\(\int x^2 \text {Chi}(b x) \, dx\) [71]

Optimal result

Integrand size = 8, antiderivative size = 49 \[ \int x^2 \text {Chi}(b x) \, dx=\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {2 \sinh (b x)}{3 b^3}-\frac {x^2 \sinh (b x)}{3 b} \]

[Out]

1/3*x^3*Chi(b*x)+2/3*x*cosh(b*x)/b^2-2/3*sinh(b*x)/b^3-1/3*x^2*sinh(b*x)/b

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6668, 12, 3377, 2717} \[ \int x^2 \text {Chi}(b x) \, dx=-\frac {2 \sinh (b x)}{3 b^3}+\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b} \]

[In]

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

[Out]

(2*x*Cosh[b*x])/(3*b^2) + (x^3*CoshIntegral[b*x])/3 - (2*Sinh[b*x])/(3*b^3) - (x^2*Sinh[b*x])/(3*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3377

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(-(c + d*x)^m)*(Cos[e + f*x]/f), x]
+ Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 6668

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} x^3 \text {Chi}(b x)-\frac {1}{3} b \int \frac {x^2 \cosh (b x)}{b} \, dx \\ & = \frac {1}{3} x^3 \text {Chi}(b x)-\frac {1}{3} \int x^2 \cosh (b x) \, dx \\ & = \frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b}+\frac {2 \int x \sinh (b x) \, dx}{3 b} \\ & = \frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {x^2 \sinh (b x)}{3 b}-\frac {2 \int \cosh (b x) \, dx}{3 b^2} \\ & = \frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {2 \sinh (b x)}{3 b^3}-\frac {x^2 \sinh (b x)}{3 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int x^2 \text {Chi}(b x) \, dx=\frac {2 x \cosh (b x)}{3 b^2}+\frac {1}{3} x^3 \text {Chi}(b x)-\frac {\left (2+b^2 x^2\right ) \sinh (b x)}{3 b^3} \]

[In]

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

[Out]

(2*x*Cosh[b*x])/(3*b^2) + (x^3*CoshIntegral[b*x])/3 - ((2 + b^2*x^2)*Sinh[b*x])/(3*b^3)

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {x^{3} \operatorname {Chi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sinh \left (b x \right )-2 b x \cosh \left (b x \right )+2 \sinh \left (b x \right )}{3 b^{3}}\)	\(42\)
derivativedivides	\(\frac {\frac {b^{3} x^{3} \operatorname {Chi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sinh \left (b x \right )}{3}+\frac {2 b x \cosh \left (b x \right )}{3}-\frac {2 \sinh \left (b x \right )}{3}}{b^{3}}\)	\(44\)
default	\(\frac {\frac {b^{3} x^{3} \operatorname {Chi}\left (b x \right )}{3}-\frac {b^{2} x^{2} \sinh \left (b x \right )}{3}+\frac {2 b x \cosh \left (b x \right )}{3}-\frac {2 \sinh \left (b x \right )}{3}}{b^{3}}\)	\(44\)

[In]

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

[Out]

1/3*x^3*Chi(b*x)-1/3/b^3*(b^2*x^2*sinh(b*x)-2*b*x*cosh(b*x)+2*sinh(b*x))

Fricas [F]

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

[In]

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

[Out]

integral(x^2*cosh_integral(b*x), x)

Sympy [A] (verification not implemented)

Time = 1.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int x^2 \text {Chi}(b x) \, dx=- \frac {x^{3} \log {\left (b x \right )}}{3} + \frac {x^{3} \log {\left (b^{2} x^{2} \right )}}{6} + \frac {x^{3} \operatorname {Chi}\left (b x\right )}{3} - \frac {x^{2} \sinh {\left (b x \right )}}{3 b} + \frac {2 x \cosh {\left (b x \right )}}{3 b^{2}} - \frac {2 \sinh {\left (b x \right )}}{3 b^{3}} \]

[In]

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

[Out]

-x**3*log(b*x)/3 + x**3*log(b**2*x**2)/6 + x**3*Chi(b*x)/3 - x**2*sinh(b*x)/(3*b) + 2*x*cosh(b*x)/(3*b**2) - 2
*sinh(b*x)/(3*b**3)

Maxima [F]

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

[In]

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

[Out]

integrate(x^2*Chi(b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2*Chi(b*x), x)

Mupad [F(-1)]

Timed out. \[ \int x^2 \text {Chi}(b x) \, dx=\frac {x^3\,\mathrm {coshint}\left (b\,x\right )}{3}-\frac {\frac {2\,\mathrm {sinh}\left (b\,x\right )}{3}+\frac {b^2\,x^2\,\mathrm {sinh}\left (b\,x\right )}{3}-\frac {2\,b\,x\,\mathrm {cosh}\left (b\,x\right )}{3}}{b^3} \]

[In]

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

[Out]

(x^3*coshint(b*x))/3 - ((2*sinh(b*x))/3 + (b^2*x^2*sinh(b*x))/3 - (2*b*x*cosh(b*x))/3)/b^3