Integrand size = 8, antiderivative size = 49 \[ \int x^2 \text {Shi}(b x) \, dx=-\frac {2 \cosh (b x)}{3 b^3}-\frac {x^2 \cosh (b x)}{3 b}+\frac {2 x \sinh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Shi}(b x) \] Output:
-2/3*cosh(b*x)/b^3-1/3*x^2*cosh(b*x)/b+2/3*x*sinh(b*x)/b^2+1/3*x^3*Shi(b*x )
Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int x^2 \text {Shi}(b x) \, dx=-\frac {\left (2+b^2 x^2\right ) \cosh (b x)}{3 b^3}+\frac {2 x \sinh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Shi}(b x) \] Input:
Integrate[x^2*SinhIntegral[b*x],x]
Output:
-1/3*((2 + b^2*x^2)*Cosh[b*x])/b^3 + (2*x*Sinh[b*x])/(3*b^2) + (x^3*SinhIn tegral[b*x])/3
Result contains complex when optimal does not.
Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.20, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.375, Rules used = {7086, 27, 3042, 26, 3777, 3042, 3777, 26, 3042, 26, 3118}
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.
\(\displaystyle \int x^2 \text {Shi}(b x) \, dx\) |
\(\Big \downarrow \) 7086 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)-\frac {1}{3} b \int \frac {x^2 \sinh (b x)}{b}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)-\frac {1}{3} \int x^2 \sinh (b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)-\frac {1}{3} \int -i x^2 \sin (i b x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \int x^2 \sin (i b x)dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \int x \cosh (b x)dx}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \int x \sin \left (i b x+\frac {\pi }{2}\right )dx}{b}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \left (\frac {x \sinh (b x)}{b}-\frac {i \int -i \sinh (b x)dx}{b}\right )}{b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \left (\frac {x \sinh (b x)}{b}-\frac {\int \sinh (b x)dx}{b}\right )}{b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \left (\frac {x \sinh (b x)}{b}-\frac {\int -i \sin (i b x)dx}{b}\right )}{b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \left (\frac {x \sinh (b x)}{b}+\frac {i \int \sin (i b x)dx}{b}\right )}{b}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{3} x^3 \text {Shi}(b x)+\frac {1}{3} i \left (\frac {i x^2 \cosh (b x)}{b}-\frac {2 i \left (\frac {x \sinh (b x)}{b}-\frac {\cosh (b x)}{b^2}\right )}{b}\right )\) |
Input:
Int[x^2*SinhIntegral[b*x],x]
Output:
(I/3)*((I*x^2*Cosh[b*x])/b - ((2*I)*(-(Cosh[b*x]/b^2) + (x*Sinh[b*x])/b))/ b) + (x^3*SinhIntegral[b*x])/3
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(SinhIntegral[a + b*x]/(d*(m + 1))), x] - Simp[b/ (d*(m + 1)) Int[(c + d*x)^(m + 1)*(Sinh[a + b*x]/(a + b*x)), x], x] /; Fr eeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.43 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86
method | result | size |
parts | \(\frac {x^{3} \operatorname {Shi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \cosh \left (b x \right )-2 b x \sinh \left (b x \right )+2 \cosh \left (b x \right )}{3 b^{3}}\) | \(42\) |
derivativedivides | \(\frac {\frac {b^{3} x^{3} \operatorname {Shi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \cosh \left (b x \right )}{3}+\frac {2 b x \sinh \left (b x \right )}{3}-\frac {2 \cosh \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
default | \(\frac {\frac {b^{3} x^{3} \operatorname {Shi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \cosh \left (b x \right )}{3}+\frac {2 b x \sinh \left (b x \right )}{3}-\frac {2 \cosh \left (b x \right )}{3}}{b^{3}}\) | \(44\) |
meijerg | \(\frac {2 \sqrt {\pi }\, \left (\frac {1}{3 \sqrt {\pi }}-\frac {\left (\frac {b^{2} x^{2}}{2}+1\right ) \cosh \left (b x \right )}{3 \sqrt {\pi }}+\frac {b x \sinh \left (b x \right )}{3 \sqrt {\pi }}+\frac {b^{3} x^{3} \operatorname {Shi}\left (b x \right )}{6 \sqrt {\pi }}\right )}{b^{3}}\) | \(60\) |
Input:
int(x^2*Shi(b*x),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*Shi(b*x)-1/3/b^3*(b^2*x^2*cosh(b*x)-2*b*x*sinh(b*x)+2*cosh(b*x))
\[ \int x^2 \text {Shi}(b x) \, dx=\int { x^{2} {\rm Shi}\left (b x\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x),x, algorithm="fricas")
Output:
integral(x^2*sinh_integral(b*x), x)
Time = 0.91 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.94 \[ \int x^2 \text {Shi}(b x) \, dx=\frac {x^{3} \operatorname {Shi}{\left (b x \right )}}{3} - \frac {x^{2} \cosh {\left (b x \right )}}{3 b} + \frac {2 x \sinh {\left (b x \right )}}{3 b^{2}} - \frac {2 \cosh {\left (b x \right )}}{3 b^{3}} \] Input:
integrate(x**2*Shi(b*x),x)
Output:
x**3*Shi(b*x)/3 - x**2*cosh(b*x)/(3*b) + 2*x*sinh(b*x)/(3*b**2) - 2*cosh(b *x)/(3*b**3)
\[ \int x^2 \text {Shi}(b x) \, dx=\int { x^{2} {\rm Shi}\left (b x\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x),x, algorithm="maxima")
Output:
integrate(x^2*Shi(b*x), x)
\[ \int x^2 \text {Shi}(b x) \, dx=\int { x^{2} {\rm Shi}\left (b x\right ) \,d x } \] Input:
integrate(x^2*Shi(b*x),x, algorithm="giac")
Output:
integrate(x^2*Shi(b*x), x)
Timed out. \[ \int x^2 \text {Shi}(b x) \, dx=\frac {x^3\,\mathrm {sinhint}\left (b\,x\right )}{3}-\frac {\frac {2\,\mathrm {cosh}\left (b\,x\right )}{3}+\frac {b^2\,x^2\,\mathrm {cosh}\left (b\,x\right )}{3}-\frac {2\,b\,x\,\mathrm {sinh}\left (b\,x\right )}{3}}{b^3} \] Input:
int(x^2*sinhint(b*x),x)
Output:
(x^3*sinhint(b*x))/3 - ((2*cosh(b*x))/3 + (b^2*x^2*cosh(b*x))/3 - (2*b*x*s inh(b*x))/3)/b^3
Time = 0.24 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.88 \[ \int x^2 \text {Shi}(b x) \, dx=\frac {-\cosh \left (b x \right ) b^{2} x^{2}-2 \cosh \left (b x \right )+\mathit {shi} \left (b x \right ) b^{3} x^{3}+2 \sinh \left (b x \right ) b x}{3 b^{3}} \] Input:
int(x^2*Shi(b*x),x)
Output:
( - cosh(b*x)*b**2*x**2 - 2*cosh(b*x) + shi(b*x)*b**3*x**3 + 2*sinh(b*x)*b *x)/(3*b**3)