Integrand size = 12, antiderivative size = 44 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}(b x)^2+b \operatorname {CosIntegral}(2 b x)-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]
Time = 0.01 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.00 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\frac {1}{2} b \operatorname {CosIntegral}(b x)^2+b \operatorname {CosIntegral}(2 b x)-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}-\frac {\sin (2 b x)}{2 x} \]
(b*CosIntegral[b*x]^2)/2 + b*CosIntegral[2*b*x] - (CosIntegral[b*x]*Sin[b* x])/x - Sin[2*b*x]/(2*x)
Time = 0.53 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7076, 27, 4906, 27, 3042, 3778, 3042, 3783, 7237}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx\) |
\(\Big \downarrow \) 7076 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+b \int \frac {\cos (b x) \sin (b x)}{b x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\int \frac {\cos (b x) \sin (b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\int \frac {\sin (2 b x)}{2 x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \left (2 b \int \frac {\cos (2 b x)}{x}dx-\frac {\sin (2 b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx+\frac {1}{2} \left (2 b \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x}dx-\frac {\sin (2 b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}\) |
\(\Big \downarrow \) 3783 |
\(\displaystyle b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}+\frac {1}{2} \left (2 b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{x}\right )\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {1}{2} b \operatorname {CosIntegral}(b x)^2-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}+\frac {1}{2} \left (2 b \operatorname {CosIntegral}(2 b x)-\frac {\sin (2 b x)}{x}\right )\) |
(b*CosIntegral[b*x]^2)/2 - (CosIntegral[b*x]*Sin[b*x])/x + (2*b*CosIntegra l[2*b*x] - Sin[2*b*x]/x)/2
3.2.8.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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] - Simp[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]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(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] && IG tQ[p, 0]
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] + (-Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)*Cos [a + b*x]*CosIntegral[c + d*x], x], x] - Simp[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]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
\[\int \frac {\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )}{x^{2}}d x\]
\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \]
\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \]
\[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^2} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^2} \,d x \]