\(\int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx\) [119]

Optimal result

Integrand size = 12, antiderivative size = 89 \[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=-\frac {3 x}{4 b^2}+\frac {2 x \cos (b x) \operatorname {CosIntegral}(b x)}{b^2}-\frac {5 \cos (b x) \sin (b x)}{4 b^3}-\frac {2 \operatorname {CosIntegral}(b x) \sin (b x)}{b^3}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {x \sin ^2(b x)}{2 b^2}+\frac {\text {Si}(2 b x)}{b^3} \] Output:

-3/4*x/b^2+2*x*cos(b*x)*Ci(b*x)/b^2-5/4*cos(b*x)*sin(b*x)/b^3-2*Ci(b*x)*si 
n(b*x)/b^3+x^2*Ci(b*x)*sin(b*x)/b-1/2*x*sin(b*x)^2/b^2+Si(2*b*x)/b^3
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72 \[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {-8 b x+2 b x \cos (2 b x)+8 \operatorname {CosIntegral}(b x) \left (2 b x \cos (b x)+\left (-2+b^2 x^2\right ) \sin (b x)\right )-5 \sin (2 b x)+8 \text {Si}(2 b x)}{8 b^3} \] Input:

Integrate[x^2*Cos[b*x]*CosIntegral[b*x],x]
 

Output:

(-8*b*x + 2*b*x*Cos[2*b*x] + 8*CosIntegral[b*x]*(2*b*x*Cos[b*x] + (-2 + b^ 
2*x^2)*Sin[b*x]) - 5*Sin[2*b*x] + 8*SinIntegral[2*b*x])/(8*b^3)
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.55, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.417, Rules used = {7068, 27, 3924, 3042, 3115, 24, 7074, 27, 3042, 3115, 24, 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.

\(\displaystyle \int x^2 \operatorname {CosIntegral}(b x) \cos (b x) \, dx\)

\(\Big \downarrow \) 7068

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\int \frac {x \cos (b x) \sin (b x)}{b}dx+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\int x \cos (b x) \sin (b x)dx}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3924

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin ^2(b x)dx}{2 b}}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\int \sin (b x)^2dx}{2 b}}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {\int 1dx}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {2 \int x \operatorname {CosIntegral}(b x) \sin (b x)dx}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 7074

\(\displaystyle -\frac {2 \left (\frac {\int \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\int \frac {\cos ^2(b x)}{b}dx-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {\int \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int \cos ^2(b x)dx}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (\frac {\int \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\int \sin \left (b x+\frac {\pi }{2}\right )^2dx}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3115

\(\displaystyle -\frac {2 \left (\frac {\int \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}+\frac {\frac {\int 1dx}{2}+\frac {\sin (b x) \cos (b x)}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 24

\(\displaystyle -\frac {2 \left (\frac {\int \cos (b x) \operatorname {CosIntegral}(b x)dx}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 7066

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\int \frac {\cos (b x) \sin (b x)}{b x}dx}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\cos (b x) \sin (b x)}{x}dx}{b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{2 x}dx}{b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\int \frac {\sin (2 b x)}{x}dx}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

\(\Big \downarrow \) 3780

\(\displaystyle -\frac {2 \left (\frac {\frac {\operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\text {Si}(2 b x)}{2 b}}{b}-\frac {x \operatorname {CosIntegral}(b x) \cos (b x)}{b}+\frac {\frac {\sin (b x) \cos (b x)}{2 b}+\frac {x}{2}}{b}\right )}{b}+\frac {x^2 \operatorname {CosIntegral}(b x) \sin (b x)}{b}-\frac {\frac {x \sin ^2(b x)}{2 b}-\frac {\frac {x}{2}-\frac {\sin (b x) \cos (b x)}{2 b}}{2 b}}{b}\)

Input:

Int[x^2*Cos[b*x]*CosIntegral[b*x],x]
 

Output:

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

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n)   Int[(b*Sin 
[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 
2*n]
 

rule 3780

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]
 

rule 3924

Int[Cos[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sin[(a_.) + (b_.)*(x_)^(n_.)]^ 
(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sin[a + b*x^n]^(p + 1)/(b*n*(p + 1) 
)), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sin[a + b*x^n]^(p + 
 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
 

rule 4906

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]
 

rule 7066

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]
 

rule 7068

Int[Cos[(a_.) + (b_.)*(x_)]*CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)* 
(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^m*Sin[a + b*x]*(CosIntegral[c + d* 
x]/b), x] + (-Simp[d/b   Int[(e + f*x)^m*Sin[a + b*x]*(Cos[c + d*x]/(c + d* 
x)), x], x] - Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Sin[a + b*x]*CosIntegral 
[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

rule 7074

Int[CosIntegral[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))^(m_.)*Sin[(a_.) + 
(b_.)*(x_)], x_Symbol] :> Simp[(-(e + f*x)^m)*Cos[a + b*x]*(CosIntegral[c + 
 d*x]/b), x] + (Simp[d/b   Int[(e + f*x)^m*Cos[a + b*x]*(Cos[c + d*x]/(c + 
d*x)), x], x] + Simp[f*(m/b)   Int[(e + f*x)^(m - 1)*Cos[a + b*x]*CosIntegr 
al[c + d*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]
 

Maple [A] (verified)

