3.36 \(\int \frac {\sin (c+d x)}{(a+b x)^3} \, dx\)

Optimal. Leaf size=104 \[ -\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (x d+\frac {a d}{b}\right )}{2 b^3}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^3}-\frac {d \cos (c+d x)}{2 b^2 (a+b x)}-\frac {\sin (c+d x)}{2 b (a+b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3297, 3303, 3299, 3302} \[ -\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {CosIntegral}\left (\frac {a d}{b}+d x\right )}{2 b^3}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (x d+\frac {a d}{b}\right )}{2 b^3}-\frac {d \cos (c+d x)}{2 b^2 (a+b x)}-\frac {\sin (c+d x)}{2 b (a+b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x)^3,x]

[Out]

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

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]

Rubi steps

\begin {align*} \int \frac {\sin (c+d x)}{(a+b x)^3} \, dx &=-\frac {\sin (c+d x)}{2 b (a+b x)^2}+\frac {d \int \frac {\cos (c+d x)}{(a+b x)^2} \, dx}{2 b}\\ &=-\frac {d \cos (c+d x)}{2 b^2 (a+b x)}-\frac {\sin (c+d x)}{2 b (a+b x)^2}-\frac {d^2 \int \frac {\sin (c+d x)}{a+b x} \, dx}{2 b^2}\\ &=-\frac {d \cos (c+d x)}{2 b^2 (a+b x)}-\frac {\sin (c+d x)}{2 b (a+b x)^2}-\frac {\left (d^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}{2 b^2}-\frac {\left (d^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}{2 b^2}\\ &=-\frac {d \cos (c+d x)}{2 b^2 (a+b x)}-\frac {d^2 \text {Ci}\left (\frac {a d}{b}+d x\right ) \sin \left (c-\frac {a d}{b}\right )}{2 b^3}-\frac {\sin (c+d x)}{2 b (a+b x)^2}-\frac {d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (\frac {a d}{b}+d x\right )}{2 b^3}\\ \end {align*}

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Mathematica [A]  time = 0.74, size = 87, normalized size = 0.84 \[ -\frac {d^2 \sin \left (c-\frac {a d}{b}\right ) \text {Ci}\left (d \left (\frac {a}{b}+x\right )\right )+d^2 \cos \left (c-\frac {a d}{b}\right ) \text {Si}\left (d \left (\frac {a}{b}+x\right )\right )+\frac {b (d (a+b x) \cos (c+d x)+b \sin (c+d x))}{(a+b x)^2}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(a + b*x)^3,x]

[Out]

-1/2*(d^2*CosIntegral[d*(a/b + x)]*Sin[c - (a*d)/b] + (b*(d*(a + b*x)*Cos[c + d*x] + b*Sin[c + d*x]))/(a + b*x
)^2 + d^2*Cos[c - (a*d)/b]*SinIntegral[d*(a/b + x)])/b^3

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fricas [B]  time = 0.62, size = 210, normalized size = 2.02 \[ -\frac {2 \, b^{2} \sin \left (d x + c\right ) + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \cos \left (-\frac {b c - a d}{b}\right ) \operatorname {Si}\left (\frac {b d x + a d}{b}\right ) + 2 \, {\left (b^{2} d x + a b d\right )} \cos \left (d x + c\right ) - {\left ({\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \operatorname {Ci}\left (\frac {b d x + a d}{b}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \operatorname {Ci}\left (-\frac {b d x + a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{4 \, {\left (b^{5} x^{2} + 2 \, a b^{4} x + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/4*(2*b^2*sin(d*x + c) + 2*(b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos(-(b*c - a*d)/b)*sin_integral((b*d*x + a
*d)/b) + 2*(b^2*d*x + a*b*d)*cos(d*x + c) - ((b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos_integral((b*d*x + a*d)/
b) + (b^2*d^2*x^2 + 2*a*b*d^2*x + a^2*d^2)*cos_integral(-(b*d*x + a*d)/b))*sin(-(b*c - a*d)/b))/(b^5*x^2 + 2*a
*b^4*x + a^2*b^3)

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giac [C]  time = 1.15, size = 5727, normalized size = 55.07 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="giac")

[Out]

