3.64 \(\int \frac {\sin (c+d x)}{x^3 (a+b x^2)} \, dx\)

Optimal. Leaf size=270 \[ -\frac {b \sin (c) \text {Ci}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Ci}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}+\frac {b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Ci}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {d^2 \sin (c) \text {Ci}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x} \]

[Out]

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

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Rubi [A]  time = 0.51, antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3345, 3297, 3303, 3299, 3302} \[ -\frac {b \sin (c) \text {CosIntegral}(d x)}{a^2}+\frac {b \sin \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}+d x\right )}{2 a^2}+\frac {b \sin \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {CosIntegral}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}-\frac {b \cos (c) \text {Si}(d x)}{a^2}-\frac {b \cos \left (\frac {\sqrt {-a} d}{\sqrt {b}}+c\right ) \text {Si}\left (\frac {\sqrt {-a} d}{\sqrt {b}}-d x\right )}{2 a^2}+\frac {b \cos \left (c-\frac {\sqrt {-a} d}{\sqrt {b}}\right ) \text {Si}\left (x d+\frac {\sqrt {-a} d}{\sqrt {b}}\right )}{2 a^2}-\frac {d^2 \sin (c) \text {CosIntegral}(d x)}{2 a}-\frac {d^2 \cos (c) \text {Si}(d x)}{2 a}-\frac {\sin (c+d x)}{2 a x^2}-\frac {d \cos (c+d x)}{2 a x} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]/(x^3*(a + b*x^2)),x]

[Out]

-(d*Cos[c + d*x])/(2*a*x) - (b*CosIntegral[d*x]*Sin[c])/a^2 - (d^2*CosIntegral[d*x]*Sin[c])/(2*a) + (b*CosInte
gral[(Sqrt[-a]*d)/Sqrt[b] + d*x]*Sin[c - (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) + (b*CosIntegral[(Sqrt[-a]*d)/Sqrt[b]
- d*x]*Sin[c + (Sqrt[-a]*d)/Sqrt[b]])/(2*a^2) - Sin[c + d*x]/(2*a*x^2) - (b*Cos[c]*SinIntegral[d*x])/a^2 - (d^
2*Cos[c]*SinIntegral[d*x])/(2*a) - (b*Cos[c + (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] - d*x])/(
2*a^2) + (b*Cos[c - (Sqrt[-a]*d)/Sqrt[b]]*SinIntegral[(Sqrt[-a]*d)/Sqrt[b] + d*x])/(2*a^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 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 3345

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*Sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Int[ExpandIntegrand[Sin[c +
d*x], x^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && ILtQ[p, 0] && IGtQ[n, 0] && (EqQ[n, 2] || EqQ
[p, -1]) && IntegerQ[m]

Rubi steps

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

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Mathematica [C]  time = 0.69, size = 247, normalized size = 0.91 \[ -\frac {x^2 \sin (c) \left (a d^2+2 b\right ) \text {Ci}(d x)-b x^2 \sin \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-b x^2 \sin \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Ci}\left (d \left (x-\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )-b x^2 \cos \left (c-\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (d \left (x+\frac {i \sqrt {a}}{\sqrt {b}}\right )\right )+b x^2 \cos \left (c+\frac {i \sqrt {a} d}{\sqrt {b}}\right ) \text {Si}\left (\frac {i \sqrt {a} d}{\sqrt {b}}-d x\right )+a d^2 x^2 \cos (c) \text {Si}(d x)+a \sin (c+d x)+a d x \cos (c+d x)+2 b x^2 \cos (c) \text {Si}(d x)}{2 a^2 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sin[c + d*x]/(x^3*(a + b*x^2)),x]

[Out]

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

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fricas [C]  time = 0.91, size = 231, normalized size = 0.86 \[ \frac {i \, {\left (a d^{2} + 2 \, b\right )} x^{2} {\rm Ei}\left (i \, d x\right ) e^{\left (i \, c\right )} - i \, {\left (a d^{2} + 2 \, b\right )} x^{2} {\rm Ei}\left (-i \, d x\right ) e^{\left (-i \, c\right )} - i \, b x^{2} {\rm Ei}\left (i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} - i \, b x^{2} {\rm Ei}\left (i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x - \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c + \sqrt {\frac {a d^{2}}{b}}\right )} + i \, b x^{2} {\rm Ei}\left (-i \, d x + \sqrt {\frac {a d^{2}}{b}}\right ) e^{\left (-i \, c - \sqrt {\frac {a d^{2}}{b}}\right )} - 2 \, a d x \cos \left (d x + c\right ) - 2 \, a \sin \left (d x + c\right )}{4 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^3/(b*x^2+a),x, algorithm="fricas")

[Out]

1/4*(I*(a*d^2 + 2*b)*x^2*Ei(I*d*x)*e^(I*c) - I*(a*d^2 + 2*b)*x^2*Ei(-I*d*x)*e^(-I*c) - I*b*x^2*Ei(I*d*x - sqrt
(a*d^2/b))*e^(I*c + sqrt(a*d^2/b)) - I*b*x^2*Ei(I*d*x + sqrt(a*d^2/b))*e^(I*c - sqrt(a*d^2/b)) + I*b*x^2*Ei(-I
*d*x - sqrt(a*d^2/b))*e^(-I*c + sqrt(a*d^2/b)) + I*b*x^2*Ei(-I*d*x + sqrt(a*d^2/b))*e^(-I*c - sqrt(a*d^2/b)) -
2*a*d*x*cos(d*x + c) - 2*a*sin(d*x + c))/(a^2*x^2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^3/(b*x^2+a),x, algorithm="giac")

[Out]

integrate(sin(d*x + c)/((b*x^2 + a)*x^3), x)

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maple [A]  time = 0.06, size = 259, normalized size = 0.96 \[ d^{2} \left (-\frac {\sin \left (d x +c \right )}{2 a \,x^{2} d^{2}}-\frac {\cos \left (d x +c \right )}{2 a x d}+\frac {b \left (\Si \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}+c b}{b}\right )+\Ci \left (d x +c -\frac {d \sqrt {-a b}+c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}+c b}{b}\right )\right )}{2 d^{2} a^{2}}+\frac {b \left (\Si \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \cos \left (\frac {d \sqrt {-a b}-c b}{b}\right )-\Ci \left (d x +c +\frac {d \sqrt {-a b}-c b}{b}\right ) \sin \left (\frac {d \sqrt {-a b}-c b}{b}\right )\right )}{2 d^{2} a^{2}}-\frac {\left (a \,d^{2}+2 b \right ) \left (\Si \left (d x \right ) \cos \relax (c )+\Ci \left (d x \right ) \sin \relax (c )\right )}{2 a^{2} d^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)/x^3/(b*x^2+a),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin \left (d x + c\right )}{{\left (b x^{2} + a\right )} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x^3/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)/((b*x^2 + a)*x^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sin \left (c+d\,x\right )}{x^3\,\left (b\,x^2+a\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)/(x^3*(a + b*x^2)),x)

[Out]

int(sin(c + d*x)/(x^3*(a + b*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (c + d x \right )}}{x^{3} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)/x**3/(b*x**2+a),x)

[Out]

Integral(sin(c + d*x)/(x**3*(a + b*x**2)), x)

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