3.2.13 \(\int x^3 \cosh (b x) \text {Chi}(b x) \, dx\) [113]

3.2.13.1 Optimal result

Integrand size = 12, antiderivative size = 142 \[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\frac {x^2}{2 b^2}-\frac {3 \cosh ^2(b x)}{4 b^4}-\frac {6 \cosh (b x) \text {Chi}(b x)}{b^4}-\frac {3 x^2 \cosh (b x) \text {Chi}(b x)}{b^2}+\frac {3 \text {Chi}(2 b x)}{b^4}+\frac {3 \log (x)}{b^4}+\frac {2 x \cosh (b x) \sinh (b x)}{b^3}+\frac {6 x \text {Chi}(b x) \sinh (b x)}{b^3}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}-\frac {13 \sinh ^2(b x)}{4 b^4}-\frac {x^2 \sinh ^2(b x)}{2 b^2} \]

output

1/2*x^2/b^2+3*Chi(2*b*x)/b^4-6*Chi(b*x)*cosh(b*x)/b^4-3*x^2*Chi(b*x)*cosh( 
b*x)/b^2-3/4*cosh(b*x)^2/b^4+3*ln(x)/b^4+6*x*Chi(b*x)*sinh(b*x)/b^3+x^3*Ch 
i(b*x)*sinh(b*x)/b+2*x*cosh(b*x)*sinh(b*x)/b^3-13/4*sinh(b*x)^2/b^4-1/2*x^ 
2*sinh(b*x)^2/b^2
 

3.2.13.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.66 \[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\frac {3 b^2 x^2-8 \cosh (2 b x)-b^2 x^2 \cosh (2 b x)+12 \text {Chi}(2 b x)+12 \log (x)+4 \text {Chi}(b x) \left (-3 \left (2+b^2 x^2\right ) \cosh (b x)+b x \left (6+b^2 x^2\right ) \sinh (b x)\right )+4 b x \sinh (2 b x)}{4 b^4} \]

input

Integrate[x^3*Cosh[b*x]*CoshIntegral[b*x],x]
 

output

(3*b^2*x^2 - 8*Cosh[2*b*x] - b^2*x^2*Cosh[2*b*x] + 12*CoshIntegral[2*b*x] 
+ 12*Log[x] + 4*CoshIntegral[b*x]*(-3*(2 + b^2*x^2)*Cosh[b*x] + b*x*(6 + b 
^2*x^2)*Sinh[b*x]) + 4*b*x*Sinh[2*b*x])/(4*b^4)
 

3.2.13.3 Rubi [A] (verified)

Time = 1.26 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.50, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.917, Rules used = {7097, 27, 5895, 3042, 25, 3791, 15, 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.

\(\displaystyle \int x^3 \text {Chi}(b x) \cosh (b x) \, dx\)

\(\Big \downarrow \) 7097

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\int \frac {x^2 \cosh (b x) \sinh (b x)}{b}dx+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int x^2 \cosh (b x) \sinh (b x)dx}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 5895

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\frac {x^2 \sinh ^2(b x)}{2 b}-\frac {\int x \sinh ^2(b x)dx}{b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\frac {x^2 \sinh ^2(b x)}{2 b}-\frac {\int -x \sin (i b x)^2dx}{b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\frac {x^2 \sinh ^2(b x)}{2 b}+\frac {\int x \sin (i b x)^2dx}{b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle -\frac {\frac {\frac {\int xdx}{2}+\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}-\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \int x^2 \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 7103

\(\displaystyle -\frac {3 \left (-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\int \frac {x \cosh ^2(b x)}{b}dx+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\frac {\int x \cosh ^2(b x)dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\frac {\int x \sin \left (i b x+\frac {\pi }{2}\right )^2dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3791

\(\displaystyle -\frac {3 \left (-\frac {\frac {\int xdx}{2}-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}}{b}-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \left (-\frac {2 \int x \cosh (b x) \text {Chi}(b x)dx}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 7097

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\int \frac {\cosh (b x) \sinh (b x)}{b}dx+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int \cosh (b x) \sinh (b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\int -i \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {i \int \cos (i b x) \sin (i b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3044

\(\displaystyle -\frac {3 \left (-\frac {2 \left (\frac {\int i \sinh (b x)d(i \sinh (b x))}{b^2}-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\int \text {Chi}(b x) \sinh (b x)dx}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 7101

