3.5 \(\int \frac {b+c+\sin (x)}{a-b \cos (x)} \, dx\)

Optimal. Leaf size=58 \[ \frac {2 (b+c) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a-b \cos (x))}{b} \]

[Out]

ln(a-b*cos(x))/b+2*(b+c)*arctan((a+b)^(1/2)*tan(1/2*x)/(a-b)^(1/2))/(a-b)^(1/2)/(a+b)^(1/2)

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Rubi [A]  time = 0.13, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4401, 2659, 205, 2668, 31} \[ \frac {2 (b+c) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a-b \cos (x))}{b} \]

Antiderivative was successfully verified.

[In]

Int[(b + c + Sin[x])/(a - b*Cos[x]),x]

[Out]

(2*(b + c)*ArcTan[(Sqrt[a + b]*Tan[x/2])/Sqrt[a - b]])/(Sqrt[a - b]*Sqrt[a + b]) + Log[a - b*Cos[x]]/b

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 4401

Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /;  !InertTrigFreeQ[u]

Rubi steps

\begin {align*} \int \frac {b+c+\sin (x)}{a-b \cos (x)} \, dx &=\int \left (\frac {-b-c}{-a+b \cos (x)}+\frac {\sin (x)}{a-b \cos (x)}\right ) \, dx\\ &=(-b-c) \int \frac {1}{-a+b \cos (x)} \, dx+\int \frac {\sin (x)}{a-b \cos (x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,-b \cos (x)\right )}{b}-(2 (b+c)) \operatorname {Subst}\left (\int \frac {1}{-a+b+(-a-b) x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {2 (b+c) \tan ^{-1}\left (\frac {\sqrt {a+b} \tan \left (\frac {x}{2}\right )}{\sqrt {a-b}}\right )}{\sqrt {a-b} \sqrt {a+b}}+\frac {\log (a-b \cos (x))}{b}\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 55, normalized size = 0.95 \[ \frac {\log (a-b \cos (x))}{b}-\frac {2 (b+c) \tanh ^{-1}\left (\frac {(a+b) \tan \left (\frac {x}{2}\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(b + c + Sin[x])/(a - b*Cos[x]),x]

[Out]

(-2*(b + c)*ArcTanh[((a + b)*Tan[x/2])/Sqrt[-a^2 + b^2]])/Sqrt[-a^2 + b^2] + Log[a - b*Cos[x]]/b

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fricas [A]  time = 0.56, size = 235, normalized size = 4.05 \[ \left [-\frac {\sqrt {-a^{2} + b^{2}} {\left (b^{2} + b c\right )} \log \left (-\frac {2 \, a b \cos \relax (x) - {\left (2 \, a^{2} - b^{2}\right )} \cos \relax (x)^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \relax (x) - b\right )} \sin \relax (x) + a^{2} - 2 \, b^{2}}{b^{2} \cos \relax (x)^{2} - 2 \, a b \cos \relax (x) + a^{2}}\right ) - {\left (a^{2} - b^{2}\right )} \log \left (b^{2} \cos \relax (x)^{2} - 2 \, a b \cos \relax (x) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}, \frac {2 \, \sqrt {a^{2} - b^{2}} {\left (b^{2} + b c\right )} \arctan \left (-\frac {a \cos \relax (x) - b}{\sqrt {a^{2} - b^{2}} \sin \relax (x)}\right ) + {\left (a^{2} - b^{2}\right )} \log \left (b^{2} \cos \relax (x)^{2} - 2 \, a b \cos \relax (x) + a^{2}\right )}{2 \, {\left (a^{2} b - b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+c+sin(x))/(a-b*cos(x)),x, algorithm="fricas")

[Out]

