3.96 \(\int \text {Chi}(a+b x)^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.07, 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 = {6535, 6541, 5448, 12, 3298} \[ \frac {(a+b x) \text {Chi}(a+b x)^2}{b}-\frac {2 \text {Chi}(a+b x) \sinh (a+b x)}{b}+\frac {\text {Shi}(2 a+2 b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[CoshIntegral[a + b*x]^2,x]

[Out]

((a + b*x)*CoshIntegral[a + b*x]^2)/b - (2*CoshIntegral[a + b*x]*Sinh[a + b*x])/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 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 6535

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

Rule 6541

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

Rubi steps

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

Antiderivative was successfully verified.

[In]

Integrate[CoshIntegral[a + b*x]^2,x]

[Out]

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

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fricas [F]  time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\operatorname {Chi}\left (b x + a\right )^{2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(cosh_integral(b*x + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(b*x + a)^2, x)

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maple [A]  time = 0.00, size = 43, normalized size = 0.90 \[ \frac {\left (b x +a \right ) \Chi \left (b x +a \right )^{2}-2 \Chi \left (b x +a \right ) \sinh \left (b x +a \right )+\Shi \left (2 b x +2 a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(Chi(b*x + a)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {coshint}\left (a+b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coshint(a + b*x)^2,x)

[Out]

int(coshint(a + b*x)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(Chi(a + b*x)**2, x)

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