\(\int x^m \text {Chi}(b x) \, dx\) [69]

Optimal result

Integrand size = 8, antiderivative size = 76 \[ \int x^m \text {Chi}(b x) \, dx=\frac {x^{1+m} \text {Chi}(b x)}{1+m}-\frac {x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b (1+m)}+\frac {x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b (1+m)} \]

[Out]

x^(1+m)*Chi(b*x)/(1+m)-1/2*x^m*GAMMA(1+m,-b*x)/b/(1+m)/((-b*x)^m)+1/2*x^m*GAMMA(1+m,b*x)/b/(1+m)/((b*x)^m)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 76, 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, 3388, 2212} \[ \int x^m \text {Chi}(b x) \, dx=\frac {x^{m+1} \text {Chi}(b x)}{m+1}-\frac {x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b (m+1)}+\frac {x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b (m+1)} \]

[In]

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

[Out]

(x^(1 + m)*CoshIntegral[b*x])/(1 + m) - (x^m*Gamma[1 + m, -(b*x)])/(2*b*(1 + m)*(-(b*x))^m) + (x^m*Gamma[1 + m
, b*x])/(2*b*(1 + m)*(b*x)^m)

Rule 12

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

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

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 {x^{1+m} \text {Chi}(b x)}{1+m}-\frac {b \int \frac {x^m \cosh (b x)}{b} \, dx}{1+m} \\ & = \frac {x^{1+m} \text {Chi}(b x)}{1+m}-\frac {\int x^m \cosh (b x) \, dx}{1+m} \\ & = \frac {x^{1+m} \text {Chi}(b x)}{1+m}-\frac {\int e^{-b x} x^m \, dx}{2 (1+m)}-\frac {\int e^{b x} x^m \, dx}{2 (1+m)} \\ & = \frac {x^{1+m} \text {Chi}(b x)}{1+m}-\frac {x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b (1+m)}+\frac {x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b (1+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.87 \[ \int x^m \text {Chi}(b x) \, dx=\frac {x^m \left (2 x \text {Chi}(b x)+\frac {\left (-b^2 x^2\right )^{-m} \left (-(b x)^m \Gamma (1+m,-b x)+(-b x)^m \Gamma (1+m,b x)\right )}{b}\right )}{2 (1+m)} \]

[In]

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

[Out]

(x^m*(2*x*CoshIntegral[b*x] + (-((b*x)^m*Gamma[1 + m, -(b*x)]) + (-(b*x))^m*Gamma[1 + m, b*x])/(b*(-(b^2*x^2))
^m)))/(2*(1 + m))

Maple [F]

\[\int x^{m} \operatorname {Chi}\left (b x \right )d x\]

[In]

int(x^m*Chi(b*x),x)

[Out]

int(x^m*Chi(b*x),x)

Fricas [F]

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

[In]

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

[Out]

integral(x^m*cosh_integral(b*x), x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (60) = 120\).

Time = 1.27 (sec) , antiderivative size = 695, normalized size of antiderivative = 9.14 \[ \int x^m \text {Chi}(b x) \, dx=\frac {4 \cdot 2^{m} b b^{- m - 1} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b b^{- m - 1} m x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {4 \cdot 2^{m} b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \log {\left (b^{2} x^{2} \right )} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} - \frac {8 \cdot 2^{m} b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {8 \cdot 2^{m} \gamma b b^{- m - 1} x \sqrt {e^{- 2 m \log {\left (2 \right )}} e^{m \log {\left (b^{2} x^{2} \right )}}} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {b^{- m - 1} b^{m + 3} m^{2} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {2 b^{- m - 1} b^{m + 3} m x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {b^{- m - 1} b^{m + 3} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{3}F_{4}\left (\begin {matrix} 1, 1, \frac {m}{2} + \frac {3}{2} \\ \frac {3}{2}, 2, 2, \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b^{2} x^{2}}{4}} \right )}}{8 m^{2} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 16 m \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) + 8 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} \]

[In]

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

[Out]

4*2**m*b*b**(-m - 1)*m*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*log(b**2*x**2)*gamma(m/2 + 5/2)/(8*m**2*
gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) + 8*2**m*EulerGamma*b*b**(-m - 1)*m*x*sqrt(exp(
-2*m*log(2))*exp(m*log(b**2*x**2)))*gamma(m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamm
a(m/2 + 5/2)) + 4*2**m*b*b**(-m - 1)*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*log(b**2*x**2)*gamma(m/2 +
5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - 8*2**m*b*b**(-m - 1)*x*sqrt(exp
(-2*m*log(2))*exp(m*log(b**2*x**2)))*gamma(m/2 + 5/2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gam
ma(m/2 + 5/2)) + 8*2**m*EulerGamma*b*b**(-m - 1)*x*sqrt(exp(-2*m*log(2))*exp(m*log(b**2*x**2)))*gamma(m/2 + 5/
2)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) + b**(-m - 1)*b**(m + 3)*m**2*x**(m
+ 3)*gamma(m/2 + 3/2)*hyper((1, 1, m/2 + 3/2), (3/2, 2, 2, m/2 + 5/2), b**2*x**2/4)/(8*m**2*gamma(m/2 + 5/2) +
16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) + 2*b**(-m - 1)*b**(m + 3)*m*x**(m + 3)*gamma(m/2 + 3/2)*hyper((1
, 1, m/2 + 3/2), (3/2, 2, 2, m/2 + 5/2), b**2*x**2/4)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gam
ma(m/2 + 5/2)) + b**(-m - 1)*b**(m + 3)*x**(m + 3)*gamma(m/2 + 3/2)*hyper((1, 1, m/2 + 3/2), (3/2, 2, 2, m/2 +
5/2), b**2*x**2/4)/(8*m**2*gamma(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2))

Maxima [F]

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

[In]

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

[Out]

integrate(x^m*Chi(b*x), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^m*Chi(b*x), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^m*coshint(b*x), x)