Optimal. Leaf size=54 \[ \frac{2 \tan ^{-1}\left (\frac{(p-q) \tan \left (\frac{x}{2}\right )+r}{\sqrt{p^2-q^2-r^2}}\right )}{\sqrt{p^2-q^2-r^2}} \]
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Rubi [A] time = 0.0681303, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3124, 618, 204} \[ \frac{2 \tan ^{-1}\left (\frac{(p-q) \tan \left (\frac{x}{2}\right )+r}{\sqrt{p^2-q^2-r^2}}\right )}{\sqrt{p^2-q^2-r^2}} \]
Antiderivative was successfully verified.
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Rule 3124
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{p+q \cos (x)+r \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{p+q+2 r x+(p-q) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\left (4 \operatorname{Subst}\left (\int \frac{1}{-4 \left (p^2-q^2-r^2\right )-x^2} \, dx,x,2 r+2 (p-q) \tan \left (\frac{x}{2}\right )\right )\right )\\ &=\frac{2 \tan ^{-1}\left (\frac{r+(p-q) \tan \left (\frac{x}{2}\right )}{\sqrt{p^2-q^2-r^2}}\right )}{\sqrt{p^2-q^2-r^2}}\\ \end{align*}
Mathematica [A] time = 0.0812852, size = 50, normalized size = 0.93 \[ -\frac{2 \tanh ^{-1}\left (\frac{(p-q) \tan \left (\frac{x}{2}\right )+r}{\sqrt{-p^2+q^2+r^2}}\right )}{\sqrt{-p^2+q^2+r^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 53, normalized size = 1. \begin{align*} 2\,{\frac{1}{\sqrt{{p}^{2}-{q}^{2}-{r}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( p-q \right ) \tan \left ( x/2 \right ) +2\,r}{\sqrt{{p}^{2}-{q}^{2}-{r}^{2}}}} \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 = 1.89938, size = 813, normalized size = 15.06 \begin{align*} \left [-\frac{\sqrt{-p^{2} + q^{2} + r^{2}} \log \left (-\frac{p^{2} q^{2} - 2 \, q^{4} - r^{4} -{\left (p^{2} + 3 \, q^{2}\right )} r^{2} -{\left (2 \, p^{2} q^{2} - q^{4} - 2 \, p^{2} r^{2} + r^{4}\right )} \cos \left (x\right )^{2} - 2 \,{\left (p q^{3} + p q r^{2}\right )} \cos \left (x\right ) - 2 \,{\left (p q^{2} r + p r^{3} -{\left (q r^{3} -{\left (2 \, p^{2} q - q^{3}\right )} r\right )} \cos \left (x\right )\right )} \sin \left (x\right ) + 2 \,{\left (2 \, p q r \cos \left (x\right )^{2} - p q r +{\left (q^{2} r + r^{3}\right )} \cos \left (x\right ) -{\left (q^{3} + q r^{2} +{\left (p q^{2} - p r^{2}\right )} \cos \left (x\right )\right )} \sin \left (x\right )\right )} \sqrt{-p^{2} + q^{2} + r^{2}}}{2 \, p q \cos \left (x\right ) +{\left (q^{2} - r^{2}\right )} \cos \left (x\right )^{2} + p^{2} + r^{2} + 2 \,{\left (q r \cos \left (x\right ) + p r\right )} \sin \left (x\right )}\right )}{2 \,{\left (p^{2} - q^{2} - r^{2}\right )}}, \frac{\arctan \left (-\frac{{\left (p q \cos \left (x\right ) + p r \sin \left (x\right ) + q^{2} + r^{2}\right )} \sqrt{p^{2} - q^{2} - r^{2}}}{{\left (r^{3} -{\left (p^{2} - q^{2}\right )} r\right )} \cos \left (x\right ) +{\left (p^{2} q - q^{3} - q r^{2}\right )} \sin \left (x\right )}\right )}{\sqrt{p^{2} - q^{2} - r^{2}}}\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 = 1.08755, size = 97, normalized size = 1.8 \begin{align*} -\frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, p + 2 \, q\right ) + \arctan \left (-\frac{p \tan \left (\frac{1}{2} \, x\right ) - q \tan \left (\frac{1}{2} \, x\right ) + r}{\sqrt{p^{2} - q^{2} - r^{2}}}\right )\right )}}{\sqrt{p^{2} - q^{2} - r^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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