Optimal. Leaf size=23 \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]
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Rubi [A] time = 0.0893867, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {3288, 2754, 8} \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]
Antiderivative was successfully verified.
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Rule 3288
Rule 2754
Rule 8
Rubi steps
\begin{align*} \int \frac{a+b \sin (d+e x)}{b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx &=\left (4 a^2\right ) \int \frac{a+b \sin (d+e x)}{\left (2 a b+2 a^2 \sin (d+e x)\right )^2} \, dx\\ &=-\frac{\cos (d+e x)}{e (b+a \sin (d+e x))}+\frac{\int 0 \, dx}{a^2-b^2}\\ &=-\frac{\cos (d+e x)}{e (b+a \sin (d+e x))}\\ \end{align*}
Mathematica [A] time = 0.0623663, size = 23, normalized size = 1. \[ -\frac{\cos (d+e x)}{e (a \sin (d+e x)+b)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.092, size = 52, normalized size = 2.3 \begin{align*} 2\,{\frac{1}{e \left ( b \left ( \tan \left ( d/2+1/2\,ex \right ) \right ) ^{2}+2\,a\tan \left ( d/2+1/2\,ex \right ) +b \right ) } \left ( -{\frac{a\tan \left ( d/2+1/2\,ex \right ) }{b}}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67657, size = 54, normalized size = 2.35 \begin{align*} -\frac{\cos \left (e x + d\right )}{a e \sin \left (e x + d\right ) + b e} \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 [B] time = 1.19627, size = 70, normalized size = 3.04 \begin{align*} -\frac{2 \,{\left (a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b\right )} e^{\left (-1\right )}}{{\left (b \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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