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

Optimal. Leaf size=188 \[ -\frac {2 b \sin (c) \text {Ci}(d x)}{a^3}+\frac {2 b \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \cos (c) \text {Ci}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x} \]

[Out]

d*Ci(d*x)*cos(c)/a^2+d*Ci(a*d/b+d*x)*cos(-c+a*d/b)/a^2-2*b*cos(c)*Si(d*x)/a^3+2*b*cos(-c+a*d/b)*Si(a*d/b+d*x)/
a^3-2*b*Ci(d*x)*sin(c)/a^3-d*Si(d*x)*sin(c)/a^2-2*b*Ci(a*d/b+d*x)*sin(-c+a*d/b)/a^3+d*Si(a*d/b+d*x)*sin(-c+a*d
/b)/a^2-sin(d*x+c)/a^2/x-b*sin(d*x+c)/a^2/(b*x+a)

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Rubi [A]  time = 0.51, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6742, 3297, 3303, 3299, 3302} \[ -\frac {2 b \sin (c) \text {CosIntegral}(d x)}{a^3}+\frac {2 b \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^2}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{a^3}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \cos (c) \text {CosIntegral}(d x)}{a^2}-\frac {d \sin (c) \text {Si}(d x)}{a^2}-\frac {\sin (c+d x)}{a^2 x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^2*(a + b*x)^2),x]

[Out]

(d*Cos[c]*CosIntegral[d*x])/a^2 + (d*Cos[c - (a*d)/b]*CosIntegral[(a*d)/b + d*x])/a^2 - (2*b*CosIntegral[d*x]*
Sin[c])/a^3 + (2*b*CosIntegral[(a*d)/b + d*x]*Sin[c - (a*d)/b])/a^3 - Sin[c + d*x]/(a^2*x) - (b*Sin[c + d*x])/
(a^2*(a + b*x)) - (2*b*Cos[c]*SinIntegral[d*x])/a^3 - (d*Sin[c]*SinIntegral[d*x])/a^2 + (2*b*Cos[c - (a*d)/b]*
SinIntegral[(a*d)/b + d*x])/a^3 - (d*Sin[c - (a*d)/b]*SinIntegral[(a*d)/b + d*x])/a^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 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^2 (a+b x)^2} \, dx &=\int \left (\frac {\sin (c+d x)}{a^2 x^2}-\frac {2 b \sin (c+d x)}{a^3 x}+\frac {b^2 \sin (c+d x)}{a^2 (a+b x)^2}+\frac {2 b^2 \sin (c+d x)}{a^3 (a+b x)}\right ) \, dx\\ &=\frac {\int \frac {\sin (c+d x)}{x^2} \, dx}{a^2}-\frac {(2 b) \int \frac {\sin (c+d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2\right ) \int \frac {\sin (c+d x)}{a+b x} \, dx}{a^3}+\frac {b^2 \int \frac {\sin (c+d x)}{(a+b x)^2} \, dx}{a^2}\\ &=-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}+\frac {d \int \frac {\cos (c+d x)}{x} \, dx}{a^2}+\frac {(b d) \int \frac {\cos (c+d x)}{a+b x} \, dx}{a^2}-\frac {(2 b \cos (c)) \int \frac {\sin (d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2 \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^3}-\frac {(2 b \sin (c)) \int \frac {\cos (d x)}{x} \, dx}{a^3}+\frac {\left (2 b^2 \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^3}\\ &=-\frac {2 b \text {Ci}(d x) \sin (c)}{a^3}+\frac {2 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}+\frac {(d \cos (c)) \int \frac {\cos (d x)}{x} \, dx}{a^2}+\frac {\left (b 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^2}-\frac {(d \sin (c)) \int \frac {\sin (d x)}{x} \, dx}{a^2}-\frac {\left (b 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^2}\\ &=\frac {d \cos (c) \text {Ci}(d x)}{a^2}+\frac {d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (\frac {a d}{b}+d x\right )}{a^2}-\frac {2 b \text {Ci}(d x) \sin (c)}{a^3}+\frac {2 b \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{a^3}-\frac {\sin (c+d x)}{a^2 x}-\frac {b \sin (c+d x)}{a^2 (a+b x)}-\frac {2 b \cos (c) \text {Si}(d x)}{a^3}-\frac {d \sin (c) \text {Si}(d x)}{a^2}+\frac {2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^3}-\frac {d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{a^2}\\ \end {align*}