Time = 5.78 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74

method	result	size
derivativedivides	\(\frac {\operatorname {Ci}\left (b x \right ) \left (b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )\right )+\frac {\cos \left (b x \right )^{2} b x}{2}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{4}-\frac {5 b x}{4}+\operatorname {Si}\left (2 b x \right )}{b^{3}}\)	\(66\)
default	\(\frac {\operatorname {Ci}\left (b x \right ) \left (b^{2} x^{2} \sin \left (b x \right )-2 \sin \left (b x \right )+2 b x \cos \left (b x \right )\right )+\frac {\cos \left (b x \right )^{2} b x}{2}-\frac {5 \sin \left (b x \right ) \cos \left (b x \right )}{4}-\frac {5 b x}{4}+\operatorname {Si}\left (2 b x \right )}{b^{3}}\)	\(66\)

Input:

int(x^2*cos(b*x)*Ci(b*x),x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(Ci(b*x)*(b^2*x^2*sin(b*x)-2*sin(b*x)+2*b*x*cos(b*x))+1/2*cos(b*x)^2 
*b*x-5/4*sin(b*x)*cos(b*x)-5/4*b*x+Si(2*b*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (83) = 166\).

Time = 0.12 (sec) , antiderivative size = 297, normalized size of antiderivative = 3.34 \[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\frac {4 \, \pi ^{2} b^{2} x \cos \left (b x\right ) \operatorname {C}\left (b x\right ) - 2 \, b \cos \left (\frac {1}{2} \, \pi b^{2} x^{2}\right ) \cos \left (b x\right ) + 2 \, {\left (\pi ^{2} b^{3} x^{2} - 2 \, \pi ^{2} b\right )} \operatorname {C}\left (b x\right ) \sin \left (b x\right ) + \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (2 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left (\pi \cos \left (\frac {1}{2 \, \pi }\right ) - {\left (2 \, \pi ^{2} - 1\right )} \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {C}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) + \sqrt {b^{2}} {\left ({\left (2 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x + 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - \sqrt {b^{2}} {\left ({\left (2 \, \pi ^{2} - 1\right )} \cos \left (\frac {1}{2 \, \pi }\right ) + \pi \sin \left (\frac {1}{2 \, \pi }\right )\right )} \operatorname {S}\left (\frac {{\left (\pi b x - 1\right )} \sqrt {b^{2}}}{\pi b}\right ) - 2 \, {\left (\pi b^{2} x \sin \left (b x\right ) + 2 \, \pi b \cos \left (b x\right )\right )} \sin \left (\frac {1}{2} \, \pi b^{2} x^{2}\right )}{2 \, \pi ^{2} b^{4}} \] Input:

integrate(x^2*cos(b*x)*fresnel_cos(b*x),x, algorithm="fricas")
 

Output:

1/2*(4*pi^2*b^2*x*cos(b*x)*fresnel_cos(b*x) - 2*b*cos(1/2*pi*b^2*x^2)*cos( 
b*x) + 2*(pi^2*b^3*x^2 - 2*pi^2*b)*fresnel_cos(b*x)*sin(b*x) + sqrt(b^2)*( 
pi*cos(1/2/pi) - (2*pi^2 - 1)*sin(1/2/pi))*fresnel_cos((pi*b*x + 1)*sqrt(b 
^2)/(pi*b)) - sqrt(b^2)*(pi*cos(1/2/pi) - (2*pi^2 - 1)*sin(1/2/pi))*fresne 
l_cos((pi*b*x - 1)*sqrt(b^2)/(pi*b)) + sqrt(b^2)*((2*pi^2 - 1)*cos(1/2/pi) 
 + pi*sin(1/2/pi))*fresnel_sin((pi*b*x + 1)*sqrt(b^2)/(pi*b)) - sqrt(b^2)* 
((2*pi^2 - 1)*cos(1/2/pi) + pi*sin(1/2/pi))*fresnel_sin((pi*b*x - 1)*sqrt( 
b^2)/(pi*b)) - 2*(pi*b^2*x*sin(b*x) + 2*pi*b*cos(b*x))*sin(1/2*pi*b^2*x^2) 
)/(pi^2*b^4)
 

Sympy [F]

\[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x^{2} \cos {\left (b x \right )} \operatorname {Ci}{\left (b x \right )}\, dx \] Input:

integrate(x**2*cos(b*x)*Ci(b*x),x)
 

Output:

Integral(x**2*cos(b*x)*Ci(b*x), x)
 

Maxima [F]

\[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x^{2} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \] Input:

integrate(x^2*cos(b*x)*fresnel_cos(b*x),x, algorithm="maxima")
 

Output:

integrate(x^2*cos(b*x)*fresnel_cos(b*x), x)
 

Giac [F]

\[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int { x^{2} \cos \left (b x\right ) \operatorname {C}\left (b x\right ) \,d x } \] Input:

integrate(x^2*cos(b*x)*fresnel_cos(b*x),x, algorithm="giac")
 

Output:

integrate(x^2*cos(b*x)*fresnel_cos(b*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int x^2\,\mathrm {cosint}\left (b\,x\right )\,\cos \left (b\,x\right ) \,d x \] Input:

int(x^2*cosint(b*x)*cos(b*x),x)
 

Output:

int(x^2*cosint(b*x)*cos(b*x), x)
 

Reduce [F]

\[ \int x^2 \cos (b x) \operatorname {CosIntegral}(b x) \, dx=\int \mathit {ci} \left (b x \right ) \cos \left (b x \right ) x^{2}d x \] Input:

int(x^2*cos(b*x)*Ci(b*x),x)
 

Output:

int(ci(b*x)*cos(b*x)*x**2,x)