3.45 \(\int \frac{(a+b x^2) \sin (c+d x)}{x^2} \, dx\)
Optimal. Leaf size=44 \[ a d \cos (c) \text{CosIntegral}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}-\frac{b \cos (c+d x)}{d} \]
[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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Rubi [A] time = 0.107446, antiderivative size = 44, normalized size of antiderivative = 1.,
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 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]
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 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{\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*}
Mathematica [A] time = 0.0951011, size = 44, normalized size = 1. \[ a d \cos (c) \text{CosIntegral}(d x)-a d \sin (c) \text{Si}(d x)-\frac{a \sin (c+d x)}{x}-\frac{b \cos (c+d x)}{d} \]
Antiderivative was successfully verified.
[In]
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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Maple [A] time = 0.013, size = 48, normalized size = 1.1 \begin{align*} d \left ( -{\frac{b\cos \left ( dx+c \right ) }{{d}^{2}}}+a \left ( -{\frac{\sin \left ( dx+c \right ) }{dx}}-{\it Si} \left ( dx \right ) \sin \left ( c \right ) +{\it Ci} \left ( dx \right ) \cos \left ( c \right ) \right ) \right ) \end{align*}
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 = 1.9033, size = 1265, normalized size = 28.75 \begin{align*} \text{result too large to display} \end{align*}
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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Fricas [A] time = 1.74233, size = 212, normalized size = 4.82 \begin{align*} -\frac{2 \, a d^{2} x \sin \left (c\right ) \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 \left (c\right )}{2 \, d x} \end{align*}
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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{2}\right ) \sin{\left (c + d x \right )}}{x^{2}}\, dx \end{align*}
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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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
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]
Exception raised: TypeError