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

Optimal. Leaf size=154 \[ -\frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^2}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b^2}+\frac {a \text {Shi}(a+b x) \cosh (a+b x)}{b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b} \]

[Out]

-1/2*Chi(2*b*x+2*a)/b^2+1/4*cosh(2*b*x+2*a)/b^2+1/2*ln(b*x+a)/b^2+a*cosh(b*x+a)*Shi(b*x+a)/b^2-x*cosh(b*x+a)*S
hi(b*x+a)/b-1/2*a*(b*x+a)*Shi(b*x+a)^2/b^2+1/2*x*(b*x+a)*Shi(b*x+a)^2/b-a*Shi(2*b*x+2*a)/b^2+Shi(b*x+a)*sinh(b
*x+a)/b^2

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Rubi [A]  time = 0.33, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 14, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.400, Rules used = {6538, 6542, 5617, 6741, 6742, 2638, 3298, 6546, 3312, 3301, 6534, 6540, 5448, 12} \[ -\frac {\text {Chi}(2 a+2 b x)}{2 b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}-\frac {a \text {Shi}(2 a+2 b x)}{b^2}+\frac {\text {Shi}(a+b x) \sinh (a+b x)}{b^2}+\frac {a \text {Shi}(a+b x) \cosh (a+b x)}{b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {\cosh (2 a+2 b x)}{4 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {x \text {Shi}(a+b x) \cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[x*SinhIntegral[a + b*x]^2,x]

[Out]

Cosh[2*a + 2*b*x]/(4*b^2) - CoshIntegral[2*a + 2*b*x]/(2*b^2) + Log[a + b*x]/(2*b^2) + (a*Cosh[a + b*x]*SinhIn
tegral[a + b*x])/b^2 - (x*Cosh[a + b*x]*SinhIntegral[a + b*x])/b + (Sinh[a + b*x]*SinhIntegral[a + b*x])/b^2 -
(a*(a + b*x)*SinhIntegral[a + b*x]^2)/(2*b^2) + (x*(a + b*x)*SinhIntegral[a + b*x]^2)/(2*b) - (a*SinhIntegral
[2*a + 2*b*x])/b^2

Rule 12

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, 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]

Rule 3301

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[(c*f*fz)/d
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 3312

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

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 5617

Int[Cosh[w_]^(p_.)*(u_.)*Sinh[v_]^(p_.), x_Symbol] :> Dist[1/2^p, Int[u*Sinh[2*v]^p, x], x] /; EqQ[w, v] && In
tegerQ[p]

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 6538

Int[((c_.) + (d_.)*(x_))^(m_.)*SinhIntegral[(a_) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*(c + d*x)^m*Sin
hIntegral[a + b*x]^2)/(b*(m + 1)), x] + (-Dist[2/(m + 1), Int[(c + d*x)^m*Sinh[a + b*x]*SinhIntegral[a + b*x],
x], x] + Dist[((b*c - a*d)*m)/(b*(m + 1)), Int[(c + d*x)^(m - 1)*SinhIntegral[a + b*x]^2, x], x]) /; FreeQ[{a
, b, c, d}, x] && IGtQ[m, 0]

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 6542

Int[((e_.) + (f_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]*SinhIntegral[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[((
e + f*x)^m*Cosh[a + b*x]*SinhIntegral[c + d*x])/b, x] + (-Dist[d/b, Int[((e + f*x)^m*Cosh[a + b*x]*Sinh[c + d*
x])/(c + d*x), x], x] - Dist[(f*m)/b, Int[(e + f*x)^(m - 1)*Cosh[a + b*x]*SinhIntegral[c + d*x], x], x]) /; Fr
eeQ[{a, b, c, d, e, f}, x] && IGtQ[m, 0]

