3.6 \(\int \frac{\sin (a+b x)}{(c+d x)^2} \, dx\)
Optimal. Leaf size=72 \[ \frac{b \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{b \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sin (a+b x)}{d (c+d x)} \]
[Out]
(b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2 - Sin[a + b*x]/(d*(c + d*x)) - (b*Sin[a - (b*c)/d]*SinInte
gral[(b*c)/d + b*x])/d^2
________________________________________________________________________________________
Rubi [A] time = 0.108748, 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} \[ \frac{b \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{b \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sin (a+b x)}{d (c+d x)} \]
Antiderivative was successfully verified.
[In]
Int[Sin[a + b*x]/(c + d*x)^2,x]
[Out]
(b*Cos[a - (b*c)/d]*CosIntegral[(b*c)/d + b*x])/d^2 - Sin[a + b*x]/(d*(c + d*x)) - (b*Sin[a - (b*c)/d]*SinInte
gral[(b*c)/d + b*x])/d^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 (a+b x)}{(c+d x)^2} \, dx &=-\frac{\sin (a+b x)}{d (c+d x)}+\frac{b \int \frac{\cos (a+b x)}{c+d x} \, dx}{d}\\ &=-\frac{\sin (a+b x)}{d (c+d x)}+\frac{\left (b \cos \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{\left (b \sin \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{b \cos \left (a-\frac{b c}{d}\right ) \text{Ci}\left (\frac{b c}{d}+b x\right )}{d^2}-\frac{\sin (a+b x)}{d (c+d x)}-\frac{b \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.210034, size = 66, normalized size = 0.92 \[ \frac{b \cos \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (b \left (\frac{c}{d}+x\right )\right )-b \sin \left (a-\frac{b c}{d}\right ) \text{Si}\left (b \left (\frac{c}{d}+x\right )\right )-\frac{d \sin (a+b x)}{c+d x}}{d^2} \]
Antiderivative was successfully verified.
[In]
Integrate[Sin[a + b*x]/(c + d*x)^2,x]
[Out]
(b*Cos[a - (b*c)/d]*CosIntegral[b*(c/d + x)] - (d*Sin[a + b*x])/(c + d*x) - b*Sin[a - (b*c)/d]*SinIntegral[b*(
c/d + x)])/d^2
________________________________________________________________________________________
Maple [A] time = 0.01, size = 107, normalized size = 1.5 \begin{align*} b \left ( -{\frac{\sin \left ( bx+a \right ) }{ \left ( \left ( bx+a \right ) d-da+cb \right ) d}}+{\frac{1}{d} \left ({\frac{1}{d}{\it Si} \left ( bx+a+{\frac{-da+cb}{d}} \right ) \sin \left ({\frac{-da+cb}{d}} \right ) }+{\frac{1}{d}{\it Ci} \left ( bx+a+{\frac{-da+cb}{d}} \right ) \cos \left ({\frac{-da+cb}{d}} \right ) } \right ) } \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(sin(b*x+a)/(d*x+c)^2,x)
[Out]
b*(-sin(b*x+a)/((b*x+a)*d-d*a+c*b)/d+(Si(b*x+a+(-a*d+b*c)/d)*sin((-a*d+b*c)/d)/d+Ci(b*x+a+(-a*d+b*c)/d)*cos((-
a*d+b*c)/d)/d)/d)
________________________________________________________________________________________
Maxima [C] time = 1.3256, size = 221, normalized size = 3.07 \begin{align*} -\frac{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 )} \cos \left (-\frac{b c - a d}{d}\right ) + 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 )} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)/(d*x+c)^2,x, algorithm="maxima")
[Out]
-1/2*(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))*cos(-(b*c - a*d)/d) + 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))*sin(-(b*c - a*d)/d))/((b*c*d + (b*x + a)*d^2 - a*d^2)*b)
________________________________________________________________________________________
Fricas [A] time = 1.64332, size = 302, normalized size = 4.19 \begin{align*} -\frac{2 \,{\left (b d x + b c\right )} \sin \left (-\frac{b c - a d}{d}\right ) \operatorname{Si}\left (\frac{b d x + b c}{d}\right ) -{\left ({\left (b d x + b c\right )} \operatorname{Ci}\left (\frac{b d x + b c}{d}\right ) +{\left (b d x + b c\right )} \operatorname{Ci}\left (-\frac{b d x + b c}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + 2 \, d \sin \left (b x + a\right )}{2 \,{\left (d^{3} x + c d^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)/(d*x+c)^2,x, algorithm="fricas")
