3.13 \(\int \text {Shi}(b x)^2 \, dx\)

Optimal. Leaf size=31 \[ x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b}-\frac {2 \text {Shi}(b x) \cosh (b x)}{b} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {6534, 6540, 12, 5448, 3298} \[ x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b}-\frac {2 \text {Shi}(b x) \cosh (b x)}{b} \]

Antiderivative was successfully verified.

[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 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 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 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]

Rubi steps

\begin {align*} \int \text {Shi}(b x)^2 \, dx &=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]  time = 0.01, size = 31, normalized size = 1.00 \[ x \text {Shi}(b x)^2+\frac {\text {Shi}(2 b x)}{b}-\frac {2 \text {Shi}(b x) \cosh (b x)}{b} \]

Antiderivative was successfully verified.

[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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sinh_integral(b*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)^2, x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 30, normalized size = 0.97 \[ \frac {b x \Shi \left (b x \right )^{2}-2 \cosh \left (b x \right ) \Shi \left (b x \right )+\Shi \left (2 b x \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Shi(b*x)^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \[ \int {\mathrm {sinhint}\left (b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(sinhint(b*x)^2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(Shi(b*x)**2, x)

________________________________________________________________________________________