Optimal. Leaf size=49 \[ -\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{c e (c \cos (d+e x)-b \sin (d+e x))} \]
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Rubi [A] time = 0.036152, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {3114} \[ -\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{c e (c \cos (d+e x)-b \sin (d+e x))} \]
Antiderivative was successfully verified.
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Rule 3114
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b^2+c^2}+b \cos (d+e x)+c \sin (d+e x)} \, dx &=-\frac{c-\sqrt{b^2+c^2} \sin (d+e x)}{c e (c \cos (d+e x)-b \sin (d+e x))}\\ \end{align*}
Mathematica [A] time = 0.0995726, size = 49, normalized size = 1. \[ \frac{\sqrt{b^2+c^2} \sin (d+e x)-c}{c e (c \cos (d+e x)-b \sin (d+e x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.074, size = 50, normalized size = 1. \begin{align*} -2\,{\frac{\sqrt{{b}^{2}+{c}^{2}}+b}{{c}^{2}e} \left ( \tan \left ( d/2+1/2\,ex \right ) +{\frac{\sqrt{{b}^{2}+{c}^{2}}}{c}}+{\frac{b}{c}} \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999555, size = 54, normalized size = 1.1 \begin{align*} -\frac{2}{{\left (c - \frac{{\left (b - \sqrt{b^{2} + c^{2}}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.15614, size = 173, normalized size = 3.53 \begin{align*} -\frac{b^{2} + c^{2} - \sqrt{b^{2} + c^{2}}{\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right )\right )}}{{\left (b^{2} c + c^{3}\right )} e \cos \left (e x + d\right ) -{\left (b^{3} + b c^{2}\right )} e \sin \left (e x + d\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 = 1.1434, size = 58, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (b + \sqrt{b^{2} + c^{2}}\right )} e^{\left (-1\right )}}{{\left (c \tan \left (\frac{1}{2} \, x e + \frac{1}{2} \, d\right ) + b + \sqrt{b^{2} + c^{2}}\right )} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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