Rule 6546

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

Rule 6741

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int x \text {Shi}(a+b x)^2 \, dx &=\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {a \int \text {Shi}(a+b x)^2 \, dx}{2 b}-\int x \sinh (a+b x) \text {Shi}(a+b x) \, dx\\ &=-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}+\frac {\int \cosh (a+b x) \text {Shi}(a+b x) \, dx}{b}+\frac {a \int \sinh (a+b x) \text {Shi}(a+b x) \, dx}{b}+\int \frac {x \cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx\\ &=\frac {a \cosh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}+\frac {1}{2} \int \frac {x \sinh (2 (a+b x))}{a+b x} \, dx-\frac {\int \frac {\sinh ^2(a+b x)}{a+b x} \, dx}{b}-\frac {a \int \frac {\cosh (a+b x) \sinh (a+b x)}{a+b x} \, dx}{b}\\ &=\frac {a \cosh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}+\frac {1}{2} \int \frac {x \sinh (2 a+2 b x)}{a+b x} \, dx+\frac {\int \left (\frac {1}{2 (a+b x)}-\frac {\cosh (2 a+2 b x)}{2 (a+b x)}\right ) \, dx}{b}-\frac {a \int \frac {\sinh (2 a+2 b x)}{2 (a+b x)} \, dx}{b}\\ &=\frac {\log (a+b x)}{2 b^2}+\frac {a \cosh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}+\frac {1}{2} \int \left (\frac {\sinh (2 a+2 b x)}{b}+\frac {a \sinh (2 a+2 b x)}{b (-a-b x)}\right ) \, dx-\frac {\int \frac {\cosh (2 a+2 b x)}{a+b x} \, dx}{2 b}-\frac {a \int \frac {\sinh (2 a+2 b x)}{a+b x} \, dx}{2 b}\\ &=-\frac {\text {Chi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \cosh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {a \text {Shi}(2 a+2 b x)}{2 b^2}+\frac {\int \sinh (2 a+2 b x) \, dx}{2 b}+\frac {a \int \frac {\sinh (2 a+2 b x)}{-a-b x} \, dx}{2 b}\\ &=\frac {\cosh (2 a+2 b x)}{4 b^2}-\frac {\text {Chi}(2 a+2 b x)}{2 b^2}+\frac {\log (a+b x)}{2 b^2}+\frac {a \cosh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {x \cosh (a+b x) \text {Shi}(a+b x)}{b}+\frac {\sinh (a+b x) \text {Shi}(a+b x)}{b^2}-\frac {a (a+b x) \text {Shi}(a+b x)^2}{2 b^2}+\frac {x (a+b x) \text {Shi}(a+b x)^2}{2 b}-\frac {a \text {Shi}(2 a+2 b x)}{b^2}\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 95, normalized size = 0.62 \[ \frac {-2 \left (a^2-b^2 x^2\right ) \text {Shi}(a+b x)^2-2 \text {Chi}(2 (a+b x))-4 a \text {Shi}(2 (a+b x))+4 \text {Shi}(a+b x) (\sinh (a+b x)+(a-b x) \cosh (a+b x))+2 \log (a+b x)+\cosh (2 (a+b x))}{4 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*SinhIntegral[a + b*x]^2,x]

[Out]

(Cosh[2*(a + b*x)] - 2*CoshIntegral[2*(a + b*x)] + 2*Log[a + b*x] + 4*((a - b*x)*Cosh[a + b*x] + Sinh[a + b*x]
)*SinhIntegral[a + b*x] - 2*(a^2 - b^2*x^2)*SinhIntegral[a + b*x]^2 - 4*a*SinhIntegral[2*(a + b*x)])/(4*b^2)

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fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x \operatorname {Shi}\left (b x + a\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x*sinh_integral(b*x + a)^2, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Shi}\left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Shi(b*x + a)^2, x)

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maple [A]  time = 0.03, size = 135, normalized size = 0.88 \[ \frac {x^{2} \Shi \left (b x +a \right )^{2}}{2}-\frac {\Shi \left (b x +a \right )^{2} a^{2}}{2 b^{2}}-\frac {x \cosh \left (b x +a \right ) \Shi \left (b x +a \right )}{b}+\frac {a \cosh \left (b x +a \right ) \Shi \left (b x +a \right )}{b^{2}}+\frac {\Shi \left (b x +a \right ) \sinh \left (b x +a \right )}{b^{2}}+\frac {\cosh ^{2}\left (b x +a \right )}{2 b^{2}}+\frac {\ln \left (b x +a \right )}{2 b^{2}}-\frac {\Chi \left (2 b x +2 a \right )}{2 b^{2}}-\frac {a \Shi \left (2 b x +2 a \right )}{b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x {\rm Shi}\left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x*Shi(b*x + a)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,{\mathrm {sinhint}\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*sinhint(a + b*x)^2,x)

[Out]

int(x*sinhint(a + b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \operatorname {Shi}^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*Shi(a + b*x)**2, x)

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