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Mathematica [A]  time = 2.05, size = 184, normalized size = 0.98 \[ -\frac {-2 b \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )-a d \cos \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+a d \sin \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )-2 b \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {a \sin (c) (a+2 b x) \cos (d x)}{x (a+b x)}+\frac {a \cos (c) (a+2 b x) \sin (d x)}{x (a+b x)}-a d \cos (c) \text {Ci}(d x)+a d \sin (c) \text {Si}(d x)+2 b \sin (c) \text {Ci}(d x)+2 b \cos (c) \text {Si}(d x)}{a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-(a*d*Cos[c]*CosIntegral[d*x]) - a*d*Cos[c - (a*d)/b]*CosIntegral[d*(a/b + x)] + (a*(a + 2*b*x)*Cos[d*x]*Si
n[c])/(x*(a + b*x)) + 2*b*CosIntegral[d*x]*Sin[c] - 2*b*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (a*(a + 2*
b*x)*Cos[c]*Sin[d*x])/(x*(a + b*x)) + 2*b*Cos[c]*SinIntegral[d*x] + a*d*Sin[c]*SinIntegral[d*x] - 2*b*Cos[c -
(a*d)/b]*SinIntegral[d*(a/b + x)] + a*d*Sin[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/a^3)

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fricas [A]  time = 0.77, size = 355, normalized size = 1.89 \[ \frac {{\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (d x\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (-d x\right ) - 4 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (d x\right )\right )} \cos \relax (c) + {\left ({\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) + 4 \, {\left (b^{2} x^{2} + a b x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) - 2 \, {\left (2 \, a b x + a^{2}\right )} \sin \left (d x + c\right ) - 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (d x\right ) + {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (-d x\right ) + {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (d x\right )\right )} \sin \relax (c) - 2 \, {\left ({\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} x^{2} + a b x\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right ) - {\left (a b d x^{2} + a^{2} d x\right )} \operatorname {Si}\left (\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left (a^{3} b x^{2} + a^{4} x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(((a*b*d*x^2 + a^2*d*x)*cos_integral(d*x) + (a*b*d*x^2 + a^2*d*x)*cos_integral(-d*x) - 4*(b^2*x^2 + a*b*x)
*sin_integral(d*x))*cos(c) + ((a*b*d*x^2 + a^2*d*x)*cos_integral((b*d*x + a*d)/b) + (a*b*d*x^2 + a^2*d*x)*cos_
integral(-(b*d*x + a*d)/b) + 4*(b^2*x^2 + a*b*x)*sin_integral((b*d*x + a*d)/b))*cos(-(b*c - a*d)/b) - 2*(2*a*b
*x + a^2)*sin(d*x + c) - 2*((b^2*x^2 + a*b*x)*cos_integral(d*x) + (b^2*x^2 + a*b*x)*cos_integral(-d*x) + (a*b*
d*x^2 + a^2*d*x)*sin_integral(d*x))*sin(c) - 2*((b^2*x^2 + a*b*x)*cos_integral((b*d*x + a*d)/b) + (b^2*x^2 + a
*b*x)*cos_integral(-(b*d*x + a*d)/b) - (a*b*d*x^2 + a^2*d*x)*sin_integral((b*d*x + a*d)/b))*sin(-(b*c - a*d)/b
))/(a^3*b*x^2 + a^4*x)

