Optimal. Leaf size=116 \[ \frac {2 \left (A \left (b^2+c^2\right )-a c C\right ) \tan ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )+c}{\sqrt {a^2-b^2-c^2}}\right )}{\left (b^2+c^2\right ) \sqrt {a^2-b^2-c^2}}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac {c C x}{b^2+c^2} \]
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Rubi [A] time = 0.11, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {3137, 3124, 618, 204} \[ \frac {2 \left (A \left (b^2+c^2\right )-a c C\right ) \tan ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )+c}{\sqrt {a^2-b^2-c^2}}\right )}{\left (b^2+c^2\right ) \sqrt {a^2-b^2-c^2}}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\frac {c C x}{b^2+c^2} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 3124
Rule 3137
Rubi steps
\begin {align*} \int \frac {A+C \sin (x)}{a+b \cos (x)+c \sin (x)} \, dx &=\frac {c C x}{b^2+c^2}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\left (A-\frac {a c C}{b^2+c^2}\right ) \int \frac {1}{a+b \cos (x)+c \sin (x)} \, dx\\ &=\frac {c C x}{b^2+c^2}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}+\left (2 \left (A-\frac {a c C}{b^2+c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+2 c x+(a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {c C x}{b^2+c^2}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}-\left (4 \left (A-\frac {a c C}{b^2+c^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2-c^2\right )-x^2} \, dx,x,2 c+2 (a-b) \tan \left (\frac {x}{2}\right )\right )\\ &=\frac {c C x}{b^2+c^2}+\frac {2 \left (A-\frac {a c C}{b^2+c^2}\right ) \tan ^{-1}\left (\frac {c+(a-b) \tan \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2-c^2}}\right )}{\sqrt {a^2-b^2-c^2}}-\frac {b C \log (a+b \cos (x)+c \sin (x))}{b^2+c^2}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 96, normalized size = 0.83 \[ \frac {C (c x-b \log (a+b \cos (x)+c \sin (x)))-\frac {2 \left (A \left (b^2+c^2\right )-a c C\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {x}{2}\right )+c}{\sqrt {-a^2+b^2+c^2}}\right )}{\sqrt {-a^2+b^2+c^2}}}{b^2+c^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.59, size = 625, normalized size = 5.39 \[ \left [\frac {{\left (A b^{2} - C a c + A c^{2}\right )} \sqrt {-a^{2} + b^{2} + c^{2}} \log \left (\frac {a^{2} b^{2} - 2 \, b^{4} - c^{4} - {\left (a^{2} + 3 \, b^{2}\right )} c^{2} - {\left (2 \, a^{2} b^{2} - b^{4} - 2 \, a^{2} c^{2} + c^{4}\right )} \cos \relax (x)^{2} - 2 \, {\left (a b^{3} + a b c^{2}\right )} \cos \relax (x) - 2 \, {\left (a b^{2} c + a c^{3} - {\left (b c^{3} - {\left (2 \, a^{2} b - b^{3}\right )} c\right )} \cos \relax (x)\right )} \sin \relax (x) - 2 \, {\left (2 \, a b c \cos \relax (x)^{2} - a b c + {\left (b^{2} c + c^{3}\right )} \cos \relax (x) - {\left (b^{3} + b c^{2} + {\left (a b^{2} - a c^{2}\right )} \cos \relax (x)\right )} \sin \relax (x)\right )} \sqrt {-a^{2} + b^{2} + c^{2}}}{2 \, a b \cos \relax (x) + {\left (b^{2} - c^{2}\right )} \cos \relax (x)^{2} + a^{2} + c^{2} + 2 \, {\left (b c \cos \relax (x) + a c\right )} \sin \relax (x)}\right ) - 2 \, {\left (C c^{3} - {\left (C a^{2} - C b^{2}\right )} c\right )} x - {\left (C a^{2} b - C b^{3} - C b c^{2}\right )} \log \left (2 \, a b \cos \relax (x) + {\left (b^{2} - c^{2}\right )} \cos \relax (x)^{2} + a^{2} + c^{2} + 2 \, {\left (b c \cos \relax (x) + a c\right )} \sin \relax (x)\right )}{2 \, {\left (a^{2} b^{2} - b^{4} - c^{4} + {\left (a^{2} - 2 \, b^{2}\right )} c^{2}\right )}}, \frac {2 \, {\left (A b^{2} - C a c + A c^{2}\right )} \sqrt {a^{2} - b^{2} - c^{2}} \arctan \left (-\frac {{\left (a b \cos \relax (x) + a c \sin \relax (x) + b^{2} + c^{2}\right )} \sqrt {a^{2} - b^{2} - c^{2}}}{{\left (c^{3} - {\left (a^{2} - b^{2}\right )} c\right )} \cos \relax (x) + {\left (a^{2} b - b^{3} - b c^{2}\right )} \sin \relax (x)}\right ) - 2 \, {\left (C c^{3} - {\left (C a^{2} - C b^{2}\right )} c\right )} x - {\left (C a^{2} b - C b^{3} - C b c^{2}\right )} \log \left (2 \, a b \cos \relax (x) + {\left (b^{2} - c^{2}\right )} \cos \relax (x)^{2} + a^{2} + c^{2} + 2 \, {\left (b c \cos \relax (x) + a c\right )} \sin \relax (x)\right )}{2 \, {\left (a^{2} b^{2} - b^{4} - c^{4} + {\left (a^{2} - 2 \, b^{2}\right )} c^{2}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 177, normalized size = 1.53 \[ \frac {C c x}{b^{2} + c^{2}} - \frac {C b \log \left (-a \tan \left (\frac {1}{2} \, x\right )^{2} + b \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, c \tan \left (\frac {1}{2} \, x\right ) - a - b\right )}{b^{2} + c^{2}} + \frac {C b \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}{b^{2} + c^{2}} - \frac {2 \, {\left (A b^{2} - C a c + A c^{2}\right )} {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x\right ) - b \tan \left (\frac {1}{2} \, x\right ) + c}{\sqrt {a^{2} - b^{2} - c^{2}}}\right )\right )}}{\sqrt {a^{2} - b^{2} - c^{2}} {\left (b^{2} + c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.13, size = 542, normalized size = 4.67 \[ -\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 c \tan \left (\frac {x}{2}\right )+a +b \right ) a b C}{\left (b^{2}+c^{2}\right ) \left (a -b \right )}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 c \tan \left (\frac {x}{2}\right )+a +b \right ) b^{2} C}{\left (b^{2}+c^{2}\right ) \left (a -b \right )}+\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) A \,b^{2}}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}}+\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) A \,c^{2}}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) a c C}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) C b c}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}}+\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) c a b C}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}\, \left (a -b \right )}-\frac {2 \arctan \left (\frac {2 \left (a -b \right ) \tan \left (\frac {x}{2}\right )+2 c}{2 \sqrt {a^{2}-b^{2}-c^{2}}}\right ) c \,b^{2} C}{\left (b^{2}+c^{2}\right ) \sqrt {a^{2}-b^{2}-c^{2}}\, \left (a -b \right )}+\frac {C b \ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b^{2}+c^{2}}+\frac {2 C c \arctan \left (\tan \left (\frac {x}{2}\right )\right )}{b^{2}+c^{2}} \]
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 = 24.86, size = 1741, normalized size = 15.01 \[ \text {result too large to display} \]
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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