Optimal. Leaf size=75 \[ -\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac{(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac{(A-B) \cos (x)}{7 (\sin (x)+1)^4} \]
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Rubi [A] time = 0.0567041, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2750, 2650, 2648} \[ -\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)}-\frac{2 (3 A+4 B) \cos (x)}{105 (\sin (x)+1)^2}-\frac{(3 A+4 B) \cos (x)}{35 (\sin (x)+1)^3}-\frac{(A-B) \cos (x)}{7 (\sin (x)+1)^4} \]
Antiderivative was successfully verified.
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Rule 2750
Rule 2650
Rule 2648
Rubi steps
\begin{align*} \int \frac{A+B \sin (x)}{(1+\sin (x))^4} \, dx &=-\frac{(A-B) \cos (x)}{7 (1+\sin (x))^4}+\frac{1}{7} (3 A+4 B) \int \frac{1}{(1+\sin (x))^3} \, dx\\ &=-\frac{(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac{(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}+\frac{1}{35} (2 (3 A+4 B)) \int \frac{1}{(1+\sin (x))^2} \, dx\\ &=-\frac{(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac{(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac{2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}+\frac{1}{105} (2 (3 A+4 B)) \int \frac{1}{1+\sin (x)} \, dx\\ &=-\frac{(A-B) \cos (x)}{7 (1+\sin (x))^4}-\frac{(3 A+4 B) \cos (x)}{35 (1+\sin (x))^3}-\frac{2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))^2}-\frac{2 (3 A+4 B) \cos (x)}{105 (1+\sin (x))}\\ \end{align*}
Mathematica [A] time = 0.0510955, size = 55, normalized size = 0.73 \[ -\frac{\cos (x) \left ((6 A+8 B) \sin ^3(x)+8 (3 A+4 B) \sin ^2(x)+13 (3 A+4 B) \sin (x)+36 A+13 B\right )}{105 (\sin (x)+1)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 115, normalized size = 1.5 \begin{align*} -{\frac{-24\,A+24\,B}{3} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-6}}-{\frac{72\,A-64\,B}{5} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}-{\frac{-32\,A+24\,B}{2} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}-2\,{\frac{A}{\tan \left ( x/2 \right ) +1}}-{\frac{16\,A-16\,B}{7} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-7}}-{\frac{36\,A-20\,B}{3} \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{(-6\,A+2\,B) \left ( \tan \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.84997, size = 417, normalized size = 5.56 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.55325, size = 428, normalized size = 5.71 \begin{align*} \frac{2 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{4} + 8 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 9 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \,{\left (4 \, A + 3 \, B\right )} \cos \left (x\right ) +{\left (2 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{3} - 6 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right )^{2} - 15 \,{\left (3 \, A + 4 \, B\right )} \cos \left (x\right ) + 15 \, A - 15 \, B\right )} \sin \left (x\right ) - 15 \, A + 15 \, B}{105 \,{\left (\cos \left (x\right )^{4} - 3 \, \cos \left (x\right )^{3} - 8 \, \cos \left (x\right )^{2} -{\left (\cos \left (x\right )^{3} + 4 \, \cos \left (x\right )^{2} - 4 \, \cos \left (x\right ) - 8\right )} \sin \left (x\right ) + 4 \, \cos \left (x\right ) + 8\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 18.9729, size = 818, normalized size = 10.91 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.34411, size = 151, normalized size = 2.01 \begin{align*} -\frac{2 \,{\left (105 \, A \tan \left (\frac{1}{2} \, x\right )^{6} + 315 \, A \tan \left (\frac{1}{2} \, x\right )^{5} + 105 \, B \tan \left (\frac{1}{2} \, x\right )^{5} + 630 \, A \tan \left (\frac{1}{2} \, x\right )^{4} + 175 \, B \tan \left (\frac{1}{2} \, x\right )^{4} + 630 \, A \tan \left (\frac{1}{2} \, x\right )^{3} + 280 \, B \tan \left (\frac{1}{2} \, x\right )^{3} + 441 \, A \tan \left (\frac{1}{2} \, x\right )^{2} + 168 \, B \tan \left (\frac{1}{2} \, x\right )^{2} + 147 \, A \tan \left (\frac{1}{2} \, x\right ) + 91 \, B \tan \left (\frac{1}{2} \, x\right ) + 36 \, A + 13 \, B\right )}}{105 \,{\left (\tan \left (\frac{1}{2} \, x\right ) + 1\right )}^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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