-1/4*(b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*
x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*sin_int
egral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*t
an(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan
(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2
- 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 4*a*b*d^2*x
*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*x^2*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^
2*tan(1/2*c)^2 - 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*b^2*d^2*x^2*imag_
part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*d^2*x^2*imag_part(cos_integra
l(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/
2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c
)^2*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d
/b) - b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan
(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(
1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 +
b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - b^2*d^2*x^2*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 -
a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_in
tegral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*
x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan
(1/2*c) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b*d^2*x*imag_part
(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d
*x)^2*tan(1/2*c)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b^2*
d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a*b*d^2*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 8*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*
tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 16*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c
)*tan(1/2*a*d/b) + 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*b^2*d^2*
x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*
d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a
*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))
*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2
- 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integr
al(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^
2*tan(1/2*a*d/b)^2 + 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*b^2*d*x*tan(1
/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - b^
2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*t
an(1/2*d*x)^2 + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*b*d^2*x*real_
part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - b^2*d^2*x^2*imag_part(cos_integral(d*x + a*d/b))*
tan(1/2*c)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*b^2*d^2*x^2*sin_integral((b*
d*x + a*d)/b)*tan(1/2*c)^2 - a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a^2*d^
2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*
tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) -
4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*b^2*d^2*x^2*imag_part(cos
_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/
2*c)*tan(1/2*a*d/b) + 8*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 4*a^2*d^2*imag_p
art(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*
x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*
tan(1/2*c)*tan(1/2*a*d/b) + 4*a*b*d^2*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 4*a
*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - b^2*d^2*x^2*imag_part(cos_integra
l(d*x + a*d/b))*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 2*b^2*
d^2*x^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x)^2*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^
2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*
d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - a^2*d^2*imag_part(cos_in
tegral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2*tan
(1/2*a*d/b)^2 + 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_integral(d*x
+ a*d/b))*tan(1/2*d*x)^2 - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 + 4*a*b*d^2*x*sin_
integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 + 2*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*b
^2*d^2*x^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*t
an(1/2*d*x)^2*tan(1/2*c) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*d
^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 + 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan
(1/2*c)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*b^2*d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2
*b^2*d^2*x^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*b^2*d^2*x^2*real_part(cos_integral(-d*x -
a*d/b))*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a^2
*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 8*a*b*d^2*x*imag_part(cos_integral(
d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 8*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2
*a*d/b) + 16*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) + 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*t
an(1/2*a*d/b) - 2*a*b*d^2*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_
integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 - 4*a*b*d^2*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*b^2*
d*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 - 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)
^2 - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 8*b^2*d*x*tan(1/2*d*x)*tan(
1/2*c)*tan(1/2*a*d/b)^2 - 2*b^2*d*x*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^2*d^2*x^2*imag_part(cos_integral(d*x + a
*d/b)) - b^2*d^2*x^2*imag_part(cos_integral(-d*x - a*d/b)) + 2*b^2*d^2*x^2*sin_integral((b*d*x + a*d)/b) + a^2
*d^2*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 - a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1
/2*d*x)^2 + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2 + 4*a*b*d^2*x*real_part(cos_integral(d*x +
a*d/b))*tan(1/2*c) + 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - a^2*d^2*imag_part(cos_inte
gral(d*x + a*d/b))*tan(1/2*c)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 - 2*a^2*d^2*sin_i
ntegral((b*d*x + a*d)/b)*tan(1/2*c)^2 + 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*c)^2 - 4*a*b*d^2*x*real_part(cos_integr
al(d*x + a*d/b))*tan(1/2*a*d/b) - 4*a*b*d^2*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a^2*d^2
*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 4*a^2*d^2*imag_part(cos_integral(-d*x - a*d/
b))*tan(1/2*c)*tan(1/2*a*d/b) + 8*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b) - a^2*d^2*im
ag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*
d/b)^2 - 2*a^2*d^2*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b)^2 - 2*a*b*d*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 -
8*a*b*d*tan(1/2*d*x)*tan(1/2*c)*tan(1/2*a*d/b)^2 - 4*b^2*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*b*d*
tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 4*b^2*tan(1/2*d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + 2*a*b*d^2*x*imag_part(cos_i
ntegral(d*x + a*d/b)) - 2*a*b*d^2*x*imag_part(cos_integral(-d*x - a*d/b)) + 4*a*b*d^2*x*sin_integral((b*d*x +
a*d)/b) - 2*b^2*d*x*tan(1/2*d*x)^2 + 2*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*a^2*d^2*rea
l_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 8*b^2*d*x*tan(1/2*d*x)*tan(1/2*c) - 2*b^2*d*x*tan(1/2*c)^2 - 2
*a^2*d^2*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a^2*d^2*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*a*d/b) + 2*b^2*d*x*tan(1/2*a*d/b)^2 + a^2*d^2*imag_part(cos_integral(d*x + a*d/b)) - a^2*d^2*imag_par
t(cos_integral(-d*x - a*d/b)) + 2*a^2*d^2*sin_integral((b*d*x + a*d)/b) - 2*a*b*d*tan(1/2*d*x)^2 - 8*a*b*d*tan
(1/2*d*x)*tan(1/2*c) - 4*b^2*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*b*d*tan(1/2*c)^2 - 4*b^2*tan(1/2*d*x)*tan(1/2*c)^
2 + 2*a*b*d*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*d*x)*tan(1/2*a*d/b)^2 + 4*b^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 2*b^2
*d*x + 2*a*b*d + 4*b^2*tan(1/2*d*x) + 4*b^2*tan(1/2*c))/(b^5*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2
+ 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^5*x^2*tan(1
/2*d*x)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1
/2*a*d/b)^2 + 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*b^4*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*b^4*x*ta
n(1/2*c)^2*tan(1/2*a*d/b)^2 + b^5*x^2*tan(1/2*d*x)^2 + b^5*x^2*tan(1/2*c)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*c
)^2 + b^5*x^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a^2*b^3*tan(1/2*c)^2*tan(1/2*a*d/b)
^2 + 2*a*b^4*x*tan(1/2*d*x)^2 + 2*a*b^4*x*tan(1/2*c)^2 + 2*a*b^4*x*tan(1/2*a*d/b)^2 + b^5*x^2 + a^2*b^3*tan(1/
2*d*x)^2 + a^2*b^3*tan(1/2*c)^2 + a^2*b^3*tan(1/2*a*d/b)^2 + 2*a*b^4*x + a^2*b^3)

