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

Optimal. Leaf size=72 \[ \frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^2}-\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{\sin (c+d x)}{b (a+b x)} \]

[Out]

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

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Rubi [A]  time = 0.0973983, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3297, 3303, 3299, 3302} \[ \frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (\frac{a d}{b}+d x\right )}{b^2}-\frac{d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (x d+\frac{a d}{b}\right )}{b^2}-\frac{\sin (c+d x)}{b (a+b x)} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(a + b*x)^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.219898, size = 66, normalized size = 0.92 \[ \frac{d \cos \left (c-\frac{a d}{b}\right ) \text{CosIntegral}\left (d \left (\frac{a}{b}+x\right )\right )-d \sin \left (c-\frac{a d}{b}\right ) \text{Si}\left (d \left (\frac{a}{b}+x\right )\right )-\frac{b \sin (c+d x)}{a+b x}}{b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]/(a + b*x)^2,x]

[Out]

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

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Maple [A]  time = 0.009, size = 107, normalized size = 1.5 \begin{align*} d \left ( -{\frac{\sin \left ( dx+c \right ) }{ \left ( \left ( dx+c \right ) b+da-cb \right ) b}}+{\frac{1}{b} \left ({\frac{1}{b}{\it Si} \left ( dx+c+{\frac{da-cb}{b}} \right ) \sin \left ({\frac{da-cb}{b}} \right ) }+{\frac{1}{b}{\it Ci} \left ( dx+c+{\frac{da-cb}{b}} \right ) \cos \left ({\frac{da-cb}{b}} \right ) } \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*(-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 [C]  time = 1.37642, size = 221, normalized size = 3.07 \begin{align*} \frac{d^{2}{\left (-i \, E_{2}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + i \, E_{2}\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^{2}{\left (E_{2}\left (\frac{i \,{\left (d x + c\right )} b - i \, b c + i \, a d}{b}\right ) + E_{2}\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 )} b^{2} - b^{2} c + a b d\right )} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.73698, size = 301, normalized size = 4.18 \begin{align*} \frac{2 \,{\left (b d x + a d\right )} \sin \left (-\frac{b c - a d}{b}\right ) \operatorname{Si}\left (\frac{b d x + a d}{b}\right ) +{\left ({\left (b d x + a d\right )} \operatorname{Ci}\left (\frac{b d x + a d}{b}\right ) +{\left (b d x + a d\right )} \operatorname{Ci}\left (-\frac{b d x + a d}{b}\right )\right )} \cos \left (-\frac{b c - a d}{b}\right ) - 2 \, b \sin \left (d x + c\right )}{2 \,{\left (b^{3} x + a b^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (c + d x \right )}}{\left (a + b x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C]  time = 1.25059, size = 4131, normalized size = 57.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(b*d*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)^2 + b*d*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)^2 - 2*b*d*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*b*d*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) - 4*b*d*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*d*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)^2 - 2*b*d*x*im
ag_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*b*d*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 + a*d*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*d*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*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 - b*d*x*real_part(cos_integr
al(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*ta
n(1/2*c)*tan(1/2*a*d/b) + 4*b*d*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*a*d*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*d*imag_part(c
os_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*d*sin_integral((b*d*x + a*d)/b)*ta
n(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b) - b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a
*d/b)^2 - b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*a*d*imag_part(cos_in
tegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*a*d*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)^2 + 4*a*d*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*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + b*d*x*real_part(cos
_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 - 2*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*
d*x)^2*tan(1/2*c) + 2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*b*d*x*sin_inte