[-1/2*(sqrt(-a^2 + b^2)*(b^2 + b*c)*log(-(2*a*b*cos(x) - (2*a^2 - b^2)*cos(x)^2 - 2*sqrt(-a^2 + b^2)*(a*cos(x)
- b)*sin(x) + a^2 - 2*b^2)/(b^2*cos(x)^2 - 2*a*b*cos(x) + a^2)) - (a^2 - b^2)*log(b^2*cos(x)^2 - 2*a*b*cos(x)
+ a^2))/(a^2*b - b^3), 1/2*(2*sqrt(a^2 - b^2)*(b^2 + b*c)*arctan(-(a*cos(x) - b)/(sqrt(a^2 - b^2)*sin(x))) +
(a^2 - b^2)*log(b^2*cos(x)^2 - 2*a*b*cos(x) + a^2))/(a^2*b - b^3)]

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giac [B]  time = 0.37, size = 103, normalized size = 1.78 \[ \frac {2 \, {\left (\pi \left \lfloor \frac {x}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, x\right ) + b \tan \left (\frac {1}{2} \, x\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )} {\left (b + c\right )}}{\sqrt {a^{2} - b^{2}}} + \frac {\log \left (a \tan \left (\frac {1}{2} \, x\right )^{2} + b \tan \left (\frac {1}{2} \, x\right )^{2} + a - b\right )}{b} - \frac {\log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+c+sin(x))/(a-b*cos(x)),x, algorithm="giac")

[Out]

2*(pi*floor(1/2*x/pi + 1/2)*sgn(2*a + 2*b) + arctan((a*tan(1/2*x) + b*tan(1/2*x))/sqrt(a^2 - b^2)))*(b + c)/sq
rt(a^2 - b^2) + log(a*tan(1/2*x)^2 + b*tan(1/2*x)^2 + a - b)/b - log(tan(1/2*x)^2 + 1)/b

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maple [B]  time = 0.06, size = 110, normalized size = 1.90 \[ \frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+\left (\tan ^{2}\left (\frac {x}{2}\right )\right ) b +a -b \right )}{b}+\frac {2 b \arctan \left (\frac {\left (a +b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 \arctan \left (\frac {\left (a +b \right ) \tan \left (\frac {x}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right ) c}{\sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {\ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b+c+sin(x))/(a-b*cos(x)),x)

[Out]

1/b*ln(a*tan(1/2*x)^2+tan(1/2*x)^2*b+a-b)+2*b/((a-b)*(a+b))^(1/2)*arctan((a+b)*tan(1/2*x)/((a-b)*(a+b))^(1/2))
+2/((a-b)*(a+b))^(1/2)*arctan((a+b)*tan(1/2*x)/((a-b)*(a+b))^(1/2))*c-1/b*ln(tan(1/2*x)^2+1)

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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.

[In]

integrate((b+c+sin(x))/(a-b*cos(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

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mupad [B]  time = 11.90, size = 2213, normalized size = 38.16 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b + c + sin(x))/(a - b*cos(x)),x)

[Out]

