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

Optimal. Leaf size=44 \[ a d \cos (c) \text {Ci}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}-\frac {b \cos (c+d x)}{d} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3339, 2638, 3297, 3303, 3299, 3302} \[ a d \cos (c) \text {CosIntegral}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}-\frac {b \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)*Sin[c + d*x])/x^2,x]

[Out]

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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 3339

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegran
d[Sin[c + d*x], (e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

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Mathematica [A]  time = 0.10, size = 44, normalized size = 1.00 \[ a d \cos (c) \text {Ci}(d x)-a d \sin (c) \text {Si}(d x)-\frac {a \sin (c+d x)}{x}-\frac {b \cos (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 68, normalized size = 1.55 \[ -\frac {2 \, a d^{2} x \sin \relax (c) \operatorname {Si}\left (d x\right ) + 2 \, b x \cos \left (d x + c\right ) + 2 \, a d \sin \left (d x + c\right ) - {\left (a d^{2} x \operatorname {Ci}\left (d x\right ) + a d^{2} x \operatorname {Ci}\left (-d x\right )\right )} \cos \relax (c)}{2 \, d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [C]  time = 0.35, size = 411, normalized size = 9.34 \[ -\frac {a d^{2} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{2} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{2} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - a d^{2} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} - a d^{2} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, d x\right )^{2} + a d^{2} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + a d^{2} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right )^{2} + 2 \, a d^{2} x \Im \left (\operatorname {Ci}\left (d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, a d^{2} x \Im \left (\operatorname {Ci}\left (-d x\right ) \right ) \tan \left (\frac {1}{2} \, c\right ) + 4 \, a d^{2} x \operatorname {Si}\left (d x\right ) \tan \left (\frac {1}{2} \, c\right ) + 2 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} - a d^{2} x \Re \left (\operatorname {Ci}\left (d x\right ) \right ) - a d^{2} x \Re \left (\operatorname {Ci}\left (-d x\right ) \right ) - 4 \, a d \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right ) - 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right )^{2} - 2 \, b x \tan \left (\frac {1}{2} \, d x\right )^{2} - 8 \, b x \tan \left (\frac {1}{2} \, d x\right ) \tan \left (\frac {1}{2} \, c\right ) - 2 \, b x \tan \left (\frac {1}{2} \, c\right )^{2} + 4 \, a d \tan \left (\frac {1}{2} \, d x\right ) + 4 \, a d \tan \left (\frac {1}{2} \, c\right ) + 2 \, b x}{2 \, {\left (d x \tan \left (\frac {1}{2} \, d x\right )^{2} \tan \left (\frac {1}{2} \, c\right )^{2} + d x \tan \left (\frac {1}{2} \, d x\right )^{2} + d x \tan \left (\frac {1}{2} \, c\right )^{2} + d x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(a*d^2*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c)^2 + a*d^2*x*real_part(cos_integral(-d*x))
*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*a*d^2*x*imag_part(cos_integral(d*x))*tan(1/2*d*x)^2*tan(1/2*c) - 2*a*d^2*x*im
ag_part(cos_integral(-d*x))*tan(1/2*d*x)^2*tan(1/2*c) + 4*a*d^2*x*sin_integral(d*x)*tan(1/2*d*x)^2*tan(1/2*c)
- a*d^2*x*real_part(cos_integral(d*x))*tan(1/2*d*x)^2 - a*d^2*x*real_part(cos_integral(-d*x))*tan(1/2*d*x)^2 +
a*d^2*x*real_part(cos_integral(d*x))*tan(1/2*c)^2 + a*d^2*x*real_part(cos_integral(-d*x))*tan(1/2*c)^2 + 2*a*
d^2*x*imag_part(cos_integral(d*x))*tan(1/2*c) - 2*a*d^2*x*imag_part(cos_integral(-d*x))*tan(1/2*c) + 4*a*d^2*x
*sin_integral(d*x)*tan(1/2*c) + 2*b*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - a*d^2*x*real_part(cos_integral(d*x)) - a*d
^2*x*real_part(cos_integral(-d*x)) - 4*a*d*tan(1/2*d*x)^2*tan(1/2*c) - 4*a*d*tan(1/2*d*x)*tan(1/2*c)^2 - 2*b*x
*tan(1/2*d*x)^2 - 8*b*x*tan(1/2*d*x)*tan(1/2*c) - 2*b*x*tan(1/2*c)^2 + 4*a*d*tan(1/2*d*x) + 4*a*d*tan(1/2*c) +
2*b*x)/(d*x*tan(1/2*d*x)^2*tan(1/2*c)^2 + d*x*tan(1/2*d*x)^2 + d*x*tan(1/2*c)^2 + d*x)