gral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*c) - a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(
1/2*c)^2 - a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*b*d*x*imag_part(cos_integ
ral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2
*tan(1/2*a*d/b) + 4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*d*real_part(cos_in
tegral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) + 4*a*d*real_part(cos_integral(-d*x - a*d/b))*ta
n(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b) - 2*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/2*a*d
/b) + 2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*b*d*x*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)
^2 - a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + 2*b*d*x*imag_part(cos_integra
l(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 - 2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2
*a*d/b)^2 + 4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + a*d*real_part(cos_integral(d*x
+ a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/
b)^2 + b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + b*d*x*real_part(cos_integral(-d*x - a*d/b))
*tan(1/2*d*x)^2 - 2*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) + 2*a*d*imag_part(cos_i
ntegral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/2*
c) - b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - b*d*x*real_part(cos_integral(-d*x - a*d/b))*tan
(1/2*c)^2 + 2*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) - 2*a*d*imag_part(cos_int
egral(-d*x - a*d/b))*tan(1/2*d*x)^2*tan(1/2*a*d/b) + 4*a*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*d*x)^2*tan(1/
2*a*d/b) + 4*b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) + 4*b*d*x*real_part(cos_inte
gral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - 2*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2*tan(1/
2*a*d/b) + 2*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2*tan(1/2*a*d/b) - 4*a*d*sin_integral((b*d*x
+ a*d)/b)*tan(1/2*c)^2*tan(1/2*a*d/b) - b*d*x*real_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - b*d*x*r
eal_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + 2*a*d*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c)*
tan(1/2*a*d/b)^2 - 2*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*a*d*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b*tan(1/2*d*x)^2*tan(1/2*c)*tan(1/2*a*d/b)^2 + 4*b*tan(1/2*
d*x)*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*d*real_part(cos_integral(d*x + a*d/b))*tan(1/2*d*x)^2 + a*d*real_part(c
os_integral(-d*x - a*d/b))*tan(1/2*d*x)^2 - 2*b*d*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*c) + 2*b*d*x*
imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*c) - a*d*real
_part(cos_integral(d*x + a*d/b))*tan(1/2*c)^2 - a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)^2 + 2*b*d
*x*imag_part(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*b*d*x*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2
*a*d/b) + 4*b*d*x*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) + 4*a*d*real_part(cos_integral(d*x + a*d/b))*ta
n(1/2*c)*tan(1/2*a*d/b) + 4*a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*c)*tan(1/2*a*d/b) - a*d*real_par
t(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b)^2 - a*d*real_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b)^2 + b
*d*x*real_part(cos_integral(d*x + a*d/b)) + b*d*x*real_part(cos_integral(-d*x - a*d/b)) - 2*a*d*imag_part(cos_
integral(d*x + a*d/b))*tan(1/2*c) + 2*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*c) - 4*a*d*sin_integra
l((b*d*x + a*d)/b)*tan(1/2*c) + 4*b*tan(1/2*d*x)^2*tan(1/2*c) + 4*b*tan(1/2*d*x)*tan(1/2*c)^2 + 2*a*d*imag_par
t(cos_integral(d*x + a*d/b))*tan(1/2*a*d/b) - 2*a*d*imag_part(cos_integral(-d*x - a*d/b))*tan(1/2*a*d/b) + 4*a
*d*sin_integral((b*d*x + a*d)/b)*tan(1/2*a*d/b) - 4*b*tan(1/2*d*x)*tan(1/2*a*d/b)^2 - 4*b*tan(1/2*c)*tan(1/2*a
*d/b)^2 + a*d*real_part(cos_integral(d*x + a*d/b)) + a*d*real_part(cos_integral(-d*x - a*d/b)) - 4*b*tan(1/2*d
*x) - 4*b*tan(1/2*c))/(b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*d*x)^2*tan(1/2*c)^2*
tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + b^3*x*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*c
)^2*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*b^2*tan(1/2*d*x)^2*tan(1/2*a*d/b)^2 + a*b^2*tan(1
/2*c)^2*tan(1/2*a*d/b)^2 + b^3*x*tan(1/2*d*x)^2 + b^3*x*tan(1/2*c)^2 + b^3*x*tan(1/2*a*d/b)^2 + a*b^2*tan(1/2*
d*x)^2 + a*b^2*tan(1/2*c)^2 + a*b^2*tan(1/2*a*d/b)^2 + b^3*x + a*b^2)