Integrand size = 6, antiderivative size = 31 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=x \operatorname {CosIntegral}(b x)^2-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \] Output:
x*Ci(b*x)^2-2*Ci(b*x)*sin(b*x)/b+Si(2*b*x)/b
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=x \operatorname {CosIntegral}(b x)^2-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}+\frac {\text {Si}(2 b x)}{b} \] Input:
Integrate[CosIntegral[b*x]^2,x]
Output:
x*CosIntegral[b*x]^2 - (2*CosIntegral[b*x]*Sin[b*x])/b + SinIntegral[2*b*x ]/b
Time = 0.35 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {7060, 7066, 27, 4906, 27, 3042, 3780}
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.
\(\displaystyle \int \operatorname {CosIntegral}(b x)^2 \, dx\) |
\(\Big \downarrow \) 7060 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \int \cos (b x) \operatorname {CosIntegral}(b x)dx\) |
\(\Big \downarrow \) 7066 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\int \frac {\cos (b x) \sin (b x)}{b x}dx\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos (b x) \sin (b x)}{x}dx}{b}\right )\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{2 x}dx}{b}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}\right )\) |
\(\Big \downarrow \) 3780 |
\(\displaystyle x \operatorname {CosIntegral}(b x)^2-2 \left (\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b}\right )\) |
Input:
Int[CosIntegral[b*x]^2,x]
Output:
x*CosIntegral[b*x]^2 - 2*((CosIntegral[b*x]*Sin[b*x])/b - SinIntegral[2*b* x]/(2*b))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[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[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(CosInte gral[a + b*x]^2/b), x] - Simp[2 Int[Cos[a + b*x]*CosIntegral[a + b*x], x] , x] /; FreeQ[{a, b}, x]
Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> S imp[Sin[a + b*x]*(CosIntegral[c + d*x]/b), x] - Simp[d/b Int[Sin[a + b*x] *(Cos[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
Time = 6.79 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97
method | result | size |
derivativedivides | \(\frac {\operatorname {Ci}\left (b x \right )^{2} b x -2 \,\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )+\operatorname {Si}\left (2 b x \right )}{b}\) | \(30\) |
default | \(\frac {\operatorname {Ci}\left (b x \right )^{2} b x -2 \,\operatorname {Ci}\left (b x \right ) \sin \left (b x \right )+\operatorname {Si}\left (2 b x \right )}{b}\) | \(30\) |
Input:
int(Ci(b*x)^2,x,method=_RETURNVERBOSE)
Output:
1/b*(Ci(b*x)^2*b*x-2*Ci(b*x)*sin(b*x)+Si(2*b*x))
Time = 0.09 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.90 \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\frac {2 \, \pi b^{2} x \operatorname {C}\left (b x\right )^{2} - 4 \, b \operatorname {C}\left (b x\right ) \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) + \sqrt {2} \sqrt {b^{2}} \operatorname {S}\left (\sqrt {2} \sqrt {b^{2}} x\right )}{2 \, \pi b^{2}} \] Input:
integrate(fresnel_cos(b*x)^2,x, algorithm="fricas")
Output:
1/2*(2*pi*b^2*x*fresnel_cos(b*x)^2 - 4*b*fresnel_cos(b*x)*sin(1/2*pi*b^2*x ^2) + sqrt(2)*sqrt(b^2)*fresnel_sin(sqrt(2)*sqrt(b^2)*x))/(pi*b^2)
\[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\int \operatorname {Ci}^{2}{\left (b x \right )}\, dx \] Input:
integrate(Ci(b*x)**2,x)
Output:
Integral(Ci(b*x)**2, x)
\[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\int { \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:
integrate(fresnel_cos(b*x)^2,x, algorithm="maxima")
Output:
integrate(fresnel_cos(b*x)^2, x)
\[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\int { \operatorname {C}\left (b x\right )^{2} \,d x } \] Input:
integrate(fresnel_cos(b*x)^2,x, algorithm="giac")
Output:
integrate(fresnel_cos(b*x)^2, x)
Timed out. \[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\int {\mathrm {cosint}\left (b\,x\right )}^2 \,d x \] Input:
int(cosint(b*x)^2,x)
Output:
int(cosint(b*x)^2, x)
\[ \int \operatorname {CosIntegral}(b x)^2 \, dx=\int \mathit {ci} \left (b x \right )^{2}d x \] Input:
int(Ci(b*x)^2,x)
Output:
int(ci(b*x)**2,x)