Optimal. Leaf size=82 \[ \frac {4 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac {11 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3} \]
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Rubi [A] time = 0.14, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2966, 2650, 2648} \[ \frac {4 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)}-\frac {11 A \cos (c+d x)}{15 a^3 d (\sin (c+d x)+1)^2}+\frac {2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2650
Rule 2966
Rubi steps
\begin {align*} \int \frac {\sin (c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (-\frac {2 A}{a^3 (1+\sin (c+d x))^3}+\frac {3 A}{a^3 (1+\sin (c+d x))^2}-\frac {A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=-\frac {A \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}-\frac {(2 A) \int \frac {1}{(1+\sin (c+d x))^3} \, dx}{a^3}+\frac {(3 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{a^3}\\ &=\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {A \cos (c+d x)}{a^3 d (1+\sin (c+d x))^2}+\frac {A \cos (c+d x)}{a^3 d (1+\sin (c+d x))}-\frac {(4 A) \int \frac {1}{(1+\sin (c+d x))^2} \, dx}{5 a^3}+\frac {A \int \frac {1}{1+\sin (c+d x)} \, dx}{a^3}\\ &=\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {11 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}-\frac {(4 A) \int \frac {1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=\frac {2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}-\frac {11 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))^2}+\frac {4 A \cos (c+d x)}{15 a^3 d (1+\sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.46, size = 107, normalized size = 1.30 \[ -\frac {A \left (15 \sin \left (2 c+\frac {3 d x}{2}\right )-4 \sin \left (2 c+\frac {5 d x}{2}\right )+15 \cos \left (c+\frac {d x}{2}\right )-5 \cos \left (c+\frac {3 d x}{2}\right )+25 \sin \left (\frac {d x}{2}\right )\right )}{30 a^3 d \left (\sin \left (\frac {c}{2}\right )+\cos \left (\frac {c}{2}\right )\right ) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 156, normalized size = 1.90 \[ \frac {4 \, A \cos \left (d x + c\right )^{3} + 7 \, A \cos \left (d x + c\right )^{2} - 3 \, A \cos \left (d x + c\right ) - {\left (4 \, A \cos \left (d x + c\right )^{2} - 3 \, A \cos \left (d x + c\right ) - 6 \, A\right )} \sin \left (d x + c\right ) - 6 \, A}{15 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \cos \left (d x + c\right ) - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 63, normalized size = 0.77 \[ -\frac {2 \, {\left (15 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 5 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + A\right )}}{15 \, a^{3} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 71, normalized size = 0.87 \[ \frac {4 A \left (-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {5}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {4}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}\right )}{d \,a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 348, normalized size = 4.24 \[ \frac {2 \, {\left (\frac {2 \, A {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}} - \frac {3 \, A {\left (\frac {5 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {5 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {5 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}}\right )}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 13.45, size = 110, normalized size = 1.34 \[ -\frac {2\,A\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left ({\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+5\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-5\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+15\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}{15\,a^3\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 14.11, size = 461, normalized size = 5.62 \[ \begin {cases} - \frac {30 A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} + \frac {10 A \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {10 A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} - \frac {2 A}{15 a^{3} d \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 150 a^{3} d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 75 a^{3} d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} + 15 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \left (- A \sin {\relax (c )} + A\right ) \sin {\relax (c )}}{\left (a \sin {\relax (c )} + a\right )^{3}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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