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\int \frac {\cosh ^2(b x)}{b x}dx}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\cosh ^2(b x)}{x}dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {3 \left (-\frac {2 \left (-\frac {\frac {\text {Chi}(b x) \cosh (b x)}{b}-\frac {\int \frac {\sin \left (i b x+\frac {\pi }{2}\right )^2}{x}dx}{b}}{b}-\frac {\sinh ^2(b x)}{2 b^2}+\frac {x \text {Chi}(b x) \sinh (b x)}{b}\right )}{b}-\frac {-\frac {\cosh ^2(b x)}{4 b^2}+\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \text {Chi}(b x) \cosh (b x)}{b}\right )}{b}-\frac {\frac {\frac {\sinh ^2(b x)}{4 b^2}-\frac {x \sinh (b x) \cosh (b x)}{2 b}+\frac {x^2}{4}}{b}+\frac {x^2 \sinh ^2(b x)}{2 b}}{b}+\frac {x^3 \text {Chi}(b x) \sinh (b x)}{b}\)

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

input

Int[x^3*Cosh[b*x]*CoshIntegral[b*x],x]
 

output

(x^3*CoshIntegral[b*x]*Sinh[b*x])/b - (3*((x^2*Cosh[b*x]*CoshIntegral[b*x] 
)/b - (x^2/4 - Cosh[b*x]^2/(4*b^2) + (x*Cosh[b*x]*Sinh[b*x])/(2*b))/b - (2 
*(-(((Cosh[b*x]*CoshIntegral[b*x])/b - (CoshIntegral[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))/b - 
((x^2*Sinh[b*x]^2)/(2*b) + (x^2/4 - (x*Cosh[b*x]*Sinh[b*x])/(2*b) + Sinh[b 
*x]^2/(4*b^2))/b)/b
 

3.2.13.3.1 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 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 5895

Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_)^(n_.) 
]^(p_.), x_Symbol] :> Simp[x^(m - n + 1)*(Sinh[a + b*x^n]^(p + 1)/(b*n*(p + 
 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1))   Int[x^(m - n)*Sinh[a + b*x^n]^ 
(p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -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]
 

3.2.13.4 Maple [A] (verified)

Time = 1.25 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )+b^{2} x^{2}-4 \cosh \left (b x \right )^{2}+3 \ln \left (b x \right )+3 \,\operatorname {Chi}\left (2 b x \right )}{b^{4}}\)	\(103\)
default	\(\frac {\operatorname {Chi}\left (b x \right ) \left (b^{3} x^{3} \sinh \left (b x \right )-3 b^{2} x^{2} \cosh \left (b x \right )+6 b x \sinh \left (b x \right )-6 \cosh \left (b x \right )\right )-\frac {b^{2} x^{2} \cosh \left (b x \right )^{2}}{2}+2 b x \cosh \left (b x \right ) \sinh \left (b x \right )+b^{2} x^{2}-4 \cosh \left (b x \right )^{2}+3 \ln \left (b x \right )+3 \,\operatorname {Chi}\left (2 b x \right )}{b^{4}}\)	\(103\)

input

int(x^3*Chi(b*x)*cosh(b*x),x,method=_RETURNVERBOSE)
 

output

1/b^4*(Chi(b*x)*(b^3*x^3*sinh(b*x)-3*b^2*x^2*cosh(b*x)+6*b*x*sinh(b*x)-6*c 
osh(b*x))-1/2*b^2*x^2*cosh(b*x)^2+2*b*x*cosh(b*x)*sinh(b*x)+b^2*x^2-4*cosh 
(b*x)^2+3*ln(b*x)+3*Chi(2*b*x))
 

3.2.13.5 Fricas [F]

\[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

input

integrate(x^3*Chi(b*x)*cosh(b*x),x, algorithm="fricas")
 

output

integral(x^3*cosh(b*x)*cosh_integral(b*x), x)
 

3.2.13.6 Sympy [F]

\[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\int x^{3} \cosh {\left (b x \right )} \operatorname {Chi}\left (b x\right )\, dx \]

input

integrate(x**3*Chi(b*x)*cosh(b*x),x)
 

output

Integral(x**3*cosh(b*x)*Chi(b*x), x)
 

3.2.13.7 Maxima [F]

\[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

input

integrate(x^3*Chi(b*x)*cosh(b*x),x, algorithm="maxima")
 

output

integrate(x^3*Chi(b*x)*cosh(b*x), x)
 

3.2.13.8 Giac [F]

\[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\int { x^{3} {\rm Chi}\left (b x\right ) \cosh \left (b x\right ) \,d x } \]

input

integrate(x^3*Chi(b*x)*cosh(b*x),x, algorithm="giac")
 

output

integrate(x^3*Chi(b*x)*cosh(b*x), x)
 

3.2.13.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \cosh (b x) \text {Chi}(b x) \, dx=\int x^3\,\mathrm {coshint}\left (b\,x\right )\,\mathrm {cosh}\left (b\,x\right ) \,d x \]

input

int(x^3*coshint(b*x)*cosh(b*x),x)
 

output

int(x^3*coshint(b*x)*cosh(b*x), x)