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

Optimal. Leaf size=101 \[ -\frac {\sin \left (\frac {b}{d}\right ) (b c-a d) \text {Ci}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {\cos \left (\frac {b}{d}\right ) (b c-a d) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cos \left (\frac {a+b x}{c+d x}\right )}{d} \]

[Out]

(d*x+c)*cos((b*x+a)/(d*x+c))/d+(-a*d+b*c)*cos(b/d)*Si((-a*d+b*c)/d/(d*x+c))/d^2-(-a*d+b*c)*Ci((-a*d+b*c)/d/(d*
x+c))*sin(b/d)/d^2

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4564, 3297, 3303, 3299, 3302} \[ -\frac {\sin \left (\frac {b}{d}\right ) (b c-a d) \text {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {\cos \left (\frac {b}{d}\right ) (b c-a d) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cos \left (\frac {a+b x}{c+d x}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[(a + b*x)/(c + d*x)],x]

[Out]

((c + d*x)*Cos[(a + b*x)/(c + d*x)])/d - ((b*c - a*d)*CosIntegral[(b*c - a*d)/(d*(c + d*x))]*Sin[b/d])/d^2 + (
(b*c - a*d)*Cos[b/d]*SinIntegral[(b*c - a*d)/(d*(c + d*x))])/d^2

Rule 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 3302

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosIntegral[e - Pi/2 + f*x]/d, x] /; FreeQ
[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]

Rule 3303

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

Rule 4564

Int[Cos[((e_.)*((a_.) + (b_.)*(x_)))/((c_.) + (d_.)*(x_))]^(n_.), x_Symbol] :> -Dist[d^(-1), Subst[Int[Cos[(b*
e)/d - (e*(b*c - a*d)*x)/d]^n/x^2, x], x, 1/(c + d*x)], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && NeQ[b*c
- a*d, 0]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 5.51, size = 260, normalized size = 2.57 \[ \frac {-4 \sin \left (\frac {b}{d}\right ) (b c-a d) \text {Ci}\left (\frac {a d-b c}{d (c+d x)}\right )+d \exp \left (-\frac {i (a d+2 b c+b d x)}{d (c+d x)}\right ) \left (2 c \left (e^{2 i \left (\frac {a}{c+d x}+\frac {b}{d}\right )}+e^{\frac {2 i b c}{d (c+d x)}}\right )+d x \left (1+e^{\frac {2 i b}{d}}\right ) \left (e^{\frac {2 i a}{c+d x}}+e^{\frac {2 i b c}{d (c+d x)}}\right )-4 d x \sin \left (\frac {b}{d}\right ) e^{\frac {i (a d+2 b c+b d x)}{d (c+d x)}} \sin \left (\frac {a d-b c}{d (c+d x)}\right )\right )-4 \cos \left (\frac {b}{d}\right ) (b c-a d) \text {Si}\left (\frac {a d-b c}{d (c+d x)}\right )}{4 d^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Cos[(a + b*x)/(c + d*x)],x]

[Out]

(-4*(b*c - a*d)*CosIntegral[(-(b*c) + a*d)/(d*(c + d*x))]*Sin[b/d] + (d*(2*c*(E^(((2*I)*b*c)/(d*(c + d*x))) +
E^((2*I)*(b/d + a/(c + d*x)))) + d*(1 + E^(((2*I)*b)/d))*(E^(((2*I)*a)/(c + d*x)) + E^(((2*I)*b*c)/(d*(c + d*x
))))*x - 4*d*E^((I*(2*b*c + a*d + b*d*x))/(d*(c + d*x)))*x*Sin[b/d]*Sin[(-(b*c) + a*d)/(d*(c + d*x))]))/E^((I*
(2*b*c + a*d + b*d*x))/(d*(c + d*x))) - 4*(b*c - a*d)*Cos[b/d]*SinIntegral[(-(b*c) + a*d)/(d*(c + d*x))])/(4*d
^2)

