3.1 \(\int x^m \text{Shi}(b x) \, dx\)

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

[Out]

-(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) + (x^(1 +
m)*SinhIntegral[b*x])/(1 + m)

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Rubi [A]  time = 0.0737281, 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 = {6532, 12, 3308, 2181} \[ -\frac{x^m (-b x)^{-m} \text{Gamma}(m+1,-b x)}{2 b (m+1)}-\frac{x^m (b x)^{-m} \text{Gamma}(m+1,b x)}{2 b (m+1)}+\frac{x^{m+1} \text{Shi}(b x)}{m+1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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) + (x^(1 +
m)*SinhIntegral[b*x])/(1 + m)

Rule 6532

Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*SinhInte
gral[a + b*x])/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[((c + d*x)^(m + 1)*Sinh[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 3308

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

Rubi steps

\begin{align*} \int x^m \text{Shi}(b x) \, dx &=\frac{x^{1+m} \text{Shi}(b x)}{1+m}-\frac{b \int \frac{x^m \sinh (b x)}{b} \, dx}{1+m}\\ &=\frac{x^{1+m} \text{Shi}(b x)}{1+m}-\frac{\int x^m \sinh (b x) \, dx}{1+m}\\ &=\frac{x^{1+m} \text{Shi}(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^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)}+\frac{x^{1+m} \text{Shi}(b x)}{1+m}\\ \end{align*}

Mathematica [A]  time = 0.0888748, size = 56, normalized size = 0.74 \[ -\frac{x^m \left ((-b x)^{-m} \text{Gamma}(m+1,-b x)+(b x)^{-m} \text{Gamma}(m+1,b x)-2 b x \text{Shi}(b x)\right )}{2 b (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.104, size = 37, normalized size = 0.5 \begin{align*}{\frac{b{x}^{2+m}}{2+m}{\mbox{$_2$F$_3$}({\frac{1}{2}},1+{\frac{m}{2}};\,{\frac{3}{2}},{\frac{3}{2}},2+{\frac{m}{2}};\,{\frac{{b}^{2}{x}^{2}}{4}})}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b/(2+m)*x^(2+m)*hypergeom([1/2,1+1/2*m],[3/2,3/2,2+1/2*m],1/4*b^2*x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Shi(b*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^m*sinh_integral(b*x), x)

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Sympy [A]  time = 1.03627, size = 44, normalized size = 0.58 \begin{align*} \frac{b x^{2} x^{m} \Gamma \left (\frac{m}{2} + 1\right ){{}_{2}F_{3}\left (\begin{matrix} \frac{1}{2}, \frac{m}{2} + 1 \\ \frac{3}{2}, \frac{3}{2}, \frac{m}{2} + 2 \end{matrix}\middle |{\frac{b^{2} x^{2}}{4}} \right )}}{2 \Gamma \left (\frac{m}{2} + 2\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x**2*x**m*gamma(m/2 + 1)*hyper((1/2, m/2 + 1), (3/2, 3/2, m/2 + 2), b**2*x**2/4)/(2*gamma(m/2 + 2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^m*Shi(b*x), x)