\(\int \text {Shi}(b x)^2 \, dx\) [13]

Optimal result

Integrand size = 6, antiderivative size = 31 \[ \int \text {Shi}(b x)^2 \, dx=-\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6669, 6675, 12, 5556, 3379} \[ \int \text {Shi}(b x)^2 \, dx=x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b}-\frac {2 \text {Shi}(b x) \cosh (b x)}{b} \]

[In]

Int[SinhIntegral[b*x]^2,x]

[Out]

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

Rule 12

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

Rule 3379

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 5556

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 6669

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 6675

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]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \text {Shi}(b x)^2 \, dx=-\frac {2 \cosh (b x) \text {Shi}(b x)}{b}+x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97

method	result	size
derivativedivides	\(\frac {\operatorname {Shi}\left (b x \right )^{2} b x -2 \cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )+\operatorname {Shi}\left (2 b x \right )}{b}\)	\(30\)
default	\(\frac {\operatorname {Shi}\left (b x \right )^{2} b x -2 \cosh \left (b x \right ) \operatorname {Shi}\left (b x \right )+\operatorname {Shi}\left (2 b x \right )}{b}\)	\(30\)

[In]

int(Shi(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

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

[Out]

integral(sinh_integral(b*x)^2, x)

Sympy [F]

\[ \int \text {Shi}(b x)^2 \, dx=\int \operatorname {Shi}^{2}{\left (b x \right )}\, dx \]

[In]

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

[Out]

Integral(Shi(b*x)**2, x)

Maxima [F]

\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(Shi(b*x)^2, x)

Giac [F]

\[ \int \text {Shi}(b x)^2 \, dx=\int { {\rm Shi}\left (b x\right )^{2} \,d x } \]

[In]

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

[Out]

integrate(Shi(b*x)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \text {Shi}(b x)^2 \, dx=\int {\mathrm {sinhint}\left (b\,x\right )}^2 \,d x \]

[In]

int(sinhint(b*x)^2,x)

[Out]

int(sinhint(b*x)^2, x)