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 = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.10, size = 49, normalized size = 1.00 \[ \frac {\sqrt {b^2+c^2} \sin (d+e x)-c}{c e (c \cos (d+e x)-b \sin (d+e x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.64, size = 75, normalized size = 1.53 \[ -\frac {b^{2} + c^{2} - \sqrt {b^{2} + c^{2}} {\left (b \cos \left (e x + d\right ) + c \sin \left (e x + d\right )\right )}}{{\left (b^{2} c + c^{3}\right )} e \cos \left (e x + d\right ) - {\left (b^{3} + b c^{2}\right )} e \sin \left (e x + d\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 43, normalized size = 0.88 \[ -\frac {2 \, {\left (b + \sqrt {b^{2} + c^{2}}\right )} e^{\left (-1\right )}}{{\left (c \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b + \sqrt {b^{2} + c^{2}}\right )} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 50, normalized size = 1.02 \[ -\frac {2 \left (\sqrt {b^{2}+c^{2}}+b \right )}{e \,c^{2} \left (\tan \left (\frac {d}{2}+\frac {e x}{2}\right )+\frac {\sqrt {b^{2}+c^{2}}}{c}+\frac {b}{c}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 40, normalized size = 0.82 \[ -\frac {2}{{\left (c - \frac {{\left (b - \sqrt {b^{2} + c^{2}}\right )} \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right )} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.82, size = 38, normalized size = 0.78 \[ \frac {2\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}{e\,\left (b+\sqrt {b^2+c^2}+c\,\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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