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 = 0.107879, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, 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 3137
Rule 3124
Rule 618
Rule 204
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*}
Mathematica [A] time = 0.277802, 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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Maple [B] time = 0.06, size = 542, normalized size = 4.7 \begin{align*} -{\frac{abC}{ \left ({b}^{2}+{c}^{2} \right ) \left ( a-b \right ) }\ln \left ( a \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-b \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,c\tan \left ( x/2 \right ) +a+b \right ) }+{\frac{{b}^{2}C}{ \left ({b}^{2}+{c}^{2} \right ) \left ( a-b \right ) }\ln \left ( a \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}-b \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+2\,c\tan \left ( x/2 \right ) +a+b \right ) }+2\,{\frac{A{b}^{2}}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }+2\,{\frac{A{c}^{2}}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{acC}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{Cbc}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }+2\,{\frac{abcC}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}} \left ( a-b \right ) }\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }-2\,{\frac{c{b}^{2}C}{ \left ({b}^{2}+{c}^{2} \right ) \sqrt{{a}^{2}-{b}^{2}-{c}^{2}} \left ( a-b \right ) }\arctan \left ( 1/2\,{\frac{2\, \left ( a-b \right ) \tan \left ( x/2 \right ) +2\,c}{\sqrt{{a}^{2}-{b}^{2}-{c}^{2}}}} \right ) }+{\frac{bC}{{b}^{2}+{c}^{2}}\ln \left ( 1+ \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+2\,{\frac{Cc\arctan \left ( \tan \left ( x/2 \right ) \right ) }{{b}^{2}+{c}^{2}}} \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 [B] time = 2.85079, size = 1374, normalized size = 11.84 \begin{align*} \text{result too large to display} \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 = 1.15184, size = 239, normalized size = 2.06 \begin{align*} \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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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