(2*atan(((((b + c)*(128*a*b^3 + 64*b^3*c + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(64*a*b^4 + 32*b^4*c + 32*
b^5 + 32*a^2*b^3 + 32*a^2*b^2*c + 64*a*b^3*c))/(2*(b^4 - a^2*b^2)) + 128*a*b^2*c + 64*a^2*b*c))/(a^2 - b^2)^(1
/2) - ((2*a^2*b - 2*b^3)*(b + c)*(64*a*b^4 + 32*b^4*c + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c + 64*a*b^3*c))/(2*(
b^4 - a^2*b^2)*(a^2 - b^2)^(1/2)))*(a^2 - b^2)*(2*b^3*c - 4*a^2 + 4*b^2 + b^4 + b^2*c^2))/((a + b)*(32*a*b + 3
2*a*c + 32*b*c + 32*b^2)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 + b^2*c^2)^2) - (tan(x/2)*(a^2 - b^2)^(3/2)*((4*b*(b +
c)*(32*a*b^3 - 128*a*b + 64*b^3*c - 64*a^2 - 64*b^2 + 32*b^4 + 32*b^2*c^2 - ((b + c)*(((b + c)*(128*a*b^3 + 6
4*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(a^2 - b^
2)^(1/2) - ((2*a^2*b - 2*b^3)*(b + c)*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2)^(1
/2))))/(a^2 - b^2)^(1/2) - ((2*a^2*b - 2*b^3)*(32*a*b^4 - 128*a^2*b - 64*a*b^2 + 64*b^4*c - 64*a^3 + 32*b^5 +
32*b^3*c^2 - ((2*a^2*b - 2*b^3)*(128*a*b^3 + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(64*a*b^4 + 128*a^2*b^3
+ 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(2*(b^4 - a^2*b^2)) + 32*a*b^2*c^2 + 64*a*b^3*c))/(2*(b^4 - a^2*b^2)) + 3
2*a*b*c^2 + 64*a*b^2*c + ((2*a^2*b - 2*b^3)*(b + c)^2*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2
)*(a^2 - b^2))))/((a + b)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 + b^2*c^2)^2) - ((((b + c)^3*(64*a*b^4 + 128*a^2*b^3
+ 64*a^3*b^2))/(a^2 - b^2)^(3/2) + ((2*a^2*b - 2*b^3)*(((b + c)*(128*a*b^3 + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b -
2*b^3)*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(a^2 - b^2)^(1/2) - ((2*a^2*b - 2*b^3)*(b
+ c)*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2)*(a^2 - b^2)^(1/2))))/(2*(b^4 - a^2*b^2)) - ((b
+ c)*(32*a*b^4 - 128*a^2*b - 64*a*b^2 + 64*b^4*c - 64*a^3 + 32*b^5 + 32*b^3*c^2 - ((2*a^2*b - 2*b^3)*(128*a*b
^3 + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(64*a*b^4 + 128*a^2*b^3 + 64*a^3*b^2))/(2*(b^4 - a^2*b^2))))/(2*
(b^4 - a^2*b^2)) + 32*a*b^2*c^2 + 64*a*b^3*c))/(a^2 - b^2)^(1/2))*(2*b^3*c - 4*a^2 + 4*b^2 + b^4 + b^2*c^2))/(
(a + b)*(a^2 - b^2)^(1/2)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 + b^2*c^2)^2)))/(32*a*b + 32*a*c + 32*b*c + 32*b^2) +
(4*b*(b + c)*(a^2 - b^2)^(3/2)*(64*a*b^2 + 32*a^2*b + 32*a^2*c + 32*b^2*c + 32*b^3 - ((b + c)^2*(64*a*b^4 + 3
2*b^4*c + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c + 64*a*b^3*c))/(a^2 - b^2) - ((2*a^2*b - 2*b^3)*(128*a*b^3 + 64*b
^3*c + 64*b^4 + 64*a^2*b^2 - ((2*a^2*b - 2*b^3)*(64*a*b^4 + 32*b^4*c + 32*b^5 + 32*a^2*b^3 + 32*a^2*b^2*c + 64
*a*b^3*c))/(2*(b^4 - a^2*b^2)) + 128*a*b^2*c + 64*a^2*b*c))/(2*(b^4 - a^2*b^2)) + 64*a*b*c))/((a + b)*(32*a*b
+ 32*a*c + 32*b*c + 32*b^2)*(2*b^3*c + 4*a^2 - 4*b^2 + b^4 + b^2*c^2)^2))*(b + c))/(a^2 - b^2)^(1/2) - (log((6
4*a*b^2 + 32*a^2*b + 32*a^2*c + 32*b^2*c + 32*b^3 - 32*tan(x/2)*(a + b)*(b*c^2 - 2*b - 2*a + 2*b^2*c + b^3) -
((b*(-(b + c)^2/(a^2 - b^2))^(1/2) - 1)*(128*a*b^3 + 64*b^3*c + 64*b^4 + 64*a^2*b^2 - 32*(b*(-(b + c)^2/(a^2 -
b^2))^(1/2) - 1)*(a + b)^2*(b*c - 2*b*tan(x/2) - 2*a*tan(x/2) + b^2 + 2*a*b*tan(x/2)*(-(b + c)^2/(a^2 - b^2))
^(1/2)) - 32*tan(x/2)*(a + b)*(2*b^3*c - 2*a*b - 2*a^2 + b^4 + b^2*c^2) + 128*a*b^2*c + 64*a^2*b*c))/b + 64*a*
b*c)*(64*a*b^2 + 32*a^2*b + 32*a^2*c + 32*b^2*c + 32*b^3 - 32*tan(x/2)*(a + b)*(b*c^2 - 2*b - 2*a + 2*b^2*c +
b^3) + 64*a*b*c + ((b*(-(b + c)^2/(a^2 - b^2))^(1/2) + 1)*(128*a*b^3 + 64*b^3*c + 64*b^4 + 64*a^2*b^2 - 32*tan
(x/2)*(a + b)*(2*b^3*c - 2*a*b - 2*a^2 + b^4 + b^2*c^2) + 128*a*b^2*c + 64*a^2*b*c - 32*(b*(-(b + c)^2/(a^2 -
b^2))^(1/2) + 1)*(a + b)^2*(2*a*tan(x/2) + 2*b*tan(x/2) - b*c - b^2 + 2*a*b*tan(x/2)*(-(b + c)^2/(a^2 - b^2))^
(1/2))))/b))*(2*a^2*b - 2*b^3))/(2*(b^4 - a^2*b^2)) - log(tan(x/2)^2 + 1)/b

