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

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

[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)

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Rubi [A]  time = 0.0771015, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, 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 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]

Rule 12

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

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 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]

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*}

Mathematica [A]  time = 0.0565109, size = 74, normalized size = 0.97 \[ \frac{x^{m+1} \text{Chi}(b x)}{m+1}-\frac{-x^{m+1} (-b x)^{-m-1} \text{Gamma}(m+1,-b x)-x^{m+1} (b x)^{-m-1} \text{Gamma}(m+1,b x)}{2 (m+1)} \]

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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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}{\it Chi} \left ( bx \right ) \, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int x^{m}{\rm Chi}\left (b x\right )\,{d x} \end{align*}

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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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \operatorname{Chi}\left (b x\right ), x\right ) \end{align*}

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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Sympy [B]  time = 2.34736, size = 649, normalized size = 8.54 \begin{align*} \text{result too large to display} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m}{\rm Chi}\left (b x\right )\,{d x} \end{align*}

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)