\(\int x \text {Chi}(b x)^2 \, dx\) [80]

Optimal result

Integrand size = 8, antiderivative size = 74 \[ \int x \text {Chi}(b x)^2 \, dx=\frac {\cosh (b x) \text {Chi}(b x)}{b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {\text {Chi}(2 b x)}{2 b^2}-\frac {\log (x)}{2 b^2}-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\sinh ^2(b x)}{2 b^2} \] Output:

cosh(b*x)*Chi(b*x)/b^2+1/2*x^2*Chi(b*x)^2-1/2*Chi(2*b*x)/b^2-1/2*ln(x)/b^2 
-x*Chi(b*x)*sinh(b*x)/b+1/2*sinh(b*x)^2/b^2
 

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.77 \[ \int x \text {Chi}(b x)^2 \, dx=\frac {\cosh (2 b x)+2 b^2 x^2 \text {Chi}(b x)^2-2 \text {Chi}(2 b x)-2 \log (x)+4 \text {Chi}(b x) (\cosh (b x)-b x \sinh (b x))}{4 b^2} \] Input:

Integrate[x*CoshIntegral[b*x]^2,x]
 

Output:

(Cosh[2*b*x] + 2*b^2*x^2*CoshIntegral[b*x]^2 - 2*CoshIntegral[2*b*x] - 2*L 
og[x] + 4*CoshIntegral[b*x]*(Cosh[b*x] - b*x*Sinh[b*x]))/(4*b^2)
 

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {7091, 7097, 27, 3042, 26, 3044, 15, 7101, 27, 3042, 3793, 2009}

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.

\(\displaystyle \int x \text {Chi}(b x)^2 \, dx\)

\(\Big \downarrow \) 7091

\(\displaystyle \frac {1}{2} x^2 \text {Chi}(b x)^2-\int x \cosh (b x) \text {Chi}(b x)dx\)

\(\Big \downarrow \) 7097

\(\displaystyle \frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\int \frac {\cosh (b x) \sinh (b x)}{b}dx+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\int \cosh (b x) \sinh (b x)dx}{b}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\int -i \cos (i b x) \sin (i b x)dx}{b}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {i \int \cos (i b x) \sin (i b x)dx}{b}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3044

\(\displaystyle -\frac {\int i \sinh (b x)d(i \sinh (b x))}{b^2}+\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 7101

\(\displaystyle \frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\int \frac {\cosh ^2(b x)}{b x}dx}{b}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{x}dx}{b}}{b}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sin \left (i b x+\frac {\pi }{2}\right )^2}{x}dx}{b}}{b}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \left (\frac {\cosh (2 b x)}{2 x}+\frac {1}{2 x}\right )dx}{b}}{b}+\frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sinh ^2(b x)}{2 b^2}+\frac {1}{2} x^2 \text {Chi}(b x)^2-\frac {x \text {Chi}(b x) \sinh (b x)}{b}+\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\frac {\text {Chi}(2 b x)}{2}+\frac {\log (x)}{2}}{b}}{b}\)

Input:

Int[x*CoshIntegral[b*x]^2,x]
 

Output:

(x^2*CoshIntegral[b*x]^2)/2 + ((Cosh[b*x]*CoshIntegral[b*x])/b - (CoshInte 
gral[2*b*x]/2 + Log[x]/2)/b)/b - (x*CoshIntegral[b*x]*Sinh[b*x])/b + Sinh[ 
b*x]^2/(2*b^2)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

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

rule 3044

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ 
Symbol] :> Simp[1/(a*f)   Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a 
*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&  !(I 
ntegerQ[(m - 1)/2] && LtQ[0, m, n])
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 7091

Int[CoshIntegral[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(Cos 
hIntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Cosh[b*x]*CoshInte 
gral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

rule 7097

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

rule 7101

Int[CoshIntegral[(c_.) + (d_.)*(x_)]*Sinh[(a_.) + (b_.)*(x_)], x_Symbol] :> 
 Simp[Cosh[a + b*x]*(CoshIntegral[c + d*x]/b), x] - Simp[d/b   Int[Cosh[a + 
 b*x]*(Cosh[c + d*x]/(c + d*x)), x], x] /; FreeQ[{a, b, c, d}, x]
 

Maple [A] (verified)

Time = 0.51 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )^{2}}{2}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b x \sinh \left (b x \right )}{2}-\frac {\cosh \left (b x \right )}{2}\right )+\frac {\cosh \left (b x \right )^{2}}{2}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\)	\(62\)
default	\(\frac {\frac {b^{2} x^{2} \operatorname {Chi}\left (b x \right )^{2}}{2}-2 \,\operatorname {Chi}\left (b x \right ) \left (\frac {b x \sinh \left (b x \right )}{2}-\frac {\cosh \left (b x \right )}{2}\right )+\frac {\cosh \left (b x \right )^{2}}{2}-\frac {\ln \left (b x \right )}{2}-\frac {\operatorname {Chi}\left (2 b x \right )}{2}}{b^{2}}\)	\(62\)

Input:

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

Output:

1/b^2*(1/2*b^2*x^2*Chi(b*x)^2-2*Chi(b*x)*(1/2*b*x*sinh(b*x)-1/2*cosh(b*x)) 
+1/2*cosh(b*x)^2-1/2*ln(b*x)-1/2*Chi(2*b*x))
 

Fricas [F]

\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \] Input:

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

Output:

integral(x*cosh_integral(b*x)^2, x)
 

Sympy [F]

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

integrate(x*Chi(b*x)**2,x)
 

Output:

Integral(x*Chi(b*x)**2, x)
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \] Input:

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

Output:

integrate(x*Chi(b*x)^2, x)
 

Giac [F]

\[ \int x \text {Chi}(b x)^2 \, dx=\int { x {\rm Chi}\left (b x\right )^{2} \,d x } \] Input:

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

Output:

integrate(x*Chi(b*x)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int x \text {Chi}(b x)^2 \, dx=\int x\,{\mathrm {coshint}\left (b\,x\right )}^2 \,d x \] Input:

int(x*coshint(b*x)^2,x)
 

Output:

int(x*coshint(b*x)^2, x)
 

Reduce [F]

\[ \int x \text {Chi}(b x)^2 \, dx=\int \chi \left (b x \right )^{2} x d x \] Input:

int(x*Chi(b*x)^2,x)
 

Output:

int(chi(b*x)**2*x,x)