3.453 \(\int \frac {\cos (c+d x)}{a+b \cos (c+d x)} \, dx\)

Optimal. Leaf size=59 \[ \frac {x}{b}-\frac {2 a \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2735, 2659, 205} \[ \frac {x}{b}-\frac {2 a \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b d \sqrt {a-b} \sqrt {a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2735

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*x)/d
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 58, normalized size = 0.98 \[ \frac {\frac {2 a \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {b^2-a^2}}\right )}{\sqrt {b^2-a^2}}+c+d x}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.23, size = 223, normalized size = 3.78 \[ \left [\frac {2 \, {\left (a^{2} - b^{2}\right )} d x - \sqrt {-a^{2} + b^{2}} a \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right )}{2 \, {\left (a^{2} b - b^{3}\right )} d}, \frac {{\left (a^{2} - b^{2}\right )} d x - \sqrt {a^{2} - b^{2}} a \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right )}{{\left (a^{2} b - b^{3}\right )} d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 0.55, size = 240, normalized size = 4.07 \[ -\frac {\frac {{\left (\sqrt {a^{2} - b^{2}} {\left (2 \, a - b\right )} {\left | a - b \right |} + \sqrt {a^{2} - b^{2}} {\left | a - b \right |} {\left | b \right |}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {\frac {2 \, a + \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )}}{{\left (a^{2} - 2 \, a b + b^{2}\right )} b^{2} + {\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} {\left | b \right |}} + \frac {{\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor + \arctan \left (\frac {2 \, \sqrt {\frac {1}{2}} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {\frac {2 \, a - \sqrt {-4 \, {\left (a + b\right )} {\left (a - b\right )} + 4 \, a^{2}}}{a - b}}}\right )\right )} {\left (2 \, a - b - {\left | b \right |}\right )}}{b^{2} - a {\left | b \right |}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((sqrt(a^2 - b^2)*(2*a - b)*abs(a - b) + sqrt(a^2 - b^2)*abs(a - b)*abs(b))*(pi*floor(1/2*(d*x + c)/pi + 1/2)
+ arctan(2*sqrt(1/2)*tan(1/2*d*x + 1/2*c)/sqrt((2*a + sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))))/((a^2 - 2*
a*b + b^2)*b^2 + (a^3 - 2*a^2*b + a*b^2)*abs(b)) + (pi*floor(1/2*(d*x + c)/pi + 1/2) + arctan(2*sqrt(1/2)*tan(
1/2*d*x + 1/2*c)/sqrt((2*a - sqrt(-4*(a + b)*(a - b) + 4*a^2))/(a - b))))*(2*a - b - abs(b))/(b^2 - a*abs(b)))
/d

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maple [A]  time = 0.05, size = 67, normalized size = 1.14 \[ -\frac {2 a \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{d b \sqrt {\left (a -b \right ) \left (a +b \right )}}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(cos(d*x+c)/(a+b*cos(d*x+c)),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 = 0.78, size = 99, normalized size = 1.68 \[ \frac {2\,\mathrm {atan}\left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{b\,d}+\frac {2\,a\,\mathrm {atanh}\left (\frac {a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )-b\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\sqrt {b^2-a^2}}\right )}{b\,d\,\sqrt {b^2-a^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atan(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/(b*d) + (2*a*atanh((a*sin(c/2 + (d*x)/2) - b*sin(c/2 + (d*x)/2
))/(cos(c/2 + (d*x)/2)*(b^2 - a^2)^(1/2))))/(b*d*(b^2 - a^2)^(1/2))

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sympy [A]  time = 24.70, size = 320, normalized size = 5.42 \[ \begin {cases} \tilde {\infty } x & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac {x}{b} - \frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{b d} & \text {for}\: a = b \\\frac {x}{b} + \frac {1}{b d \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {for}\: a = - b \\\frac {\sin {\left (c + d x \right )}}{a d} & \text {for}\: b = 0 \\\frac {x \cos {\relax (c )}}{a + b \cos {\relax (c )}} & \text {for}\: d = 0 \\\frac {a d x \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}}{a b d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {a \log {\left (- \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{a b d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} + \frac {a \log {\left (\sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} + \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )}}{a b d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} - \frac {b d x \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}}{a b d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}} - b^{2} d \sqrt {- \frac {a}{a - b} - \frac {b}{a - b}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo*x, Eq(a, 0) & Eq(b, 0) & Eq(d, 0)), (x/b - tan(c/2 + d*x/2)/(b*d), Eq(a, b)), (x/b + 1/(b*d*tan
(c/2 + d*x/2)), Eq(a, -b)), (sin(c + d*x)/(a*d), Eq(b, 0)), (x*cos(c)/(a + b*cos(c)), Eq(d, 0)), (a*d*x*sqrt(-
a/(a - b) - b/(a - b))/(a*b*d*sqrt(-a/(a - b) - b/(a - b)) - b**2*d*sqrt(-a/(a - b) - b/(a - b))) - a*log(-sqr
t(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b*d*sqrt(-a/(a - b) - b/(a - b)) - b**2*d*sqrt(-a/(a - b) - b
/(a - b))) + a*log(sqrt(-a/(a - b) - b/(a - b)) + tan(c/2 + d*x/2))/(a*b*d*sqrt(-a/(a - b) - b/(a - b)) - b**2
*d*sqrt(-a/(a - b) - b/(a - b))) - b*d*x*sqrt(-a/(a - b) - b/(a - b))/(a*b*d*sqrt(-a/(a - b) - b/(a - b)) - b*
*2*d*sqrt(-a/(a - b) - b/(a - b))), True))

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