[Out]
-1/2*(2*(b*d*x + b*c)*sin(-(b*c - a*d)/d)*sin_integral((b*d*x + b*c)/d) - ((b*d*x + b*c)*cos_integral((b*d*x +
b*c)/d) + (b*d*x + b*c)*cos_integral(-(b*d*x + b*c)/d))*cos(-(b*c - a*d)/d) + 2*d*sin(b*x + a))/(d^3*x + c*d^
2)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sin{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(sin(b*x+a)/(d*x+c)**2,x)
[Out]
Integral(sin(a + b*x)/(c + d*x)**2, x)
________________________________________________________________________________________
Giac [C] time = 1.29518, 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(b*x+a)/(d*x+c)^2,x, algorithm="giac")
[Out]
1/2*(b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + b*d*x*real_part
(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(b*x
+ b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*
b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) - 4*b*d*x*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2
*b*c/d) + 2*b*d*x*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*b*d*x*im
ag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 + 4*b*d*x*sin_integral((b*d*x +
b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 + b*c*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*
tan(1/2*a)^2*tan(1/2*b*c/d)^2 + b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*
b*c/d)^2 - b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2 - b*d*x*real_part(cos_integr
al(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2 + 4*b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*ta
n(1/2*a)*tan(1/2*b*c/d) + 4*b*d*x*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/
d) - 2*b*c*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) + 2*b*c*imag_part(c
os_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) - 4*b*c*sin_integral((b*d*x + b*c)/d)*ta
n(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d) - b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*b
*c/d)^2 - b*d*x*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d)^2 + 2*b*c*imag_part(cos_in
tegral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*b*c*imag_part(cos_integral(-b*x - b*c/d))*
tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 + 4*b*c*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a)*tan
(1/2*b*c/d)^2 + b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + b*d*x*real_part(cos
_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*
b*x)^2*tan(1/2*a) + 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a) - 4*b*d*x*sin_inte
gral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*a) - b*c*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(
1/2*a)^2 - b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)^2 + 2*b*d*x*imag_part(cos_integ
ral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d) - 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2
*tan(1/2*b*c/d) + 4*b*d*x*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*b*c/d) + 4*b*c*real_part(cos_in
tegral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d) + 4*b*c*real_part(cos_integral(-b*x - b*c/d))*ta
n(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d) - 2*b*d*x*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/2*b*c
/d) + 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) - 4*b*d*x*sin_integral((b*d*x
+ b*c)/d)*tan(1/2*a)^2*tan(1/2*b*c/d) - b*c*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d)
^2 - b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d)^2 + 2*b*d*x*imag_part(cos_integra
l(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)^2 - 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2
