\(\int x^2 \text {Chi}(b x) \sinh (b x) \, dx\) [119]

Optimal result

Integrand size = 12, antiderivative size = 109 \[ \int x^2 \text {Chi}(b x) \sinh (b x) \, dx=-\frac {x^2}{4 b}+\frac {\cosh ^2(b x)}{4 b^3}+\frac {2 \cosh (b x) \text {Chi}(b x)}{b^3}+\frac {x^2 \cosh (b x) \text {Chi}(b x)}{b}-\frac {\text {Chi}(2 b x)}{b^3}-\frac {\log (x)}{b^3}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}-\frac {2 x \text {Chi}(b x) \sinh (b x)}{b^2}+\frac {\sinh ^2(b x)}{b^3} \] Output:

-1/4*x^2/b+1/4*cosh(b*x)^2/b^3+2*cosh(b*x)*Chi(b*x)/b^3+x^2*cosh(b*x)*Chi( 
b*x)/b-Chi(2*b*x)/b^3-ln(x)/b^3-1/2*x*cosh(b*x)*sinh(b*x)/b^2-2*x*Chi(b*x) 
*sinh(b*x)/b^2+sinh(b*x)^2/b^3
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.66 \[ \int x^2 \text {Chi}(b x) \sinh (b x) \, dx=-\frac {2 b^2 x^2-5 \cosh (2 b x)+8 \text {Chi}(2 b x)+8 \log (x)-8 \text {Chi}(b x) \left (\left (2+b^2 x^2\right ) \cosh (b x)-2 b x \sinh (b x)\right )+2 b x \sinh (2 b x)}{8 b^3} \] Input:

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

Output:

-1/8*(2*b^2*x^2 - 5*Cosh[2*b*x] + 8*CoshIntegral[2*b*x] + 8*Log[x] - 8*Cos 
hIntegral[b*x]*((2 + b^2*x^2)*Cosh[b*x] - 2*b*x*Sinh[b*x]) + 2*b*x*Sinh[2* 
b*x])/b^3
 

Rubi [A] (verified)

Time = 0.75 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {7103, 27, 3042, 3791, 15, 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.

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

\(\Big \downarrow \) 7103

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3791

\(\displaystyle -\frac {\frac {\int xdx}{2}-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}}{b}-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 7097

\(\displaystyle -\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b}dx+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int \cosh (b x) \sinh (b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int -i \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {i \int \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3044

\(\displaystyle -\frac {2 \left (\frac {\int i \sinh (b x)d(i \sinh (b x))}{b^2}-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 7101

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\int \frac {\cosh ^2(b x)}{b x}dx}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{x}dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sin \left (i b x+\frac {\pi }{2}\right )^2}{x}dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \left (\frac {\cosh (2 b x)}{2 x}+\frac {1}{2 x}\right )dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\frac {\text {Chi}(2 b x)}{2}+\frac {\log (x)}{2}}{b}}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\)

Input:

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

Output:

(x^2*Cosh[b*x]*CoshIntegral[b*x])/b - (x^2/4 - Cosh[b*x]^2/(4*b^2) + (x*Co 
sh[b*x]*Sinh[b*x])/(2*b))/b - (2*(-(((Cosh[b*x]*CoshIntegral[b*x])/b - (Co 
shIntegral[2*b*x]/2 + Log[x]/2)/b)/b) + (x*CoshIntegral[b*x]*Sinh[b*x])/b 
- Sinh[b*x]^2/(2*b^2)))/b
 

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 3791

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

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 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]
 

rule 7103

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

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.72

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}-\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(78\)
default	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}-\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}-\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(78\)

Input:

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

Output:

1/b^3*(Chi(b*x)*(b^2*x^2*cosh(b*x)-2*b*x*sinh(b*x)+2*cosh(b*x))-1/2*b*x*co 
sh(b*x)*sinh(b*x)-1/4*b^2*x^2+5/4*cosh(b*x)^2-ln(b*x)-Chi(2*b*x))
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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