________________________________________________________________________________________

fricas [A]  time = 4.12, size = 138, normalized size = 1.37 \[ -\frac {2 \, {\left (b c - a d\right )} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) - 2 \, {\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right ) + {\left ({\left (b c - a d\right )} \operatorname {Ci}\left (\frac {b c - a d}{d^{2} x + c d}\right ) + {\left (b c - a d\right )} \operatorname {Ci}\left (-\frac {b c - a d}{d^{2} x + c d}\right )\right )} \sin \left (\frac {b}{d}\right )}{2 \, d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [B]  time = 12.27, size = 633, normalized size = 6.27 \[ -\frac {{\left (b^{3} c^{2} \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right ) - 2 \, a b^{2} c d \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right ) - \frac {{\left (b x + a\right )} b^{2} c^{2} d \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right )}{d x + c} + a^{2} b d^{2} \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right ) + \frac {2 \, {\left (b x + a\right )} a b c d^{2} \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right )}{d x + c} - \frac {{\left (b x + a\right )} a^{2} d^{3} \operatorname {Ci}\left (-\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) \sin \left (\frac {b}{d}\right )}{d x + c} - b^{3} c^{2} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) + 2 \, a b^{2} c d \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) + \frac {{\left (b x + a\right )} b^{2} c^{2} d \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right )}{d x + c} - a^{2} b d^{2} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right ) - \frac {2 \, {\left (b x + a\right )} a b c d^{2} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right )}{d x + c} + \frac {{\left (b x + a\right )} a^{2} d^{3} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (\frac {b - \frac {{\left (b x + a\right )} d}{d x + c}}{d}\right )}{d x + c} - b^{2} c^{2} d \cos \left (\frac {b x + a}{d x + c}\right ) + 2 \, a b c d^{2} \cos \left (\frac {b x + a}{d x + c}\right ) - a^{2} d^{3} \cos \left (\frac {b x + a}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )}}{b d^{2} - \frac {{\left (b x + a\right )} d^{3}}{d x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b^3*c^2*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)*sin(b/d) - 2*a*b^2*c*d*cos_integral(-(b - (b*x + a)*d/(
d*x + c))/d)*sin(b/d) - (b*x + a)*b^2*c^2*d*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)*sin(b/d)/(d*x + c) +
a^2*b*d^2*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)*sin(b/d) + 2*(b*x + a)*a*b*c*d^2*cos_integral(-(b - (b*
x + a)*d/(d*x + c))/d)*sin(b/d)/(d*x + c) - (b*x + a)*a^2*d^3*cos_integral(-(b - (b*x + a)*d/(d*x + c))/d)*sin
(b/d)/(d*x + c) - b^3*c^2*cos(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d) + 2*a*b^2*c*d*cos(b/d)*sin_inte
gral((b - (b*x + a)*d/(d*x + c))/d) + (b*x + a)*b^2*c^2*d*cos(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d)
/(d*x + c) - a^2*b*d^2*cos(b/d)*sin_integral((b - (b*x + a)*d/(d*x + c))/d) - 2*(b*x + a)*a*b*c*d^2*cos(b/d)*s
in_integral((b - (b*x + a)*d/(d*x + c))/d)/(d*x + c) + (b*x + a)*a^2*d^3*cos(b/d)*sin_integral((b - (b*x + a)*
d/(d*x + c))/d)/(d*x + c) - b^2*c^2*d*cos((b*x + a)/(d*x + c)) + 2*a*b*c*d^2*cos((b*x + a)/(d*x + c)) - a^2*d^
3*cos((b*x + a)/(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)/(b*d^2 - (b*x + a)*d^3/(d*x + c))

________________________________________________________________________________________

maple [A]  time = 0.10, size = 142, normalized size = 1.41 \[ -\left (d a -c b \right ) \left (-\frac {\cos \left (\frac {b}{d}+\frac {d a -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {d a -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {\frac {\Si \left (\frac {d a -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}+\frac {\Ci \left (\frac {d a -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}}{d}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*d-b*c)*(-cos(b/d+(a*d-b*c)/d/(d*x+c))/((b/d+(a*d-b*c)/d/(d*x+c))*d-b)/d-(Si((a*d-b*c)/d/(d*x+c))*cos(b/d)/
d+Ci((a*d-b*c)/d/(d*x+c))*sin(b/d)/d)/d)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos \left (\frac {b x + a}{d x + c}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \cos \left (\frac {a+b\,x}{c+d\,x}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos((a + b*x)/(c + d*x)),x)

[Out]

int(cos((a + b*x)/(c + d*x)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\left (\frac {a + b x}{c + d x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(cos((a + b*x)/(c + d*x)), x)

________________________________________________________________________________________