\(\int x^2 \cosh (b x) \text {Shi}(b x) \, dx\) [51]

Optimal result

Integrand size = 12, antiderivative size = 98 \[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=\frac {x^2}{4 b}-\frac {\text {Chi}(2 b x)}{b^3}+\frac {\log (x)}{b^3}-\frac {x \cosh (b x) \sinh (b x)}{2 b^2}+\frac {5 \sinh ^2(b x)}{4 b^3}-\frac {2 x \cosh (b x) \text {Shi}(b x)}{b^2}+\frac {2 \sinh (b x) \text {Shi}(b x)}{b^3}+\frac {x^2 \sinh (b x) \text {Shi}(b x)}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.73 \[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=\frac {2 b^2 x^2+5 \cosh (2 b x)-8 \text {Chi}(2 b x)+8 \log (x)-2 b x \sinh (2 b x)+8 \left (-2 b x \cosh (b x)+\left (2+b^2 x^2\right ) \sinh (b x)\right ) \text {Shi}(b x)}{8 b^3} \] Input:

Integrate[x^2*Cosh[b*x]*SinhIntegral[b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.30, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {7102, 27, 3042, 25, 3791, 15, 7096, 27, 3042, 26, 3044, 15, 7100, 27, 3042, 25, 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.

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

\(\Big \downarrow \) 7102

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3791

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

Input:

Int[x^2*Cosh[b*x]*SinhIntegral[b*x],x]
 

Output:

(x^2/4 - (x*Cosh[b*x]*Sinh[b*x])/(2*b) + Sinh[b*x]^2/(4*b^2))/b + (x^2*Sin 
h[b*x]*SinhIntegral[b*x])/b - (2*(-1/2*Sinh[b*x]^2/b^2 + (x*Cosh[b*x]*Sinh 
Integral[b*x])/b - ((-1/2*CoshIntegral[2*b*x] + Log[x]/2)/b + (Sinh[b*x]*S 
inhIntegral[b*x])/b)/b))/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 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 7096

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

rule 7100

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

rule 7102

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

Maple [A] (verified)

Time = 0.99 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.78

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \sinh \left (b x \right )-2 b x \cosh \left (b x \right )+2 \sinh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(76\)
default	\(\frac {\operatorname {Shi}\left (b x \right ) \left (b^{2} x^{2} \sinh \left (b x \right )-2 b x \cosh \left (b x \right )+2 \sinh \left (b x \right )\right )-\frac {b x \cosh \left (b x \right ) \sinh \left (b x \right )}{2}+\frac {b^{2} x^{2}}{4}+\frac {5 \cosh \left (b x \right )^{2}}{4}+\ln \left (b x \right )-\operatorname {Chi}\left (2 b x \right )}{b^{3}}\)	\(76\)

Input:

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

Output:

1/b^3*(Shi(b*x)*(b^2*x^2*sinh(b*x)-2*b*x*cosh(b*x)+2*sinh(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]

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

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

Output:

integral(x^2*cosh(b*x)*sinh_integral(b*x), x)
 

Sympy [F]

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

integrate(x**2*cosh(b*x)*Shi(b*x),x)
 

Output:

Integral(x**2*cosh(b*x)*Shi(b*x), x)
 

Maxima [F]

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

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

Output:

integrate(x^2*Shi(b*x)*cosh(b*x), x)
 

Giac [F]

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

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

Output:

integrate(x^2*Shi(b*x)*cosh(b*x), x)
 

Mupad [F(-1)]

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

int(x^2*sinhint(b*x)*cosh(b*x),x)
 

Output:

int(x^2*sinhint(b*x)*cosh(b*x), x)
 

Reduce [F]

\[ \int x^2 \cosh (b x) \text {Shi}(b x) \, dx=\int \cosh \left (b x \right ) \mathit {shi} \left (b x \right ) x^{2}d x \] Input:

int(x^2*cosh(b*x)*Shi(b*x),x)
 

Output:

int(cosh(b*x)*shi(b*x)*x**2,x)