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sympy [A]  time = 24.57, size = 748, normalized size = 12.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b+c+sin(x))/(a-b*cos(x)),x)

[Out]

Piecewise((zoo*(-c*log(tan(x/2) - 1) + c*log(tan(x/2) + 1) - log(tan(x/2) - 1) - log(tan(x/2) + 1) + log(tan(x
/2)**2 + 1)), Eq(a, 0) & Eq(b, 0)), (-tan(x/2) - c*tan(x/2)/b - log(tan(x/2)**2 + 1)/b, Eq(a, -b)), (-1/tan(x/
2) - c/(b*tan(x/2)) - log(tan(x/2)**2 + 1)/b + 2*log(tan(x/2))/b, Eq(a, b)), ((c*x - cos(x))/a, Eq(b, 0)), (a*
sqrt(-a/(a + b) + b/(a + b))*log(-sqrt(-a/(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) +
b**2*sqrt(-a/(a + b) + b/(a + b))) + a*sqrt(-a/(a + b) + b/(a + b))*log(sqrt(-a/(a + b) + b/(a + b)) + tan(x/
2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a + b))) - a*sqrt(-a/(a + b) + b/(a + b))*lo
g(tan(x/2)**2 + 1)/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a + b))) + b**2*log(-sqrt(-a/
(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a + b))) - b**2
*log(sqrt(-a/(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a
+ b))) + b*c*log(-sqrt(-a/(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a
+ b) + b/(a + b))) - b*c*log(sqrt(-a/(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**
2*sqrt(-a/(a + b) + b/(a + b))) + b*sqrt(-a/(a + b) + b/(a + b))*log(-sqrt(-a/(a + b) + b/(a + b)) + tan(x/2))
/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a + b))) + b*sqrt(-a/(a + b) + b/(a + b))*log(s
qrt(-a/(a + b) + b/(a + b)) + tan(x/2))/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b) + b/(a + b)))
- b*sqrt(-a/(a + b) + b/(a + b))*log(tan(x/2)**2 + 1)/(a*b*sqrt(-a/(a + b) + b/(a + b)) + b**2*sqrt(-a/(a + b
) + b/(a + b))), True))

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