3.1.28 \(\int \text {Shi}(a+b x)^2 \, dx\) [28]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 48, normalized size of antiderivative = 1.00, 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 = {6669, 6675, 5556, 12, 3379} \begin {gather*} \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} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.01, size = 41, normalized size = 0.85 \begin {gather*} \frac {-2 \cosh (a+b x) \text {Shi}(a+b x)+(a+b x) \text {Shi}(a+b x)^2+\text {Shi}(2 (a+b x))}{b} \end {gather*}

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 = 0.36, size = 43, normalized size = 0.90

method	result	size
derivativedivides	\(\frac {\hyperbolicSineIntegral \left (b x +a \right )^{2} \left (b x +a \right )-2 \cosh \left (b x +a \right ) \hyperbolicSineIntegral \left (b x +a \right )+\hyperbolicSineIntegral \left (2 b x +2 a \right )}{b}\)	\(43\)
default	\(\frac {\hyperbolicSineIntegral \left (b x +a \right )^{2} \left (b x +a \right )-2 \cosh \left (b x +a \right ) \hyperbolicSineIntegral \left (b x +a \right )+\hyperbolicSineIntegral \left (2 b x +2 a \right )}{b}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {Shi}^{2}{\left (a + b x \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\mathrm {sinhint}\left (a+b\,x\right )}^2 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinhint(a + b*x)^2,x)

[Out]

int(sinhint(a + b*x)^2, x)

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