3.1.12 \(\int x \text {Shi}(b x)^2 \, dx\) [12]

3.1.12.1 Optimal result

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

output

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

3.1.12.2 Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int x \text {Shi}(b x)^2 \, dx=\frac {\cosh (2 b x)-2 \text {Chi}(2 b x)+2 \log (x)+(-4 b x \cosh (b x)+4 \sinh (b x)) \text {Shi}(b x)+2 b^2 x^2 \text {Shi}(b x)^2}{4 b^2} \]

input

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

output

(Cosh[2*b*x] - 2*CoshIntegral[2*b*x] + 2*Log[x] + (-4*b*x*Cosh[b*x] + 4*Si 
nh[b*x])*SinhIntegral[b*x] + 2*b^2*x^2*SinhIntegral[b*x]^2)/(4*b^2)
 

3.1.12.3 Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.05, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.625, Rules used = {7090, 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.

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

\(\Big \downarrow \) 7090

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

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3044

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

\(\Big \downarrow \) 15

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

\(\Big \downarrow \) 7100

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

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

output

Sinh[b*x]^2/(2*b^2) - (x*Cosh[b*x]*SinhIntegral[b*x])/b + (x^2*SinhIntegra 
l[b*x]^2)/2 + ((-1/2*CoshIntegral[2*b*x] + Log[x]/2)/b + (Sinh[b*x]*SinhIn 
tegral[b*x])/b)/b
 

3.1.12.3.1 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 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 7090

Int[(x_)^(m_.)*SinhIntegral[(b_.)*(x_)]^2, x_Symbol] :> Simp[x^(m + 1)*(Sin 
hIntegral[b*x]^2/(m + 1)), x] - Simp[2/(m + 1)   Int[x^m*Sinh[b*x]*SinhInte 
gral[b*x], x], x] /; FreeQ[b, x] && IGtQ[m, 0]
 

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]
 

3.1.12.4 Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.84

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

input

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

output

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

3.1.12.5 Fricas [F]

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

input

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

output

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

3.1.12.6 Sympy [F]

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

input

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

output

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

3.1.12.7 Maxima [F]

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

input

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

output

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

3.1.12.8 Giac [F]

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

input

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

output

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

3.1.12.9 Mupad [F(-1)]

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

input

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

output

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