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 = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 204
Rule 618
Rule 3124
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*}
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Mathematica [A] time = 0.09, 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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fricas [A] time = 0.50, size = 364, normalized size = 6.74 \[ \left [-\frac {\sqrt {-p^{2} + q^{2} + r^{2}} \log \left (-\frac {p^{2} q^{2} - 2 \, q^{4} - r^{4} - {\left (p^{2} + 3 \, q^{2}\right )} r^{2} - {\left (2 \, p^{2} q^{2} - q^{4} - 2 \, p^{2} r^{2} + r^{4}\right )} \cos \relax (x)^{2} - 2 \, {\left (p q^{3} + p q r^{2}\right )} \cos \relax (x) - 2 \, {\left (p q^{2} r + p r^{3} - {\left (q r^{3} - {\left (2 \, p^{2} q - q^{3}\right )} r\right )} \cos \relax (x)\right )} \sin \relax (x) + 2 \, {\left (2 \, p q r \cos \relax (x)^{2} - p q r + {\left (q^{2} r + r^{3}\right )} \cos \relax (x) - {\left (q^{3} + q r^{2} + {\left (p q^{2} - p r^{2}\right )} \cos \relax (x)\right )} \sin \relax (x)\right )} \sqrt {-p^{2} + q^{2} + r^{2}}}{2 \, p q \cos \relax (x) + {\left (q^{2} - r^{2}\right )} \cos \relax (x)^{2} + p^{2} + r^{2} + 2 \, {\left (q r \cos \relax (x) + p r\right )} \sin \relax (x)}\right )}{2 \, {\left (p^{2} - q^{2} - r^{2}\right )}}, \frac {\arctan \left (-\frac {{\left (p q \cos \relax (x) + p r \sin \relax (x) + q^{2} + r^{2}\right )} \sqrt {p^{2} - q^{2} - r^{2}}}{{\left (r^{3} - {\left (p^{2} - q^{2}\right )} r\right )} \cos \relax (x) + {\left (p^{2} q - q^{3} - q r^{2}\right )} \sin \relax (x)}\right )}{\sqrt {p^{2} - q^{2} - r^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.91, size = 72, normalized size = 1.33 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, p + 2 \, q\right ) + \arctan \left (-\frac {p \tan \left (\frac {1}{2} \, x\right ) - q \tan \left (\frac {1}{2} \, x\right ) + r}{\sqrt {p^{2} - q^{2} - r^{2}}}\right )\right )}}{\sqrt {p^{2} - q^{2} - r^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 53, normalized size = 0.98 \[ \frac {2 \arctan \left (\frac {2 r +2 \left (p -q \right ) \tan \left (\frac {x}{2}\right )}{2 \sqrt {p^{2}-q^{2}-r^{2}}}\right )}{\sqrt {p^{2}-q^{2}-r^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.32, size = 68, normalized size = 1.26 \[ \left \{\begin {array}{cl} \frac {\ln \left (q+r\,\mathrm {tan}\left (\frac {x}{2}\right )\right )}{r} & \text {\ if\ \ }p=q\\ \frac {2\,\mathrm {atan}\left (\frac {r+\mathrm {tan}\left (\frac {x}{2}\right )\,\left (p-q\right )}{\sqrt {p^2-q^2-r^2}}\right )}{\sqrt {p^2-q^2-r^2}} & \text {\ if\ \ }p\neq q \end {array}\right . \]
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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