3.2.27 \(\int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx\) [127]

3.2.27.1 Optimal result

Integrand size = 16, antiderivative size = 186 \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {x}{b^2}+\frac {a \cosh (2 a+2 b x)}{4 b^3}-\frac {x \cosh (2 a+2 b x)}{4 b^2}-\frac {2 x \cosh (a+b x) \text {Chi}(a+b x)}{b^2}-\frac {a \text {Chi}(2 a+2 b x)}{b^3}-\frac {a \log (a+b x)}{b^3}+\frac {\cosh (a+b x) \sinh (a+b x)}{b^3}+\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b^3}+\frac {x^2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {\sinh (2 a+2 b x)}{8 b^3}-\frac {\text {Shi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Shi}(2 a+2 b x)}{2 b^3} \]

output

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

3.2.27.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.66 \[ \int x^2 \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {8 b x+2 a \cosh (2 (a+b x))-2 b x \cosh (2 (a+b x))-8 a \text {Chi}(2 (a+b x))-8 a \log (a+b x)+8 \text {Chi}(a+b x) \left (-2 b x \cosh (a+b x)+\left (2+b^2 x^2\right ) \sinh (a+b x)\right )+5 \sinh (2 (a+b x))-8 \text {Shi}(2 (a+b x))-4 a^2 \text {Shi}(2 (a+b x))}{8 b^3} \]

input

Integrate[x^2*Cosh[a + b*x]*CoshIntegral[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*CoshIntegra 
l[2*(a + b*x)] - 8*a*Log[a + b*x] + 8*CoshIntegral[a + b*x]*(-2*b*x*Cosh[a 
 + b*x] + (2 + b^2*x^2)*Sinh[a + b*x]) + 5*Sinh[2*(a + b*x)] - 8*SinhInteg 
ral[2*(a + b*x)] - 4*a^2*SinhIntegral[2*(a + b*x)])/(8*b^3)
 

3.2.27.3 Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {7097, 6151, 7103, 7095, 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.

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

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 6151

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

\(\Big \downarrow \) 7103

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

\(\Big \downarrow \) 7095

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

\(\Big \downarrow \) 5971

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3779

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

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

3.2.27.3.1 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 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 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 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]
 

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] 
]
 

3.2.27.4 Maple [A] (verified)

Time = 2.55 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \ln \left (b x +a \right )-a \,\operatorname {Chi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {5 b x}{4}+\frac {5 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\)	\(175\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )\right )-\frac {a^{2} \operatorname {Shi}\left (2 b x +2 a \right )}{2}+a \cosh \left (b x +a \right )^{2}-a \ln \left (b x +a \right )-a \,\operatorname {Chi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{2}+\frac {5 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{4}+\frac {5 b x}{4}+\frac {5 a}{4}-\operatorname {Shi}\left (2 b x +2 a \right )}{b^{3}}\)	\(175\)

input

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

output

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

3.2.27.5 Fricas [F]

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

input

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

output

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

3.2.27.6 Sympy [F]

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

input

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

output

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

3.2.27.7 Maxima [F]

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

input

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

output

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

3.2.27.8 Giac [F]

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

input

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

output

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

3.2.27.9 Mupad [F(-1)]

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

input

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

output

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