\right )} A a^{2} c + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c - 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c + 96 \, A a^{2} c \cos \left (f x + e\right ) + 96 \, B a^{2} c \cos \left (f x + e\right )}{96 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 81, normalized size = 0.83 \begin {gather*} -\frac {8 \, {\left (A + B\right )} a^{2} c \cos \left (f x + e\right )^{3} - 3 \, {\left (4 \, A + B\right )} a^{2} c f x + 3 \, {\left (2 \, B a^{2} c \cos \left (f x + e\right )^{3} - {\left (4 \, A + B\right )} a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 396 vs.
\(2 (90) = 180\).
time = 0.24, size = 396, normalized size = 4.04 \begin {gather*} \begin {cases} - \frac {A a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {A a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} c x + \frac {A a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {A a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 A a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {A a^{2} c \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 B a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {B a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {3 B a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {B a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + \frac {5 B a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {B a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 B a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {B a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 B a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 111, normalized size = 1.13 \begin {gather*} -\frac {B a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {A a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (4 \, A a^{2} c + B a^{2} c\right )} x - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (A a^{2} c + B a^{2} c\right )} \cos \left (f x + e\right )}{4 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.70, size = 339, normalized size = 3.46 \begin {gather*} \frac {a^2\,c\,\mathrm {atan}\left (\frac {a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,A+B\right )}{4\,\left (A\,a^2\,c+\frac {B\,a^2\,c}{4}\right )}\right )\,\left (4\,A+B\right )}{4\,f}-\frac {a^2\,c\,\left (4\,A+B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (2\,A\,a^2\,c+2\,B\,a^2\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a^2\,c-\frac {B\,a^2\,c}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a^2\,c+\frac {7\,B\,a^2\,c}{4}\right )+\frac {2\,A\,a^2\,c}{3}+\frac {2\,B\,a^2\,c}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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