\(\int \cos (\frac {a+b x}{c+d x}) \, dx\) [51]

Optimal result

Integrand size = 14, antiderivative size = 101 \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {(c+d x) \cos \left (\frac {a+b x}{c+d x}\right )}{d}-\frac {(b c-a d) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right ) \sin \left (\frac {b}{d}\right )}{d^2}+\frac {(b c-a d) \cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2} \]

[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] (verified)

Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4660, 3378, 3384, 3380, 3383} \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {\sin \left (\frac {b}{d}\right ) (b c-a d) \operatorname {CosIntegral}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {\cos \left (\frac {b}{d}\right ) (b c-a d) \text {Si}\left (\frac {b c-a d}{d (c+d x)}\right )}{d^2}+\frac {(c+d x) \cos \left (\frac {a+b x}{c+d x}\right )}{d} \]

[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 3378

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 3380

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 3383

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 3384

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 4660

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*} \text {integral}& = -\frac {\text {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) \text {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 ) \text {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 ) \text {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) \operatorname {CosIntegral}\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] (verified)

Result contains complex when optimal does not.

Time = 0.79 (sec) , antiderivative size = 319, normalized size of antiderivative = 3.16 \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=\frac {c d e^{-\frac {i (a+b x)}{c+d x}}+c d e^{\frac {i (a+b x)}{c+d x}}+2 d^2 x \cos \left (\frac {b}{d}\right ) \cos \left (\frac {-b c+a d}{d (c+d x)}\right )-2 d^2 x \sin \left (\frac {b}{d}\right ) \sin \left (\frac {-b c+a d}{d (c+d x)}\right )-(b c-a d) \left (\operatorname {CosIntegral}\left (\frac {b c-a d}{c d+d^2 x}\right ) \left (-i \cos \left (\frac {b}{d}\right )+\sin \left (\frac {b}{d}\right )\right )+\operatorname {CosIntegral}\left (\frac {-b c+a d}{d (c+d x)}\right ) \left (i \cos \left (\frac {b}{d}\right )+\sin \left (\frac {b}{d}\right )\right )+\cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )+i \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {-b c+a d}{d (c+d x)}\right )-\cos \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )+i \sin \left (\frac {b}{d}\right ) \text {Si}\left (\frac {b c-a d}{c d+d^2 x}\right )\right )}{2 d^2} \]

[In]

Integrate[Cos[(a + b*x)/(c + d*x)],x]

[Out]

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

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.41

method	result	size
derivativedivides	\(-\left (a d -c b \right ) \left (-\frac {\cos \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}}{d}\right )\)	\(142\)
default	\(-\left (a d -c b \right ) \left (-\frac {\cos \left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right )}{\left (\left (\frac {b}{d}+\frac {a d -c b}{d \left (d x +c \right )}\right ) d -b \right ) d}-\frac {\frac {\operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \cos \left (\frac {b}{d}\right )}{d}+\frac {\operatorname {Ci}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) \sin \left (\frac {b}{d}\right )}{d}}{d}\right )\)	\(142\)
risch	\(\frac {i \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} a}{2 d}-\frac {i \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) {\mathrm e}^{\frac {i b}{d}} c b}{2 d^{2}}-\frac {{\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {{\mathrm e}^{-\frac {i b}{d}} \pi \,\operatorname {csgn}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\frac {{\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) a}{d}-\frac {{\mathrm e}^{-\frac {i b}{d}} \operatorname {Si}\left (\frac {a d -c b}{d \left (d x +c \right )}\right ) b c}{d^{2}}-\frac {i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) a}{2 d}+\frac {i {\mathrm e}^{-\frac {i b}{d}} \operatorname {Ei}_{1}\left (-\frac {i \left (a d -c b \right )}{d \left (d x +c \right )}\right ) b c}{2 d^{2}}+\cos \left (\frac {x b +a}{d x +c}\right ) x +\frac {\cos \left (\frac {x b +a}{d x +c}\right ) c}{d}\)	\(330\)

[In]

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

[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)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04 \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=-\frac {{\left (b c - a d\right )} \operatorname {Ci}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) \sin \left (\frac {b}{d}\right ) + {\left (b c - a d\right )} \cos \left (\frac {b}{d}\right ) \operatorname {Si}\left (-\frac {b c - a d}{d^{2} x + c d}\right ) - {\left (d^{2} x + c d\right )} \cos \left (\frac {b x + a}{d x + c}\right )}{d^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=\int \cos {\left (\frac {a + b x}{c + d x} \right )}\, dx \]

[In]

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

[Out]

Integral(cos((a + b*x)/(c + d*x)), x)

Maxima [F]

\[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=\int { \cos \left (\frac {b x + a}{d x + c}\right ) \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 633 vs. \(2 (101) = 202\).

Time = 5.26 (sec) , antiderivative size = 633, normalized size of antiderivative = 6.27 \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=-\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}} \]

[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))

Mupad [F(-1)]

Timed out. \[ \int \cos \left (\frac {a+b x}{c+d x}\right ) \, dx=\int \cos \left (\frac {a+b\,x}{c+d\,x}\right ) \,d x \]

[In]

int(cos((a + b*x)/(c + d*x)),x)

[Out]

int(cos((a + b*x)/(c + d*x)), x)