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maple [A]  time = 0.03, size = 145, normalized size = 1.39 \[ d^{2} \left (-\frac {\sin \left (d x +c \right )}{2 \left (\left (d x +c \right ) b +d a -c b \right )^{2} b}+\frac {-\frac {\cos \left (d x +c \right )}{\left (\left (d x +c \right ) b +d a -c b \right ) b}-\frac {\frac {\Si \left (d x +c +\frac {d a -c b}{b}\right ) \cos \left (\frac {d a -c b}{b}\right )}{b}-\frac {\Ci \left (d x +c +\frac {d a -c b}{b}\right ) \sin \left (\frac {d a -c b}{b}\right )}{b}}{b}}{2 b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/(b*x+a)^3,x)

[Out]

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

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maxima [C]  time = 0.46, size = 199, normalized size = 1.91 \[ \frac {d^{3} {\left (-i \, E_{3}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{3}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \cos \left (-\frac {b c - a d}{b}\right ) + d^{3} {\left (E_{3}\left (\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{3}\left (-\frac {i \, {\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right )\right )} \sin \left (-\frac {b c - a d}{b}\right )}{2 \, {\left ({\left (d x + c\right )}^{2} b^{3} + b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2} - 2 \, {\left (b^{3} c - a b^{2} d\right )} {\left (d x + c\right )}\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/(b*x+a)^3,x, algorithm="maxima")

[Out]

1/2*(d^3*(-I*exp_integral_e(3, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + I*exp_integral_e(3, -(I*(d*x + c)*b - I*b*
c + I*a*d)/b))*cos(-(b*c - a*d)/b) + d^3*(exp_integral_e(3, (I*(d*x + c)*b - I*b*c + I*a*d)/b) + exp_integral_
e(3, -(I*(d*x + c)*b - I*b*c + I*a*d)/b))*sin(-(b*c - a*d)/b))/(((d*x + c)^2*b^3 + b^3*c^2 - 2*a*b^2*c*d + a^2
*b*d^2 - 2*(b^3*c - a*b^2*d)*(d*x + c))*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sin \left (c+d\,x\right )}{{\left (a+b\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(a + b*x)^3,x)

[Out]

int(sin(c + d*x)/(a + b*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (c + d x \right )}}{\left (a + b x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/(b*x+a)**3,x)

[Out]

Integral(sin(c + d*x)/(a + b*x)**3, x)

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