Optimal. Leaf size=54 \[ \frac {(A-B+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac {x (B-C)}{a}+\frac {C \sin (c+d x)}{a d} \]
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Rubi [A] time = 0.11, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3023, 2735, 2648} \[ \frac {(A-B+C) \sin (c+d x)}{a d (\cos (c+d x)+1)}+\frac {x (B-C)}{a}+\frac {C \sin (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 2648
Rule 2735
Rule 3023
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{a+a \cos (c+d x)} \, dx &=\frac {C \sin (c+d x)}{a d}+\frac {\int \frac {a A+a (B-C) \cos (c+d x)}{a+a \cos (c+d x)} \, dx}{a}\\ &=\frac {(B-C) x}{a}+\frac {C \sin (c+d x)}{a d}+(A-B+C) \int \frac {1}{a+a \cos (c+d x)} \, dx\\ &=\frac {(B-C) x}{a}+\frac {C \sin (c+d x)}{a d}+\frac {(A-B+C) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.28, size = 136, normalized size = 2.52 \[ \frac {\sec \left (\frac {c}{2}\right ) \cos \left (\frac {1}{2} (c+d x)\right ) \left (4 A \sin \left (\frac {d x}{2}\right )+2 d x (B-C) \cos \left (c+\frac {d x}{2}\right )+2 d x (B-C) \cos \left (\frac {d x}{2}\right )-4 B \sin \left (\frac {d x}{2}\right )+C \sin \left (c+\frac {d x}{2}\right )+C \sin \left (c+\frac {3 d x}{2}\right )+C \sin \left (2 c+\frac {3 d x}{2}\right )+5 C \sin \left (\frac {d x}{2}\right )\right )}{2 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 62, normalized size = 1.15 \[ \frac {{\left (B - C\right )} d x \cos \left (d x + c\right ) + {\left (B - C\right )} d x + {\left (C \cos \left (d x + c\right ) + A - B + 2 \, C\right )} \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 88, normalized size = 1.63 \[ \frac {\frac {{\left (d x + c\right )} {\left (B - C\right )}}{a} + \frac {A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a} + \frac {2 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} a}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.12, size = 125, normalized size = 2.31 \[ \frac {A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}-\frac {B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d}+\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a d \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) B}{a d}-\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) C}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 165, normalized size = 3.06 \[ -\frac {C {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {2 \, \sin \left (d x + c\right )}{{\left (a + \frac {a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - B {\left (\frac {2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac {\sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - \frac {A \sin \left (d x + c\right )}{a {\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 65, normalized size = 1.20 \[ \frac {x\,\left (B-C\right )}{a}+\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (A-B+C\right )}{a\,d}+\frac {2\,C\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.76, size = 330, normalized size = 6.11 \[ \begin {cases} \frac {A \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {A \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {B d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {B d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {B \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {B \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {C d x \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} - \frac {C d x}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {C \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} + \frac {3 C \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{a d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + a d} & \text {for}\: d \neq 0 \\\frac {x \left (A + B \cos {\relax (c )} + C \cos ^{2}{\relax (c )}\right )}{a \cos {\relax (c )} + a} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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