3.28 \(\int \text{Shi}(a+b x)^2 \, dx\)

Optimal. Leaf size=48 \[ \frac{(a+b x) \text{Shi}(a+b x)^2}{b}+\frac{\text{Shi}(2 a+2 b x)}{b}-\frac{2 \text{Shi}(a+b x) \cosh (a+b x)}{b} \]

[Out]

(-2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + ((a + b*x)*SinhIntegral[a + b*x]^2)/b + SinhIntegral[2*a + 2*b*x]
/b

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Rubi [A]  time = 0.0732181, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6534, 6540, 5448, 12, 3298} \[ \frac{(a+b x) \text{Shi}(a+b x)^2}{b}+\frac{\text{Shi}(2 a+2 b x)}{b}-\frac{2 \text{Shi}(a+b x) \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[SinhIntegral[a + b*x]^2,x]

[Out]

(-2*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + ((a + b*x)*SinhIntegral[a + b*x]^2)/b + SinhIntegral[2*a + 2*b*x]
/b

Rule 6534

Int[SinhIntegral[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*SinhIntegral[a + b*x]^2)/b, x] - Dist[2,
Int[Sinh[a + b*x]*SinhIntegral[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 6540

Int[Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[(Cosh[a + b*x]*SinhIntegral[c
+ d*x])/b, x] - Dist[d/b, Int[(Cosh[a + b*x]*Sinh[c + d*x])/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 5448

Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int
[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n,
0] && IGtQ[p, 0]

Rule 12

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

Rule 3298

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(I*SinhIntegral[(c*f*fz)
/d + f*fz*x])/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*e - c*f*fz*I, 0]

Rubi steps

\begin{align*} \int \text{Shi}(a+b x)^2 \, dx &=\frac{(a+b x) \text{Shi}(a+b x)^2}{b}-2 \int \sinh (a+b x) \text{Shi}(a+b x) \, dx\\ &=-\frac{2 \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{(a+b x) \text{Shi}(a+b x)^2}{b}+2 \int \frac{\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=-\frac{2 \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{(a+b x) \text{Shi}(a+b x)^2}{b}+2 \int \frac{\sinh (2 a+2 b x)}{2 (a+b x)} \, dx\\ &=-\frac{2 \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{(a+b x) \text{Shi}(a+b x)^2}{b}+\int \frac{\sinh (2 a+2 b x)}{a+b x} \, dx\\ &=-\frac{2 \cosh (a+b x) \text{Shi}(a+b x)}{b}+\frac{(a+b x) \text{Shi}(a+b x)^2}{b}+\frac{\text{Shi}(2 a+2 b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0092112, size = 41, normalized size = 0.85 \[ \frac{(a+b x) \text{Shi}(a+b x)^2+\text{Shi}(2 (a+b x))-2 \text{Shi}(a+b x) \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Integrate[SinhIntegral[a + b*x]^2,x]

[Out]

(-2*Cosh[a + b*x]*SinhIntegral[a + b*x] + (a + b*x)*SinhIntegral[a + b*x]^2 + SinhIntegral[2*(a + b*x)])/b

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Maple [A]  time = 0.045, size = 43, normalized size = 0.9 \begin{align*}{\frac{ \left ( bx+a \right ) \left ({\it Shi} \left ( bx+a \right ) \right ) ^{2}-2\,\cosh \left ( bx+a \right ){\it Shi} \left ( bx+a \right ) +{\it Shi} \left ( 2\,bx+2\,a \right ) }{b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Shi(b*x+a)^2,x)

[Out]

1/b*((b*x+a)*Shi(b*x+a)^2-2*cosh(b*x+a)*Shi(b*x+a)+Shi(2*b*x+2*a))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x+a)^2,x, algorithm="maxima")

[Out]

integrate(Shi(b*x + a)^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\operatorname{Shi}\left (b x + a\right )^{2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(sinh_integral(b*x + a)^2, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{Shi}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x+a)**2,x)

[Out]

Integral(Shi(a + b*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\rm Shi}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Shi(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(Shi(b*x + a)^2, x)