Optimal. Leaf size=51 \[ -\frac {2 \tanh ^{-1}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}} \]
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Rubi [A] time = 0.07, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3166, 3124, 618, 206} \[ -\frac {2 \tanh ^{-1}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 618
Rule 3124
Rule 3166
Rubi steps
\begin {align*} \int \frac {\csc (x)}{a+b \cot (x)+c \csc (x)} \, dx &=\int \frac {1}{c+b \cos (x)+a \sin (x)} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{b+c+2 a x+(-b+c) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2-c^2\right )-x^2} \, dx,x,2 a+2 (-b+c) \tan \left (\frac {x}{2}\right )\right )\right )\\ &=-\frac {2 \tanh ^{-1}\left (\frac {a-(b-c) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 50, normalized size = 0.98 \[ -\frac {2 \tanh ^{-1}\left (\frac {a+(c-b) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.81, size = 349, normalized size = 6.84 \[ \left [\frac {\log \left (-\frac {a^{4} + 3 \, a^{2} b^{2} + 2 \, b^{4} + {\left (a^{2} - b^{2}\right )} c^{2} + 2 \, {\left (a^{2} b + b^{3}\right )} c \cos \relax (x) + {\left (a^{4} - b^{4} - 2 \, {\left (a^{2} - b^{2}\right )} c^{2}\right )} \cos \relax (x)^{2} + 2 \, {\left ({\left (a^{3} + a b^{2}\right )} c - {\left (a^{3} b + a b^{3} - 2 \, a b c^{2}\right )} \cos \relax (x)\right )} \sin \relax (x) - 2 \, {\left (2 \, a b c \cos \relax (x)^{2} - a b c + {\left (a^{3} + a b^{2}\right )} \cos \relax (x) - {\left (a^{2} b + b^{3} - {\left (a^{2} - b^{2}\right )} c \cos \relax (x)\right )} \sin \relax (x)\right )} \sqrt {a^{2} + b^{2} - c^{2}}}{2 \, b c \cos \relax (x) - {\left (a^{2} - b^{2}\right )} \cos \relax (x)^{2} + a^{2} + c^{2} + 2 \, {\left (a b \cos \relax (x) + a c\right )} \sin \relax (x)}\right )}{2 \, \sqrt {a^{2} + b^{2} - c^{2}}}, \frac {\sqrt {-a^{2} - b^{2} + c^{2}} \arctan \left (\frac {{\left (b c \cos \relax (x) + a c \sin \relax (x) + a^{2} + b^{2}\right )} \sqrt {-a^{2} - b^{2} + c^{2}}}{{\left (a^{3} + a b^{2} - a c^{2}\right )} \cos \relax (x) - {\left (a^{2} b + b^{3} - b c^{2}\right )} \sin \relax (x)}\right )}{a^{2} + b^{2} - c^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 73, normalized size = 1.43 \[ -\frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, b - 2 \, c\right ) + \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x\right ) - c \tan \left (\frac {1}{2} \, x\right ) - a}{\sqrt {-a^{2} - b^{2} + c^{2}}}\right )\right )}}{\sqrt {-a^{2} - b^{2} + c^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 53, normalized size = 1.04 \[ -\frac {2 \arctan \left (\frac {2 \left (b -c \right ) \tan \left (\frac {x}{2}\right )-2 a}{2 \sqrt {-a^{2}-b^{2}+c^{2}}}\right )}{\sqrt {-a^{2}-b^{2}+c^{2}}} \]
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 = 2.79, size = 47, normalized size = 0.92 \[ -\frac {2\,\mathrm {atanh}\left (\frac {a-\frac {\mathrm {tan}\left (\frac {x}{2}\right )\,\left (2\,b-2\,c\right )}{2}}{\sqrt {a^2+b^2-c^2}}\right )}{\sqrt {a^2+b^2-c^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc {\relax (x )}}{a + b \cot {\relax (x )} + c \csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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