3.108 \(\int \frac {\text {Ci}(b x) \sin (b x)}{x^2} \, dx\)

Optimal. Leaf size=44 \[ \frac {1}{2} b \text {Ci}(b x)^2+b \text {Ci}(2 b x)-\frac {\text {Ci}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]

[Out]

1/2*b*Ci(b*x)^2+b*Ci(2*b*x)-Ci(b*x)*sin(b*x)/x-1/2*sin(2*b*x)/x

________________________________________________________________________________________

Rubi [A]  time = 0.10, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6522, 6686, 12, 4406, 3297, 3302} \[ \frac {1}{2} b \text {CosIntegral}(b x)^2+b \text {CosIntegral}(2 b x)-\frac {\text {CosIntegral}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(CosIntegral[b*x]*Sin[b*x])/x^2,x]

[Out]

(b*CosIntegral[b*x]^2)/2 + b*CosIntegral[2*b*x] - (CosIntegral[b*x]*Sin[b*x])/x - Sin[2*b*x]/(2*x)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 4406

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

Rule 6522

Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_)*Sin[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((e +
f*x)^(m + 1)*Sin[a + b*x]*CosIntegral[c + d*x])/(f*(m + 1)), x] + (-Dist[b/(f*(m + 1)), Int[(e + f*x)^(m + 1)
*Cos[a + b*x]*CosIntegral[c + d*x], x], x] - Dist[d/(f*(m + 1)), Int[((e + f*x)^(m + 1)*Sin[a + b*x]*Cos[c + d
*x])/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \frac {\text {Ci}(b x) \sin (b x)}{x^2} \, dx &=-\frac {\text {Ci}(b x) \sin (b x)}{x}+b \int \frac {\cos (b x) \text {Ci}(b x)}{x} \, dx+b \int \frac {\cos (b x) \sin (b x)}{b x^2} \, dx\\ &=\frac {1}{2} b \text {Ci}(b x)^2-\frac {\text {Ci}(b x) \sin (b x)}{x}+\int \frac {\cos (b x) \sin (b x)}{x^2} \, dx\\ &=\frac {1}{2} b \text {Ci}(b x)^2-\frac {\text {Ci}(b x) \sin (b x)}{x}+\int \frac {\sin (2 b x)}{2 x^2} \, dx\\ &=\frac {1}{2} b \text {Ci}(b x)^2-\frac {\text {Ci}(b x) \sin (b x)}{x}+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2} \, dx\\ &=\frac {1}{2} b \text {Ci}(b x)^2-\frac {\text {Ci}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x}+b \int \frac {\cos (2 b x)}{x} \, dx\\ &=\frac {1}{2} b \text {Ci}(b x)^2+b \text {Ci}(2 b x)-\frac {\text {Ci}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 44, normalized size = 1.00 \[ \frac {1}{2} b \text {Ci}(b x)^2+b \text {Ci}(2 b x)-\frac {\text {Ci}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(CosIntegral[b*x]*Sin[b*x])/x^2,x]

[Out]

(b*CosIntegral[b*x]^2)/2 + b*CosIntegral[2*b*x] - (CosIntegral[b*x]*Sin[b*x])/x - Sin[2*b*x]/(2*x)

________________________________________________________________________________________

fricas [F]  time = 1.95, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {Ci}\left (b x\right ) \sin \left (b x\right )}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x)/x^2,x, algorithm="fricas")

[Out]

integral(cos_integral(b*x)*sin(b*x)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x)/x^2,x, algorithm="giac")

[Out]

integrate(Ci(b*x)*sin(b*x)/x^2, x)

________________________________________________________________________________________

maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {\Ci \left (b x \right ) \sin \left (b x \right )}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Ci(b*x)*sin(b*x)/x^2,x)

[Out]

int(Ci(b*x)*sin(b*x)/x^2,x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x)/x^2,x, algorithm="maxima")

[Out]

integrate(Ci(b*x)*sin(b*x)/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosint(b*x)*sin(b*x))/x^2,x)

[Out]

int((cosint(b*x)*sin(b*x))/x^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Ci(b*x)*sin(b*x)/x**2,x)

[Out]

Integral(sin(b*x)*Ci(b*x)/x**2, x)

________________________________________________________________________________________