Optimal. Leaf size=89 \[ -\frac{13 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{7 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}-\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A x}{a^3} \]
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Rubi [A] time = 0.173079, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {2966, 2650, 2648} \[ -\frac{13 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)}+\frac{7 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^2}-\frac{2 A \cos (c+d x)}{5 a^3 d (\sin (c+d x)+1)^3}-\frac{A x}{a^3} \]
Antiderivative was successfully verified.
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Rule 2966
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{\sin ^2(c+d x) (A-A \sin (c+d x))}{(a+a \sin (c+d x))^3} \, dx &=\int \left (-\frac{A}{a^3}+\frac{2 A}{a^3 (1+\sin (c+d x))^3}-\frac{5 A}{a^3 (1+\sin (c+d x))^2}+\frac{4 A}{a^3 (1+\sin (c+d x))}\right ) \, dx\\ &=-\frac{A x}{a^3}+\frac{(2 A) \int \frac{1}{(1+\sin (c+d x))^3} \, dx}{a^3}+\frac{(4 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{a^3}-\frac{(5 A) \int \frac{1}{(1+\sin (c+d x))^2} \, dx}{a^3}\\ &=-\frac{A x}{a^3}-\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{5 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))^2}-\frac{4 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{(5 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{3 a^3}\\ &=-\frac{A x}{a^3}-\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{7 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^2}-\frac{7 A \cos (c+d x)}{3 a^3 d (1+\sin (c+d x))}+\frac{(4 A) \int \frac{1}{1+\sin (c+d x)} \, dx}{15 a^3}\\ &=-\frac{A x}{a^3}-\frac{2 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^3}+\frac{7 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))^2}-\frac{13 A \cos (c+d x)}{5 a^3 d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [B] time = 0.762797, size = 189, normalized size = 2.12 \[ \frac{A \left (-50 d x \sin \left (c+\frac{d x}{2}\right )-25 d x \sin \left (c+\frac{3 d x}{2}\right )+40 \sin \left (2 c+\frac{3 d x}{2}\right )-26 \sin \left (2 c+\frac{5 d x}{2}\right )+5 d x \sin \left (3 c+\frac{5 d x}{2}\right )+110 \cos \left (c+\frac{d x}{2}\right )-90 \cos \left (c+\frac{3 d x}{2}\right )+25 d x \cos \left (2 c+\frac{3 d x}{2}\right )+5 d x \cos \left (2 c+\frac{5 d x}{2}\right )+150 \sin \left (\frac{d x}{2}\right )-50 d x \cos \left (\frac{d x}{2}\right )\right )}{20 a^3 d \left (\sin \left (\frac{c}{2}\right )+\cos \left (\frac{c}{2}\right )\right ) \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.099, size = 131, normalized size = 1.5 \begin{align*} -2\,{\frac{A\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{3}}}-{\frac{16\,A}{5\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-5}}+8\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{4}}}-4\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{3}}}-2\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}-2\,{\frac{A}{d{a}^{3} \left ( \tan \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.48544, size = 529, normalized size = 5.94 \begin{align*} -\frac{2 \,{\left (A{\left (\frac{\frac{95 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac{145 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{75 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + 22}{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{15 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}\right )} + \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}}}\right )}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.76422, size = 508, normalized size = 5.71 \begin{align*} -\frac{{\left (5 \, A d x + 13 \, A\right )} \cos \left (d x + c\right )^{3} - 20 \, A d x + 3 \,{\left (5 \, A d x - 2 \, A\right )} \cos \left (d x + c\right )^{2} -{\left (10 \, A d x + 21 \, A\right )} \cos \left (d x + c\right ) -{\left (20 \, A d x -{\left (5 \, A d x - 13 \, A\right )} \cos \left (d x + c\right )^{2} +{\left (10 \, A d x + 19 \, A\right )} \cos \left (d x + c\right ) - 2 \, A\right )} \sin \left (d x + c\right ) - 2 \, A}{5 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16008, size = 126, normalized size = 1.42 \begin{align*} -\frac{\frac{5 \,{\left (d x + c\right )} A}{a^{3}} + \frac{2 \,{\left (5 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 25 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 55 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 35 \, A \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 8 \, A\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}^{5}}}{5 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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