\(\int x^2 \sinh (a+b x) \text {Shi}(a+b x) \, dx\) [55]

Optimal result

Integrand size = 16, antiderivative size = 174 \[ \int x^2 \sinh (a+b x) \text {Shi}(a+b x) \, dx=-\frac {x}{b^2}+\frac {(a-b x) \cosh (2 a+2 b x)}{4 b^3}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}+\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{8 b^3}+\frac {2 \cosh (a+b x) \text {Shi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {2 x \sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3} \] Output:

-x/b^2+1/4*(-b*x+a)*cosh(2*b*x+2*a)/b^3-a*Chi(2*b*x+2*a)/b^3+a*ln(b*x+a)/b 
^3+cosh(b*x+a)*sinh(b*x+a)/b^3+1/8*sinh(2*b*x+2*a)/b^3+2*cosh(b*x+a)*Shi(b 
*x+a)/b^3+x^2*cosh(b*x+a)*Shi(b*x+a)/b-2*x*sinh(b*x+a)*Shi(b*x+a)/b^2-Shi( 
2*b*x+2*a)/b^3-1/2*a^2*Shi(2*b*x+2*a)/b^3
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.71 \[ \int x^2 \sinh (a+b x) \text {Shi}(a+b x) \, dx=\frac {-8 b x+2 a \cosh (2 (a+b x))-2 b x \cosh (2 (a+b x))-8 a \text {Chi}(2 (a+b x))+8 a \log (a+b x)+5 \sinh (2 (a+b x))+8 \left (\left (2+b^2 x^2\right ) \cosh (a+b x)-2 b x \sinh (a+b x)\right ) \text {Shi}(a+b x)-8 \text {Shi}(2 (a+b x))-4 a^2 \text {Shi}(2 (a+b x))}{8 b^3} \] Input:

Integrate[x^2*Sinh[a + b*x]*SinhIntegral[a + b*x],x]
 

Output:

(-8*b*x + 2*a*Cosh[2*(a + b*x)] - 2*b*x*Cosh[2*(a + b*x)] - 8*a*CoshIntegr 
al[2*(a + b*x)] + 8*a*Log[a + b*x] + 5*Sinh[2*(a + b*x)] + 8*((2 + b^2*x^2 
)*Cosh[a + b*x] - 2*b*x*Sinh[a + b*x])*SinhIntegral[a + b*x] - 8*SinhInteg 
ral[2*(a + b*x)] - 4*a^2*SinhIntegral[2*(a + b*x)])/(8*b^3)
 

Rubi [A] (verified)

Time = 2.02 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.21, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7096, 6151, 7102, 7094, 5971, 27, 3042, 26, 3779, 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.

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

\(\Big \downarrow \) 7096

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

\(\Big \downarrow \) 6151

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

\(\Big \downarrow \) 7102

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

\(\Big \downarrow \) 7094

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {a^2 \text {Shi}(2 a+2 b x)}{b^3}+\frac {\sinh (2 a+2 b x)}{4 b^3}+\frac {a \cosh (2 a+2 b x)}{2 b^3}-\frac {x \cosh (2 a+2 b x)}{2 b^2}\right )-\frac {2 \left (\frac {a \text {Chi}(2 a+2 b x)}{2 b^2}-\frac {a \log (a+b x)}{2 b^2}-\frac {\sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x \text {Shi}(a+b x) \sinh (a+b x)}{b}-\frac {\frac {\text {Shi}(a+b x) \cosh (a+b x)}{b}-\frac {\text {Shi}(2 a+2 b x)}{2 b}}{b}+\frac {x}{2 b}\right )}{b}+\frac {x^2 \text {Shi}(a+b x) \cosh (a+b x)}{b}\)

Input:

Int[x^2*Sinh[a + b*x]*SinhIntegral[a + b*x],x]
 

Output:

(x^2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + ((a*Cosh[2*a + 2*b*x])/(2*b^ 
3) - (x*Cosh[2*a + 2*b*x])/(2*b^2) + Sinh[2*a + 2*b*x]/(4*b^3) - (a^2*Sinh 
Integral[2*a + 2*b*x])/b^3)/2 - (2*(x/(2*b) + (a*CoshIntegral[2*a + 2*b*x] 
)/(2*b^2) - (a*Log[a + b*x])/(2*b^2) - (Cosh[a + b*x]*Sinh[a + b*x])/(2*b^ 
2) + (x*Sinh[a + b*x]*SinhIntegral[a + b*x])/b - ((Cosh[a + b*x]*SinhInteg 
ral[a + b*x])/b - SinhIntegral[2*a + 2*b*x]/(2*b))/b))/b
 

Defintions of rubi rules used

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 3779

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo 
l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f 
, fz}, x] && EqQ[d*e - c*f*fz*I, 0]
 

rule 5971

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + 
(b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + 
b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & 
& IGtQ[p, 0]
 

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 7094

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

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 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]
 

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 = 1.64 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \,\operatorname {Chi}\left (2 b x +2 a \right )+a \ln \left (b x +a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\)	\(174\)
default	\(\frac {\operatorname {Shi}\left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \,\operatorname {Chi}\left (2 b x +2 a \right )+a \ln \left (b x +a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )}{4}-\frac {3 b x}{4}-\frac {3 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\)	\(174\)

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(x^2*sinh(b*x + a)*sinh_integral(b*x + a), x)
 

Sympy [F]

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

integrate(x**2*sinh(b*x+a)*Shi(b*x+a),x)
 

Output:

Integral(x**2*sinh(a + b*x)*Shi(a + b*x), x)
 

Maxima [F]

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

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

Output:

integrate(x^2*Shi(b*x + a)*sinh(b*x + a), x)
 

Giac [F]

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

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

Output:

integrate(x^2*Shi(b*x + a)*sinh(b*x + a), x)
 

Mupad [F(-1)]

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

int(x^2*sinhint(a + b*x)*sinh(a + b*x),x)
 

Output:

int(x^2*sinhint(a + b*x)*sinh(a + b*x), x)
 

Reduce [F]

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

int(x^2*sinh(b*x+a)*Shi(b*x+a),x)
 

Output:

int(shi(a + b*x)*sinh(a + b*x)*x**2,x)