3.69 \(\int x^m \text {Chi}(b x) \, dx\)

Optimal. Leaf size=76 \[ \frac {x^{m+1} \text {Chi}(b x)}{m+1}-\frac {x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b (m+1)}+\frac {x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b (m+1)} \]

[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)

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Rubi [A]  time = 0.08, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6533, 12, 3307, 2181} \[ -\frac {x^m (-b x)^{-m} \text {Gamma}(m+1,-b x)}{2 b (m+1)}+\frac {x^m (b x)^{-m} \text {Gamma}(m+1,b x)}{2 b (m+1)}+\frac {x^{m+1} \text {Chi}(b x)}{m+1} \]

Antiderivative was successfully verified.

[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 2181

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

Rule 3307

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 6533

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*} \int x^m \text {Chi}(b x) \, dx &=\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*}

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Mathematica [A]  time = 0.06, size = 74, normalized size = 0.97 \[ \frac {x^{m+1} \text {Chi}(b x)}{m+1}-\frac {-x^{m+1} (-b x)^{-m-1} \Gamma (m+1,-b x)-x^{m+1} (b x)^{-m-1} \Gamma (m+1,b x)}{2 (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {Chi}\left (b x\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*cosh_integral(b*x), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\rm Chi}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Chi(b*x), x)

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[ \int x^{m} \Chi \left (b x \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} {\rm Chi}\left (b x\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Chi(b*x), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\mathrm {coshint}\left (b\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^m*coshint(b*x), x)

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sympy [B]  time = 1.56, size = 649, normalized size = 8.54 \[ \frac {4 \cdot 2^{m} b^{- m} m x \sqrt {e^{- 2 m \log {\relax (2 )}} 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^{- m} m x \sqrt {e^{- 2 m \log {\relax (2 )}} 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^{- m} x \sqrt {e^{- 2 m \log {\relax (2 )}} 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^{- m} x \sqrt {e^{- 2 m \log {\relax (2 )}} 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^{- m} x \sqrt {e^{- 2 m \log {\relax (2 )}} 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^{2} m^{2} x^{3} x^{m} \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^{2} m x^{3} x^{m} \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^{2} x^{3} x^{m} \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 )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*2**m*b**(-m)*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**(-m)*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*gamma(m/2 + 5/2)
) + 4*2**m*b**(-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*gamm
a(m/2 + 5/2) + 16*m*gamma(m/2 + 5/2) + 8*gamma(m/2 + 5/2)) - 8*2**m*b**(-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*gamma(m/2 + 5/2)) + 8*2**m*
EulerGamma*b**(-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*gamma(m/2 + 5/2)) + b**2*m**2*x**3*x**m*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**2*m*x**3*x**m*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)) + b**2*x**3*x**m*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))

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