3.31 \(\int \frac {\sin (c+d x)}{x (a+b x)^2} \, dx\)

Optimal. Leaf size=149 \[ -\frac {\sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\sin (c) \text {Ci}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}-\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\sin (c+d x)}{a (a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.41, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3303, 3299, 3302, 3297} \[ -\frac {\sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}+\frac {\sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {\cos (c) \text {Si}(d x)}{a^2}-\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a b}+\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a b}+\frac {\sin (c+d x)}{a (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x*(a + b*x)^2),x]

[Out]

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

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 6742

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.07, size = 138, normalized size = 0.93 \[ \frac {-\frac {\text {Ci}\left (d \left (\frac {a}{b}+x\right )\right ) \left (b \sin \left (c-\frac {a d}{b}\right )+a d \cos \left (c-\frac {a d}{b}\right )\right )}{b}+\frac {a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )}{b}-\cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {a \sin (c) \cos (d x)}{a+b x}+\frac {a \cos (c) \sin (d x)}{a+b x}+\sin (c) \text {Ci}(d x)+\cos (c) \text {Si}(d x)}{a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(x*(a + b*x)^2),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.70, size = 260, normalized size = 1.74 \[ \frac {2 \, a b \sin \left (d x + c\right ) + 2 \, {\left (b^{2} x + a b\right )} \cos \relax (c) \operatorname {Si}\left (d x\right ) - {\left ({\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b d x + a^{2} d\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 2 \, {\left (b^{2} x + a b\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + {\left ({\left (b^{2} x + a b\right )} \operatorname {Ci}\left (d x\right ) + {\left (b^{2} x + a b\right )} \operatorname {Ci}\left (-d x\right )\right )} \sin \relax (c) + {\left ({\left (b^{2} x + a b\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} x + a b\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - 2 \, {\left (a b d x + a^{2} d\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{2} b^{2} x + a^{3} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*a*b*sin(d*x + c) + 2*(b^2*x + a*b)*cos(c)*sin_integral(d*x) - ((a*b*d*x + a^2*d)*cos_integral((b*d*x +
a*d)/b) + (a*b*d*x + a^2*d)*cos_integral(-(b*d*x + a*d)/b) + 2*(b^2*x + a*b)*sin_integral((b*d*x + a*d)/b))*co
s(-(b*c - a*d)/b) + ((b^2*x + a*b)*cos_integral(d*x) + (b^2*x + a*b)*cos_integral(-d*x))*sin(c) + ((b^2*x + a*
b)*cos_integral((b*d*x + a*d)/b) + (b^2*x + a*b)*cos_integral(-(b*d*x + a*d)/b) - 2*(a*b*d*x + a^2*d)*sin_inte
gral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(a^2*b^2*x + a^3*b)

________________________________________________________________________________________

giac [B]  time = 0.79, size = 1281, normalized size = 8.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x +
a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a*b*c*d^2*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a)
- a*d/(b*x + a) + d) - b*c + a*d)/b) + a^2*d^3*cos(-(b*c - a*d)/b)*cos_integral(((b*x + a)*(b*c/(b*x + a) - a
*d/(b*x + a) + d) - b*c + a*d)/b) + (b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*sin(-(b*c - a*d)/b)*si
n_integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - a*b*c*d^2*sin(-(b*c - a*d)/b)*sin_
integral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a^2*d^3*sin(-(b*c - a*d)/b)*sin_inte
gral(((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x +
a) + d)*d*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) + b^2*c*d*cos_integral((b*x
+ a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) - a*b*d^2*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/
(b*x + a) + d)/b - c)*sin(c) - (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*cos_integral(((b*x + a)*(b*c/
(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + b^2*c*d*cos_integral(((b*x + a)*(b*c/(b*x
+ a) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) - a*b*d^2*cos_integral(((b*x + a)*(b*c/(b*x + a
) - a*d/(b*x + a) + d) - b*c + a*d)/b)*sin(-(b*c - a*d)/b) + (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d
*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) - b^2*c*d*cos(c)*sin_integral(-(b*x
+ a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) + a*b*d^2*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d
/(b*x + a) + d)/b + c) + (b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d*cos(-(b*c - a*d)/b)*sin_integral(((
b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) - b^2*c*d*cos(-(b*c - a*d)/b)*sin_integral(((b*x
+ a)*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a*b*d^2*cos(-(b*c - a*d)/b)*sin_integral(((b*x + a)
*(b*c/(b*x + a) - a*d/(b*x + a) + d) - b*c + a*d)/b) + a*b*d^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) +
d)/b))*b^3/(((b*x + a)*a^2*b^4*(b*c/(b*x + a) - a*d/(b*x + a) + d) - a^2*b^5*c + a^3*b^4*d)*d)

________________________________________________________________________________________

maple [A]  time = 0.03, size = 210, normalized size = 1.41 \[ -\frac {b \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{a^{2}}+\frac {\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )}{a^{2}}-\frac {d b \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x/(b*x+a)^2,x)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x + a\right )}^{2} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )}{x\,{\left (a+b\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x*(a + b*x)^2),x)

[Out]

int(sin(c + d*x)/(x*(a + b*x)^2), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (c + d x \right )}}{x \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________