Integrand size = 14, antiderivative size = 97 \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=-\frac {\cosh (2 a+2 b x)}{4 b^2}-\frac {\cosh (a+b x) \text {Chi}(a+b x)}{b^2}+\frac {\text {Chi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {a \text {Shi}(2 a+2 b x)}{2 b^2} \]
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+1/2*a*Shi(2*b*x+2*a)/b^2+x*Chi(b*x+a)*sinh(b*x+a)/b
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\frac {-\cosh (2 (a+b x))+2 \text {Chi}(2 (a+b x))+2 \log (a+b x)+4 \text {Chi}(a+b x) (-\cosh (a+b x)+b x \sinh (a+b x))+2 a \text {Shi}(2 (a+b x))}{4 b^2} \]
(-Cosh[2*(a + b*x)] + 2*CoshIntegral[2*(a + b*x)] + 2*Log[a + b*x] + 4*Cos hIntegral[a + b*x]*(-Cosh[a + b*x] + b*x*Sinh[a + b*x]) + 2*a*SinhIntegral [2*(a + b*x)])/(4*b^2)
Time = 0.94 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {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.
\(\displaystyle \int x \text {Chi}(a+b x) \cosh (a+b x) \, dx\) |
\(\Big \downarrow \) 7097 |
\(\displaystyle -\frac {\int \text {Chi}(a+b x) \sinh (a+b x)dx}{b}-\int \frac {x \cosh (a+b x) \sinh (a+b x)}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 6151 |
\(\displaystyle -\frac {\int \text {Chi}(a+b x) \sinh (a+b x)dx}{b}-\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 7101 |
\(\displaystyle -\frac {\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\int \frac {\cosh ^2(a+b x)}{a+b x}dx}{b}-\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^2}{a+b x}dx}{b}-\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\int \left (\frac {\cosh (2 a+2 b x)}{2 (a+b x)}+\frac {1}{2 (a+b x)}\right )dx}{b}-\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle -\frac {1}{2} \int \frac {x \sinh (2 a+2 b x)}{a+b x}dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle -\frac {1}{2} \int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right )dx+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {a \text {Shi}(2 a+2 b x)}{b^2}-\frac {\cosh (2 a+2 b x)}{2 b^2}\right )+\frac {x \text {Chi}(a+b x) \sinh (a+b x)}{b}-\frac {-\frac {\text {Chi}(2 a+2 b x)}{2 b}+\frac {\text {Chi}(a+b x) \cosh (a+b x)}{b}-\frac {\log (a+b x)}{2 b}}{b}\) |
-(((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 + (-1/2*Cosh[2*a + 2*b*x]/b^2 + (a*SinhIntegral[2*a + 2*b*x])/b^2)/2
3.2.28.3.1 Defintions of rubi rules used
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]))
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]
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]
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]
Time = 1.47 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}-\frac {\cosh \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(84\) |
default | \(\frac {\operatorname {Chi}\left (b x +a \right ) \left (-a \sinh \left (b x +a \right )+\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+\frac {a \,\operatorname {Shi}\left (2 b x +2 a \right )}{2}-\frac {\cosh \left (b x +a \right )^{2}}{2}+\frac {\ln \left (b x +a \right )}{2}+\frac {\operatorname {Chi}\left (2 b x +2 a \right )}{2}}{b^{2}}\) | \(84\) |
1/b^2*(Chi(b*x+a)*(-a*sinh(b*x+a)+(b*x+a)*sinh(b*x+a)-cosh(b*x+a))+1/2*a*S hi(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))
\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x \cosh {\left (a + b x \right )} \operatorname {Chi}\left (a + b x\right )\, dx \]
\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
\[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int { x {\rm Chi}\left (b x + a\right ) \cosh \left (b x + a\right ) \,d x } \]
Timed out. \[ \int x \cosh (a+b x) \text {Chi}(a+b x) \, dx=\int x\,\mathrm {coshint}\left (a+b\,x\right )\,\mathrm {cosh}\left (a+b\,x\right ) \,d x \]