3.1.95 \(\int x \text {Chi}(a+b x)^2 \, dx\) [95]

3.1.95.1 Optimal result

Integrand size = 10, antiderivative size = 154 \[ \int x \text {Chi}(a+b x)^2 \, dx=\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Chi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Chi}(a+b x)^2}{2 b}-\frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {\log (a+b x)}{2 b^2}+\frac {a \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {a \text {Shi}(2 a+2 b x)}{b^2} \]

output

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

3.1.95.2 Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.62 \[ \int x \text {Chi}(a+b x)^2 \, dx=\frac {\cosh (2 (a+b x))-2 \left (a^2-b^2 x^2\right ) \text {Chi}(a+b x)^2-2 \text {Chi}(2 (a+b x))-2 \log (a+b x)+4 \text {Chi}(a+b x) (\cosh (a+b x)+(a-b x) \sinh (a+b x))-4 a \text {Shi}(2 (a+b x))}{4 b^2} \]

input

Integrate[x*CoshIntegral[a + b*x]^2,x]
 

output

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

3.1.95.3 Rubi [A] (verified)

Time = 1.67 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.21, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.700, Rules used = {7093, 7089, 7095, 5971, 27, 3042, 26, 3779, 7097, 6151, 7101, 3042, 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.

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

\(\Big \downarrow \) 7093

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

\(\Big \downarrow \) 7089

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

\(\Big \downarrow \) 7095

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 6151

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

\(\Big \downarrow \) 7101

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

Int[x*CoshIntegral[a + b*x]^2,x]
 

output

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

3.1.95.3.1 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 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 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 7089

Int[CoshIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(CoshIn 
tegral[a + b*x]^2/b), x] - Simp[2   Int[Cosh[a + b*x]*CoshIntegral[a + b*x] 
, x], x] /; FreeQ[{a, b}, x]
 

rule 7093

Int[CoshIntegral[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(a + b*x)*(c + d*x)^m*(CoshIntegral[a + b*x]^2/(b*(m + 1))), x] + 
(-Simp[2/(m + 1)   Int[(c + d*x)^m*Cosh[a + b*x]*CoshIntegral[a + b*x], x], 
 x] + Simp[(b*c - a*d)*(m/(b*(m + 1)))   Int[(c + d*x)^(m - 1)*CoshIntegral 
[a + b*x]^2, x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
 

rule 7095

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

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 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] 
]
 

3.1.95.4 Maple [A] (verified)

Time = 0.92 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.73

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\frac {\left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\cosh \left (b x +a \right )}{2}\right )-a \,\operatorname {Shi}\left (2 b x +2 a \right )+\frac {\cosh \left (b x +a \right )^{2}}{2}-\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(113\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right )^{2} \left (-\left (b x +a \right ) a +\frac {\left (b x +a \right )^{2}}{2}\right )-2 \,\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\frac {\left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\cosh \left (b x +a \right )}{2}\right )-a \,\operatorname {Shi}\left (2 b x +2 a \right )+\frac {\cosh \left (b x +a \right )^{2}}{2}-\frac {\ln \left (b x +a \right )}{2}-\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\)	\(113\)

input

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

output

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

3.1.95.5 Fricas [F]

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

input

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

output

integral(x*cosh_integral(b*x + a)^2, x)
 

3.1.95.6 Sympy [F]

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

input

integrate(x*Chi(b*x+a)**2,x)
 

output

Integral(x*Chi(a + b*x)**2, x)
 

3.1.95.7 Maxima [F]

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

input

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

output

integrate(x*Chi(b*x + a)^2, x)
 

3.1.95.8 Giac [F]

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

input

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

output

integrate(x*Chi(b*x + a)^2, x)
 

3.1.95.9 Mupad [F(-1)]

Timed out. \[ \int x \text {Chi}(a+b x)^2 \, dx=\int x\,{\mathrm {coshint}\left (a+b\,x\right )}^2 \,d x \]

input

int(x*coshint(a + b*x)^2,x)
 

output

int(x*coshint(a + b*x)^2, x)