\(\int x \sinh (a+b x) \text {Shi}(a+b x) \, dx\) [56]

Optimal result

Integrand size = 14, antiderivative size = 97 \[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\frac {-\cosh (2 (a+b x))+2 \text {Chi}(2 (a+b x))-2 \log (a+b x)+4 (b x \cosh (a+b x)-\sinh (a+b x)) \text {Shi}(a+b x)+2 a \text {Shi}(2 (a+b x))}{4 b^2} \] Input:

Integrate[x*Sinh[a + b*x]*SinhIntegral[a + b*x],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.98 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {7096, 6151, 7100, 3042, 25, 3793, 2009, 7292, 7293, 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.

\(\displaystyle \int x \text {Shi}(a+b x) \sinh (a+b x) \, dx\)

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 6151

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

\(\Big \downarrow \) 7100

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

Input:

Int[x*Sinh[a + b*x]*SinhIntegral[a + b*x],x]
 

Output:

(x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b - (-1/2*CoshIntegral[2*a + 2*b*x 
]/b + Log[a + b*x]/(2*b) + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b)/b + (- 
1/2*Cosh[2*a + 2*b*x]/b^2 + (a*SinhIntegral[2*a + 2*b*x])/b^2)/2
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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 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 6151

Int[Cosh[w_]^(p_.)*(u_.)*Sinh[v_]^(p_.), x_Symbol] :> Simp[1/2^p   Int[u*Si 
nh[2*v]^p, x], x] /; EqQ[w, v] && IntegerQ[p]
 

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 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )-a \cosh \left (b x +a \right )\right )-\frac {\cosh \left (b x +a \right )^{2}}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(84\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )-a \cosh \left (b x +a \right )\right )-\frac {\cosh \left (b x +a \right )^{2}}{2}-\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(84\)

Input:

int(x*sinh(b*x+a)*Shi(b*x+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

\[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\int { x {\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right ) \,d x } \] Input:

integrate(x*sinh(b*x+a)*Shi(b*x+a),x, algorithm="fricas")
 

Output:

integral(x*sinh(b*x + a)*sinh_integral(b*x + a), x)
 

Sympy [F]

\[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\int x \sinh {\left (a + b x \right )} \operatorname {Shi}{\left (a + b x \right )}\, dx \] Input:

integrate(x*sinh(b*x+a)*Shi(b*x+a),x)
 

Output:

Integral(x*sinh(a + b*x)*Shi(a + b*x), x)
 

Maxima [F]

\[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\int { x {\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right ) \,d x } \] Input:

integrate(x*sinh(b*x+a)*Shi(b*x+a),x, algorithm="maxima")
 

Output:

integrate(x*Shi(b*x + a)*sinh(b*x + a), x)
 

Giac [F]

\[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\int { x {\rm Shi}\left (b x + a\right ) \sinh \left (b x + a\right ) \,d x } \] Input:

integrate(x*sinh(b*x+a)*Shi(b*x+a),x, algorithm="giac")
 

Output:

integrate(x*Shi(b*x + a)*sinh(b*x + a), x)
 

Mupad [F(-1)]

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

int(x*sinhint(a + b*x)*sinh(a + b*x),x)
 

Output:

int(x*sinhint(a + b*x)*sinh(a + b*x), x)
 

Reduce [F]

\[ \int x \sinh (a+b x) \text {Shi}(a+b x) \, dx=\int \mathit {shi} \left (b x +a \right ) \sinh \left (b x +a \right ) x d x \] Input:

int(x*sinh(b*x+a)*Shi(b*x+a),x)
 

Output:

int(shi(a + b*x)*sinh(a + b*x)*x,x)