Integrand size = 12, antiderivative size = 12 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=-\frac {b \cos ^2(b x)}{2 x}-\frac {b \cos (2 b x)}{4 x}-\frac {b \cos (b x) \operatorname {CosIntegral}(b x)}{2 x}-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}-\frac {\sin (2 b x)}{8 x^2}-b^2 \text {Si}(2 b x)-\frac {1}{2} b^2 \text {Int}\left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x},x\right ) \]
-1/2*b^2*CannotIntegrate(Ci(b*x)*sin(b*x)/x,x)-1/2*b*Ci(b*x)*cos(b*x)/x-1/ 2*b*cos(b*x)^2/x-1/4*b*cos(2*b*x)/x-b^2*Si(2*b*x)-1/2*Ci(b*x)*sin(b*x)/x^2 -1/8*sin(2*b*x)/x^2
Not integrable
Time = 1.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx \]
Not integrable
Time = 1.08 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {7076, 27, 4906, 27, 3042, 3778, 3042, 3778, 25, 3042, 3780, 7070, 27, 3042, 3794, 27, 3042, 3780, 7299}
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.
\(\displaystyle \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx\) |
\(\Big \downarrow \) 7076 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} b \int \frac {\cos (b x) \sin (b x)}{b x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\cos (b x) \sin (b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{2} \int \frac {\sin (2 b x)}{2 x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \int \frac {\sin (2 b x)}{x^3}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \int \frac {\cos (2 b x)}{x^2}dx-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \int \frac {\sin \left (2 b x+\frac {\pi }{2}\right )}{x^2}dx-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3778 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (2 b \int -\frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (-2 b \int \frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx+\frac {1}{4} \left (b \left (-2 b \int \frac {\sin (2 b x)}{x}dx-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {1}{2} b \int \frac {\cos (b x) \operatorname {CosIntegral}(b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 7070 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+b \int \frac {\cos ^2(b x)}{b x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+\int \frac {\cos ^2(b x)}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+\int \frac {\sin \left (b x+\frac {\pi }{2}\right )^2}{x^2}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3794 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx+2 b \int -\frac {\sin (2 b x)}{2 x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-b \int \frac {\sin (2 b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-b \int \frac {\sin (2 b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-b \text {Si}(2 b x)-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \frac {1}{2} b \left (-b \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x}dx-\frac {\operatorname {CosIntegral}(b x) \cos (b x)}{x}-b \text {Si}(2 b x)-\frac {\cos ^2(b x)}{x}\right )-\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{2 x^2}+\frac {1}{4} \left (b \left (-2 b \text {Si}(2 b x)-\frac {\cos (2 b x)}{x}\right )-\frac {\sin (2 b x)}{2 x^2}\right )\) |
3.2.7.3.1 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[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
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[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*Cos[a + b*x]*(CosIntegral[ c + d*x]/(f*(m + 1))), x] + (Simp[b/(f*(m + 1)) Int[(e + f*x)^(m + 1)*Sin [a + b*x]*CosIntegral[c + d*x], x], x] - Simp[d/(f*(m + 1)) Int[(e + f*x) ^(m + 1)*Cos[a + b*x]*(Cos[c + d*x]/(c + d*x)), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[m, -1]
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]
Not integrable
Time = 0.21 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00
\[\int \frac {\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )}{x^{3}}d x\]
Not integrable
Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 2.46 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\sin {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}}{x^{3}}\, dx \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int { \frac {\operatorname {C}\left (b x\right ) \sin \left (b x\right )}{x^{3}} \,d x } \]
Not integrable
Time = 5.56 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.17 \[ \int \frac {\operatorname {CosIntegral}(b x) \sin (b x)}{x^3} \, dx=\int \frac {\mathrm {cosint}\left (b\,x\right )\,\sin \left (b\,x\right )}{x^3} \,d x \]