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maple [A]  time = 0.03, size = 48, normalized size = 1.09 \[ d \left (-\frac {b \cos \left (d x +c \right )}{d^{2}}+a \left (-\frac {\sin \left (d x +c \right )}{x d}-\Si \left (d x \right ) \sin \relax (c )+\Ci \left (d x \right ) \cos \relax (c )\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [C]  time = 0.60, size = 937, normalized size = 21.30 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(((I*exp_integral_e(2, I*d*x) - I*exp_integral_e(2, -I*d*x))*cos(c)^3 + (I*exp_integral_e(2, I*d*x) - I*e
xp_integral_e(2, -I*d*x))*cos(c)*sin(c)^2 + (exp_integral_e(2, I*d*x) + exp_integral_e(2, -I*d*x))*sin(c)^3 +
(I*exp_integral_e(2, I*d*x) - I*exp_integral_e(2, -I*d*x))*cos(c) + ((exp_integral_e(2, I*d*x) + exp_integral_
e(2, -I*d*x))*cos(c)^2 + exp_integral_e(2, I*d*x) + exp_integral_e(2, -I*d*x))*sin(c))*b*c^2/((d*x + c)*(cos(c
)^2 + sin(c)^2)*d^2 - (c*cos(c)^2 + c*sin(c)^2)*d^2) - ((I*exp_integral_e(2, I*d*x) - I*exp_integral_e(2, -I*d
*x))*cos(c)^3 + (I*exp_integral_e(2, I*d*x) - I*exp_integral_e(2, -I*d*x))*cos(c)*sin(c)^2 + (exp_integral_e(2
, I*d*x) + exp_integral_e(2, -I*d*x))*sin(c)^3 + (I*exp_integral_e(2, I*d*x) - I*exp_integral_e(2, -I*d*x))*co
s(c) + ((exp_integral_e(2, I*d*x) + exp_integral_e(2, -I*d*x))*cos(c)^2 + exp_integral_e(2, I*d*x) + exp_integ
ral_e(2, -I*d*x))*sin(c))*a/(c*cos(c)^2 + c*sin(c)^2 - (d*x + c)*(cos(c)^2 + sin(c)^2)) + 2*(((b*cos(c)^2 + b*
sin(c)^2)*(d*x + c)^2 - 2*(b*c*cos(c)^2 + b*c*sin(c)^2)*(d*x + c))*cos(d*x + c)^3 + (b*c^2*(exp_integral_e(3,
I*d*x) + exp_integral_e(3, -I*d*x))*cos(c)^3 + b*c^2*(exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*co
s(c)*sin(c)^2 + b*c^2*(-I*exp_integral_e(3, I*d*x) + I*exp_integral_e(3, -I*d*x))*sin(c)^3 + b*c^2*(exp_integr
al_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*cos(c) + (b*c^2*(-I*exp_integral_e(3, I*d*x) + I*exp_integral_e(3,
-I*d*x))*cos(c)^2 + b*c^2*(-I*exp_integral_e(3, I*d*x) + I*exp_integral_e(3, -I*d*x)))*sin(c))*cos(d*x + c)^2
+ (b*c^2*(exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*cos(c)^3 + b*c^2*(exp_integral_e(3, I*d*x) +
exp_integral_e(3, -I*d*x))*cos(c)*sin(c)^2 + b*c^2*(-I*exp_integral_e(3, I*d*x) + I*exp_integral_e(3, -I*d*x))
*sin(c)^3 + b*c^2*(exp_integral_e(3, I*d*x) + exp_integral_e(3, -I*d*x))*cos(c) + ((b*cos(c)^2 + b*sin(c)^2)*(
d*x + c)^2 - 2*(b*c*cos(c)^2 + b*c*sin(c)^2)*(d*x + c))*cos(d*x + c) + (b*c^2*(-I*exp_integral_e(3, I*d*x) + I
*exp_integral_e(3, -I*d*x))*cos(c)^2 + b*c^2*(-I*exp_integral_e(3, I*d*x) + I*exp_integral_e(3, -I*d*x)))*sin(
c))*sin(d*x + c)^2 + ((b*cos(c)^2 + b*sin(c)^2)*(d*x + c)^2 - 2*(b*c*cos(c)^2 + b*c*sin(c)^2)*(d*x + c))*cos(d
*x + c))/(((d*x + c)^2*(cos(c)^2 + sin(c)^2)*d^2 - 2*(c*cos(c)^2 + c*sin(c)^2)*(d*x + c)*d^2 + (c^2*cos(c)^2 +
c^2*sin(c)^2)*d^2)*cos(d*x + c)^2 + ((d*x + c)^2*(cos(c)^2 + sin(c)^2)*d^2 - 2*(c*cos(c)^2 + c*sin(c)^2)*(d*x
+ c)*d^2 + (c^2*cos(c)^2 + c^2*sin(c)^2)*d^2)*sin(d*x + c)^2))*d

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (c+d\,x\right )\,\left (b\,x^2+a\right )}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sin(c + d*x)*(a + b*x^2))/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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