Optimal. Leaf size=24 \[ -\frac {c \cos (x)-b \sin (x)}{a+b \cos (x)+c \sin (x)} \]
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Rubi [B] time = 0.07, antiderivative size = 68, normalized size of antiderivative = 2.83, 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 = {3150} \[ -\frac {c \cos (x) \left (a^2-b^2-c^2\right )-b \sin (x) \left (a^2-b^2-c^2\right )}{\left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))} \]
Antiderivative was successfully verified.
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Rule 3150
Rubi steps
\begin {align*} \int \frac {b^2+c^2+a b \cos (x)+a c \sin (x)}{(a+b \cos (x)+c \sin (x))^2} \, dx &=-\frac {c \left (a^2-b^2-c^2\right ) \cos (x)-b \left (a^2-b^2-c^2\right ) \sin (x)}{\left (a^2-b^2-c^2\right ) (a+b \cos (x)+c \sin (x))}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 32, normalized size = 1.33 \[ \frac {a c+b^2 \sin (x)+c^2 \sin (x)}{b (a+b \cos (x)+c \sin (x))} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 24, normalized size = 1.00 \[ -\frac {c \cos \relax (x) - b \sin \relax (x)}{b \cos \relax (x) + c \sin \relax (x) + a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 68, normalized size = 2.83 \[ \frac {2 \, {\left (a b \tan \left (\frac {1}{2} \, x\right ) - b^{2} \tan \left (\frac {1}{2} \, x\right ) - c^{2} \tan \left (\frac {1}{2} \, x\right ) - a c\right )}}{{\left (a \tan \left (\frac {1}{2} \, x\right )^{2} - b \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, c \tan \left (\frac {1}{2} \, x\right ) + a + b\right )} {\left (a - b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 70, normalized size = 2.92 \[ -\frac {2 \left (-\frac {\left (a b -b^{2}-c^{2}\right ) \tan \left (\frac {x}{2}\right )}{a -b}+\frac {a c}{a -b}\right )}{a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 c \tan \left (\frac {x}{2}\right )+a +b} \]
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 = 3.03, size = 62, normalized size = 2.58 \[ -\frac {\frac {2\,a\,c}{a-b}+\frac {2\,\mathrm {tan}\left (\frac {x}{2}\right )\,\left (b^2-a\,b+c^2\right )}{a-b}}{\left (a-b\right )\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2+2\,c\,\mathrm {tan}\left (\frac {x}{2}\right )+a+b} \]
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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