*b*c/d)^2 + 4*b*d*x*sin_integral((b*d*x + b*c)/d)*tan(1/2*a)*tan(1/2*b*c/d)^2 + b*c*real_part(cos_integral(b*x
+ b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/
d)^2 + b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2 + b*d*x*real_part(cos_integral(-b*x - b*c/d))
*tan(1/2*b*x)^2 - 2*b*c*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*a) + 2*b*c*imag_part(cos_i
ntegral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*a) - 4*b*c*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/2*
a) - b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2 - b*d*x*real_part(cos_integral(-b*x - b*c/d))*tan
(1/2*a)^2 + 2*b*c*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d) - 2*b*c*imag_part(cos_int
egral(-b*x - b*c/d))*tan(1/2*b*x)^2*tan(1/2*b*c/d) + 4*b*c*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*x)^2*tan(1/
2*b*c/d) + 4*b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) + 4*b*d*x*real_part(cos_inte
gral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) - 2*b*c*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2*tan(1/
2*b*c/d) + 2*b*c*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2*tan(1/2*b*c/d) - 4*b*c*sin_integral((b*d*x
+ b*c)/d)*tan(1/2*a)^2*tan(1/2*b*c/d) - b*d*x*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d)^2 - b*d*x*r
eal_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/d)^2 + 2*b*c*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a)*
tan(1/2*b*c/d)^2 - 2*b*c*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d)^2 + 4*b*c*sin_integra
l((b*d*x + b*c)/d)*tan(1/2*a)*tan(1/2*b*c/d)^2 + 4*d*tan(1/2*b*x)^2*tan(1/2*a)*tan(1/2*b*c/d)^2 + 4*d*tan(1/2*
b*x)*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + b*c*real_part(cos_integral(b*x + b*c/d))*tan(1/2*b*x)^2 + b*c*real_part(c
os_integral(-b*x - b*c/d))*tan(1/2*b*x)^2 - 2*b*d*x*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*a) + 2*b*d*x*
imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a) - 4*b*d*x*sin_integral((b*d*x + b*c)/d)*tan(1/2*a) - b*c*real
_part(cos_integral(b*x + b*c/d))*tan(1/2*a)^2 - b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)^2 + 2*b*d
*x*imag_part(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d) - 2*b*d*x*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2
*b*c/d) + 4*b*d*x*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*c/d) + 4*b*c*real_part(cos_integral(b*x + b*c/d))*ta
n(1/2*a)*tan(1/2*b*c/d) + 4*b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*a)*tan(1/2*b*c/d) - b*c*real_par
t(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d)^2 - b*c*real_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/d)^2 + b
*d*x*real_part(cos_integral(b*x + b*c/d)) + b*d*x*real_part(cos_integral(-b*x - b*c/d)) - 2*b*c*imag_part(cos_
integral(b*x + b*c/d))*tan(1/2*a) + 2*b*c*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*a) - 4*b*c*sin_integra
l((b*d*x + b*c)/d)*tan(1/2*a) + 4*d*tan(1/2*b*x)^2*tan(1/2*a) + 4*d*tan(1/2*b*x)*tan(1/2*a)^2 + 2*b*c*imag_par
t(cos_integral(b*x + b*c/d))*tan(1/2*b*c/d) - 2*b*c*imag_part(cos_integral(-b*x - b*c/d))*tan(1/2*b*c/d) + 4*b
*c*sin_integral((b*d*x + b*c)/d)*tan(1/2*b*c/d) - 4*d*tan(1/2*b*x)*tan(1/2*b*c/d)^2 - 4*d*tan(1/2*a)*tan(1/2*b
*c/d)^2 + b*c*real_part(cos_integral(b*x + b*c/d)) + b*c*real_part(cos_integral(-b*x - b*c/d)) - 4*d*tan(1/2*b
*x) - 4*d*tan(1/2*a))/(d^3*x*tan(1/2*b*x)^2*tan(1/2*a)^2*tan(1/2*b*c/d)^2 + c*d^2*tan(1/2*b*x)^2*tan(1/2*a)^2*
tan(1/2*b*c/d)^2 + d^3*x*tan(1/2*b*x)^2*tan(1/2*a)^2 + d^3*x*tan(1/2*b*x)^2*tan(1/2*b*c/d)^2 + d^3*x*tan(1/2*a
)^2*tan(1/2*b*c/d)^2 + c*d^2*tan(1/2*b*x)^2*tan(1/2*a)^2 + c*d^2*tan(1/2*b*x)^2*tan(1/2*b*c/d)^2 + c*d^2*tan(1
/2*a)^2*tan(1/2*b*c/d)^2 + d^3*x*tan(1/2*b*x)^2 + d^3*x*tan(1/2*a)^2 + d^3*x*tan(1/2*b*c/d)^2 + c*d^2*tan(1/2*
b*x)^2 + c*d^2*tan(1/2*a)^2 + c*d^2*tan(1/2*b*c/d)^2 + d^3*x + c*d^2)