\(\int \frac {\cos (a+b x)}{(c+d x)^2} \, dx\) [6]

Optimal result

Integrand size = 14, antiderivative size = 73 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {\cos (a+b x)}{d (c+d x)}-\frac {b \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right ) \sin \left (a-\frac {b c}{d}\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3378, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {b \sin \left (a-\frac {b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (\frac {b c}{d}+b x\right )}{d^2}-\frac {\cos (a+b x)}{d (c+d x)} \]

[In]

Int[Cos[a + b*x]/(c + d*x)^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.89 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {\frac {d \cos (a+b x)}{c+d x}+b \operatorname {CosIntegral}\left (b \left (\frac {c}{d}+x\right )\right ) \sin \left (a-\frac {b c}{d}\right )+b \cos \left (a-\frac {b c}{d}\right ) \text {Si}\left (b \left (\frac {c}{d}+x\right )\right )}{d^2} \]

[In]

Integrate[Cos[a + b*x]/(c + d*x)^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.77 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.56

method	result	size
derivativedivides	\(b \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )\)	\(114\)
default	\(b \left (-\frac {\cos \left (b x +a \right )}{\left (-a d +b c +d \left (b x +a \right )\right ) d}-\frac {-\frac {\operatorname {Si}\left (-b x -a -\frac {-a d +b c}{d}\right ) \cos \left (\frac {-a d +b c}{d}\right )}{d}-\frac {\operatorname {Ci}\left (b x +a +\frac {-a d +b c}{d}\right ) \sin \left (\frac {-a d +b c}{d}\right )}{d}}{d}\right )\)	\(114\)
risch	\(\frac {i b \,{\mathrm e}^{-\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (i x b +i a -\frac {i \left (a d -b c \right )}{d}\right )}{2 d^{2}}-\frac {i b \,{\mathrm e}^{\frac {i \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (-i x b -i a -\frac {-i a d +i b c}{d}\right )}{2 d^{2}}-\frac {\left (-2 b x d -2 b c \right ) \cos \left (b x +a \right )}{2 d \left (d x +c \right ) \left (-b x d -b c \right )}\)	\(140\)

[In]

int(cos(b*x+a)/(d*x+c)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.32 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left (b d x + b c\right )} \operatorname {Ci}\left (\frac {b d x + b c}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + {\left (b d x + b c\right )} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (\frac {b d x + b c}{d}\right ) + d \cos \left (b x + a\right )}{d^{3} x + c d^{2}} \]

[In]

integrate(cos(b*x+a)/(d*x+c)^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos {\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \]

[In]

integrate(cos(b*x+a)/(d*x+c)**2,x)

[Out]

Integral(cos(a + b*x)/(c + d*x)**2, x)

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.25 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {b^{2} {\left (E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \cos \left (-\frac {b c - a d}{d}\right ) + b^{2} {\left (-i \, E_{2}\left (\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right ) + i \, E_{2}\left (-\frac {i \, b c + i \, {\left (b x + a\right )} d - i \, a d}{d}\right )\right )} \sin \left (-\frac {b c - a d}{d}\right )}{2 \, {\left (b c d + {\left (b x + a\right )} d^{2} - a d^{2}\right )} b} \]

[In]

integrate(cos(b*x+a)/(d*x+c)^2,x, algorithm="maxima")

[Out]

-1/2*(b^2*(exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + exp_integral_e(2, -(I*b*c + I*(b*x + a)*d -
I*a*d)/d))*cos(-(b*c - a*d)/d) + b^2*(-I*exp_integral_e(2, (I*b*c + I*(b*x + a)*d - I*a*d)/d) + I*exp_integral
_e(2, -(I*b*c + I*(b*x + a)*d - I*a*d)/d))*sin(-(b*c - a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 523 vs. \(2 (73) = 146\).

Time = 0.31 (sec) , antiderivative size = 523, normalized size of antiderivative = 7.16 \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=-\frac {{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) + b^{3} c \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - a b^{2} d \operatorname {Ci}\left (\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) \sin \left (-\frac {b c - a d}{d}\right ) - {\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} b^{2} \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) - b^{3} c \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + a b^{2} d \cos \left (-\frac {b c - a d}{d}\right ) \operatorname {Si}\left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} + b c - a d}{d}\right ) + b^{2} d \cos \left (-\frac {{\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )}}{d}\right )\right )} d^{2}}{{\left ({\left (d x + c\right )} {\left (b - \frac {b c}{d x + c} + \frac {a d}{d x + c}\right )} d^{4} + b c d^{4} - a d^{5}\right )} b} \]

[In]

integrate(cos(b*x+a)/(d*x+c)^2,x, algorithm="giac")

[Out]

-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos_integral(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)
) + b*c - a*d)/d)*sin(-(b*c - a*d)/d) + b^3*c*cos_integral(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*
c - a*d)/d)*sin(-(b*c - a*d)/d) - a*b^2*d*cos_integral(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c -
a*d)/d)*sin(-(b*c - a*d)/d) - (d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*b^2*cos(-(b*c - a*d)/d)*sin_integr
al(-((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) - b^3*c*cos(-(b*c - a*d)/d)*sin_integral(-(
(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) + a*b^2*d*cos(-(b*c - a*d)/d)*sin_integral(-((d*
x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c)) + b*c - a*d)/d) + b^2*d*cos(-(d*x + c)*(b - b*c/(d*x + c) + a*d/(d*
x + c))/d))*d^2/(((d*x + c)*(b - b*c/(d*x + c) + a*d/(d*x + c))*d^4 + b*c*d^4 - a*d^5)*b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos (a+b x)}{(c+d x)^2} \, dx=\int \frac {\cos \left (a+b\,x\right )}{{\left (c+d\,x\right )}^2} \,d x \]

[In]

int(cos(a + b*x)/(c + d*x)^2,x)

[Out]

int(cos(a + b*x)/(c + d*x)^2, x)