Optimal. Leaf size=103 \[ -\frac {2 (A-B) c \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {a (A-11 B) c \cos (e+f x)}{15 f \left (a^2+a^2 \sin (e+f x)\right )^2}+\frac {(A+4 B) c \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )} \]
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Rubi [A]
time = 0.15, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3046, 2936,
2829, 2727} \begin {gather*} \frac {c (A+4 B) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {a c (A-11 B) \cos (e+f x)}{15 f \left (a^2 \sin (e+f x)+a^2\right )^2}-\frac {2 c (A-B) \cos (e+f x)}{5 f (a \sin (e+f x)+a)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2829
Rule 2936
Rule 3046
Rubi steps
\begin {align*} \int \frac {(A+B \sin (e+f x)) (c-c \sin (e+f x))}{(a+a \sin (e+f x))^3} \, dx &=(a c) \int \frac {\cos ^2(e+f x) (A+B \sin (e+f x))}{(a+a \sin (e+f x))^4} \, dx\\ &=-\frac {2 (A-B) c \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}-\frac {c \int \frac {a A-6 a B+5 a B \sin (e+f x)}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {2 (A-B) c \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {(A-11 B) c \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {((A+4 B) c) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=-\frac {2 (A-B) c \cos (e+f x)}{5 f (a+a \sin (e+f x))^3}+\frac {(A-11 B) c \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}+\frac {(A+4 B) c \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.56, size = 139, normalized size = 1.35 \begin {gather*} \frac {c \left (-15 (A+B) \cos \left (e+\frac {f x}{2}\right )+5 (A+B) \cos \left (e+\frac {3 f x}{2}\right )+5 A \sin \left (\frac {f x}{2}\right )-25 B \sin \left (\frac {f x}{2}\right )-15 B \sin \left (2 e+\frac {3 f x}{2}\right )+A \sin \left (2 e+\frac {5 f x}{2}\right )+4 B \sin \left (2 e+\frac {5 f x}{2}\right )\right )}{30 a^3 f \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) \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 = 0.33, size = 115, normalized size = 1.12
method | result | size |
derivativedivides | \(\frac {2 c \left (-\frac {8 A -8 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-6 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {14 A -10 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{3}}\) | \(115\) |
default | \(\frac {2 c \left (-\frac {8 A -8 B}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {-6 A +2 B}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {A}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-16 A +16 B}{4 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {14 A -10 B}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\right )}{f \,a^{3}}\) | \(115\) |
risch | \(\frac {-\frac {10 B c \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}+2 i B c \,{\mathrm e}^{3 i \left (f x +e \right )}-\frac {2 i B c \,{\mathrm e}^{i \left (f x +e \right )}}{3}+2 B c \,{\mathrm e}^{4 i \left (f x +e \right )}-\frac {2 i A c \,{\mathrm e}^{i \left (f x +e \right )}}{3}+\frac {2 A c \,{\mathrm e}^{2 i \left (f x +e \right )}}{3}+\frac {2 A c}{15}+\frac {8 B c}{15}+2 i A c \,{\mathrm e}^{3 i \left (f x +e \right )}}{f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(126\) |
norman | \(\frac {-\frac {8 A c +2 B c}{15 f a}-\frac {2 A c \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}-\frac {2 \left (23 A c -3 B c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {2 \left (7 A c +7 B c \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (11 A c -B c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f a}-\frac {2 \left (11 A c -B c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {\left (2 A c +2 B c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 f a}-\frac {10 \left (A c +B c \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f a}-\frac {2 \left (A c +B c \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(262\) |
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. 797 vs.
\(2 (103) = 206\).
time = 0.30, size = 797, normalized size = 7.74 \begin {gather*} -\frac {2 \, {\left (\frac {A c {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {2 \, B c {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {3 \, A c {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, B c {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.67, size = 203, normalized size = 1.97 \begin {gather*} \frac {{\left (A + 4 \, B\right )} c \cos \left (f x + e\right )^{3} - {\left (2 \, A - 7 \, B\right )} c \cos \left (f x + e\right )^{2} + 3 \, {\left (A - B\right )} c \cos \left (f x + e\right ) + 6 \, {\left (A - B\right )} c - {\left ({\left (A + 4 \, B\right )} c \cos \left (f x + e\right )^{2} + 3 \, {\left (A - B\right )} c \cos \left (f x + e\right ) + 6 \, {\left (A - B\right )} c\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f\right )} \sin \left (f x + e\right )\right )}} \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. 1035 vs.
\(2 (97) = 194\).
time = 5.43, size = 1035, normalized size = 10.05 \begin {gather*} \begin {cases} - \frac {30 A c \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {30 A c \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {50 A c \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {10 A c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {8 A c}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {30 B c \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} + \frac {10 B c \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {10 B c \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} - \frac {2 B c}{15 a^{3} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 150 a^{3} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 75 a^{3} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 15 a^{3} f} & \text {for}\: f \neq 0 \\\frac {x \left (A + B \sin {\left (e \right )}\right ) \left (- c \sin {\left (e \right )} + c\right )}{\left (a \sin {\left (e \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.52, size = 138, normalized size = 1.34 \begin {gather*} -\frac {2 \, {\left (15 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 15 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 15 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 25 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 5 \, A c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, B c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, A c + B c\right )}}{15 \, a^{3} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.03, size = 172, normalized size = 1.67 \begin {gather*} \frac {2\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {41\,A\,c}{4}-\frac {B\,c}{4}-\frac {11\,A\,c\,\cos \left (e+f\,x\right )}{2}+\frac {B\,c\,\cos \left (e+f\,x\right )}{2}+5\,A\,c\,\sin \left (e+f\,x\right )+5\,B\,c\,\sin \left (e+f\,x\right )-\frac {3\,A\,c\,\cos \left (2\,e+2\,f\,x\right )}{4}+\frac {3\,B\,c\,\cos \left (2\,e+2\,f\,x\right )}{4}-\frac {5\,A\,c\,\sin \left (2\,e+2\,f\,x\right )}{4}-\frac {5\,B\,c\,\sin \left (2\,e+2\,f\,x\right )}{4}\right )}{15\,a^3\,f\,\left (\frac {5\,\sqrt {2}\,\cos \left (\frac {3\,e}{2}+\frac {\pi }{4}+\frac {3\,f\,x}{2}\right )}{4}-\frac {5\,\sqrt {2}\,\cos \left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\right )}{2}+\frac {\sqrt {2}\,\cos \left (\frac {5\,e}{2}-\frac {\pi }{4}+\frac {5\,f\,x}{2}\right )}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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