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giac [B]  time = 1.74, size = 3180, normalized size = 16.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*cos(c)*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b
*x + a) + d)/b - c)/b - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d^2*cos(c)*cos_integral((b*x + a)*
(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c) + a*b*c^2*d^2*cos(c)*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b
*x + a) + d)/b - c) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*cos(c)*cos_integral((b*x + a)*(b*c
/(b*x + a) - a*d/(b*x + a) + d)/b - c)/b - a^2*c*d^3*cos(c)*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x +
a) + d)/b - c) + (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*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)/b - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) +
d)*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*
b*c^2*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) + (b
*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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 - a^2*c*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)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d^2*sin(c)*sin_int
egral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)/b - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) +
d)*c*d^2*sin(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) + a*b*c^2*d^2*sin(c)*sin_i
ntegral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) +
d)*d^3*sin(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c)/b - a^2*c*d^3*sin(c)*sin_int
egral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b + c) + (b*x + a)^2*a*(b*c/(b*x + a) - a*d/(b*x + a) + d
)^2*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)/b - 2*
(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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*b*c^2*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) + (b*x + a)*a^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^3*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)/b - a^2*c*d^3*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) - 2*(b*x + a)^2*(b*c/(b*x +
a) - a*d/(b*x + a) + d)^2*d*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) + 4*(b*x
+ a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b -
c)*sin(c) - 2*b^2*c^2*d*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) - 2*(b*x + a)
*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*s
in(c) + 2*a*b*c*d^2*cos_integral((b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b - c)*sin(c) - 2*(b*x + a)^2*(
b*c/(b*x + a) - a*d/(b*x + a) + d)^2*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) + 4*(b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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) - 2*b^2*c^2*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) - 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x +
a) + d)*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) +
2*a*b*c*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) +
2*(b*x + a)^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*d*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*
x + a) + d)/b + c) - 4*(b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c*d*cos(c)*sin_integral(-(b*x + a)*(b*c
/(b*x + a) - a*d/(b*x + a) + d)/b + c) + 2*b^2*c^2*d*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x
+ a) + d)/b + c) + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*cos(c)*sin_integral(-(b*x + a)*(b*c/(
b*x + a) - a*d/(b*x + a) + d)/b + c) - 2*a*b*c*d^2*cos(c)*sin_integral(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x +
a) + d)/b + c) + 2*(b*x + a)^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)^2*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) - 4*(b*x + a)*b*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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) + 2*b^2*c^2
*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) + 2*(b*x +
a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*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) - 2*a*b*c*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) + 2*(b*x + a)*a*(b*c/(b*x + a) - a*d/(b*x + a) + d)*d^2*sin(-(b*x + a)*(b*c/(b*
x + a) - a*d/(b*x + a) + d)/b) - 2*a*b*c*d^2*sin(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b) + a^2*d^3*s
in(-(b*x + a)*(b*c/(b*x + a) - a*d/(b*x + a) + d)/b))*b^2/(((b*x + a)^2*a^3*b*(b*c/(b*x + a) - a*d/(b*x + a) +
d)^2 - 2*(b*x + a)*a^3*b^2*(b*c/(b*x + a) - a*d/(b*x + a) + d)*c + a^3*b^3*c^2 + (b*x + a)*a^4*b*(b*c/(b*x +
a) - a*d/(b*x + a) + d)*d - a^4*b^2*c*d)*d)

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maple [A]  time = 0.03, size = 256, normalized size = 1.36 \[ d \left (\frac {-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )}{a^{2}}+\frac {2 b^{2} \left (\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}\right )}{d \,a^{3}}-\frac {2 b \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{d \,a^{3}}+\frac {b^{2} \left (-\frac {\sin \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}+\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}+\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}}{b}\right )}{a^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(1/a^2*(-sin(d*x+c)/x/d-Si(d*x)*sin(c)+Ci(d*x)*cos(c))+2/d*b^2/a^3*(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)-2/d/a^3*b*(Si(d*x)*cos(c)+Ci(d*x)*sin(c))+b^2/a^2*(-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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x^2*(a + b*x)^2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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