3.2.23 \(\int x^2 \text {Chi}(a+b x) \sinh (a+b x) \, dx\) [123]

3.2.23.1 Optimal result

Integrand size = 16, antiderivative size = 220 \[ \int x^2 \text {Chi}(a+b x) \sinh (a+b x) \, dx=\frac {a x}{2 b^2}-\frac {x^2}{4 b}+\frac {\cosh ^2(a+b x)}{4 b^3}+\frac {\cosh (2 a+2 b x)}{2 b^3}+\frac {2 \cosh (a+b x) \text {Chi}(a+b x)}{b^3}+\frac {x^2 \cosh (a+b x) \text {Chi}(a+b x)}{b}-\frac {\text {Chi}(2 a+2 b x)}{b^3}-\frac {a^2 \text {Chi}(2 a+2 b x)}{2 b^3}-\frac {\log (a+b x)}{b^3}-\frac {a^2 \log (a+b x)}{2 b^3}+\frac {a \cosh (a+b x) \sinh (a+b x)}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}-\frac {2 x \text {Chi}(a+b x) \sinh (a+b x)}{b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^3} \]

output

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

3.2.23.2 Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.61 \[ \int x^2 \text {Chi}(a+b x) \sinh (a+b x) \, dx=-\frac {-4 a b x+2 b^2 x^2-5 \cosh (2 (a+b x))+4 \left (2+a^2\right ) \text {Chi}(2 (a+b x))+8 \log (a+b x)+4 a^2 \log (a+b x)-8 \text {Chi}(a+b x) \left (\left (2+b^2 x^2\right ) \cosh (a+b x)-2 b x \sinh (a+b x)\right )-2 a \sinh (2 (a+b x))+2 b x \sinh (2 (a+b x))+8 a \text {Shi}(2 (a+b x))}{8 b^3} \]

input

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

output

-1/8*(-4*a*b*x + 2*b^2*x^2 - 5*Cosh[2*(a + b*x)] + 4*(2 + a^2)*CoshIntegra 
l[2*(a + b*x)] + 8*Log[a + b*x] + 4*a^2*Log[a + b*x] - 8*CoshIntegral[a + 
b*x]*((2 + b^2*x^2)*Cosh[a + b*x] - 2*b*x*Sinh[a + b*x]) - 2*a*Sinh[2*(a + 
 b*x)] + 2*b*x*Sinh[2*(a + b*x)] + 8*a*SinhIntegral[2*(a + b*x)])/b^3
 

3.2.23.3 Rubi [A] (verified)

Time = 2.01 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {7103, 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.

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

\(\Big \downarrow \) 7103

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

\(\Big \downarrow \) 7097

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

\(\Big \downarrow \) 6151

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

\(\Big \downarrow \) 7101

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

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

input

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

output

(a*x)/(2*b^2) - x^2/(4*b) + Cosh[a + b*x]^2/(4*b^3) + (x^2*Cosh[a + b*x]*C 
oshIntegral[a + b*x])/b - (a^2*CoshIntegral[2*a + 2*b*x])/(2*b^3) - (a^2*L 
og[a + b*x])/(2*b^3) + (a*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^3) - (x*Cosh[a 
 + b*x]*Sinh[a + b*x])/(2*b^2) - (2*(-(((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*Cosh 
Integral[a + b*x]*Sinh[a + b*x])/b + (-1/2*Cosh[2*a + 2*b*x]/b^2 + (a*Sinh 
Integral[2*a + 2*b*x])/b^2)/2))/b
 

3.2.23.3.1 Defintions of rubi rules used

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 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]
 

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] 
]
 

3.2.23.4 Maple [A] (verified)

Time = 1.74 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Chi}\left (2 b x +2 a \right )}{2}+\cosh \left (b x +a \right ) \sinh \left (b x +a \right ) a +\left (b x +a \right ) a -a \,\operatorname {Shi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \cosh \left (b x +a \right )^{2}}{4}-\ln \left (b x +a \right )-\operatorname {Chi}\left (2 b x +2 a \right )}{b^{3}}\)	\(198\)
default	\(\frac {\operatorname {Chi}\left (b x +a \right ) \left (a^{2} \cosh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \cosh \left (b x +a \right )-\sinh \left (b x +a \right )\right )+\left (b x +a \right )^{2} \cosh \left (b x +a \right )-2 \left (b x +a \right ) \sinh \left (b x +a \right )+2 \cosh \left (b x +a \right )\right )-\frac {a^{2} \ln \left (b x +a \right )}{2}-\frac {a^{2} \operatorname {Chi}\left (2 b x +2 a \right )}{2}+\cosh \left (b x +a \right ) \sinh \left (b x +a \right ) a +\left (b x +a \right ) a -a \,\operatorname {Shi}\left (2 b x +2 a \right )-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{2}-\frac {\left (b x +a \right )^{2}}{4}+\frac {5 \cosh \left (b x +a \right )^{2}}{4}-\ln \left (b x +a \right )-\operatorname {Chi}\left (2 b x +2 a \right )}{b^{3}}\)	\(198\)

input

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

output

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

3.2.23.5 Fricas [F]

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

input

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

output

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

3.2.23.6 Sympy [F]

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

input

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

output

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

3.2.23.7 Maxima [F]

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

input

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

output

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

3.2.23.8 Giac [F]

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

input

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

output

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

3.2.23.9 Mupad [F(-1)]

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

input

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

output

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