3.107 \(\int \frac{\cosh (b x) \text{Chi}(b x)}{x^3} \, dx\)

Optimal. Leaf size=96 \[ \frac{1}{4} b^2 \text{Chi}(b x)^2+b^2 \text{Chi}(2 b x)-\frac{\text{Chi}(b x) \cosh (b x)}{2 x^2}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}-\frac{\cosh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]

[Out]

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

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Rubi [A]  time = 0.213058, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6545, 6551, 6686, 12, 5448, 3297, 3301, 3314, 29, 3312} \[ \frac{1}{4} b^2 \text{Chi}(b x)^2+b^2 \text{Chi}(2 b x)-\frac{\text{Chi}(b x) \cosh (b x)}{2 x^2}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}-\frac{\cosh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[b*x]*CoshIntegral[b*x])/x^3,x]

[Out]

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

Rule 6545

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

Rule 6551

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 12

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

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 3297

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[((c + d*x)^(m + 1)*Sin[e + f*x])/(d*(
m + 1)), x] - Dist[f/(d*(m + 1)), Int[(c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[
m, -1]

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 3314

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(b*Si
n[e + f*x])^n)/(d*(m + 1)), x] + (Dist[(b^2*f^2*n*(n - 1))/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin
[e + f*x])^(n - 2), x], x] - Dist[(f^2*n^2)/(d^2*(m + 1)*(m + 2)), Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x
], x] - Simp[(b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*(b*Sin[e + f*x])^(n - 1))/(d^2*(m + 1)*(m + 2)), x]) /; Fre
eQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

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])
)

Rubi steps

\begin{align*} \int \frac{\cosh (b x) \text{Chi}(b x)}{x^3} \, dx &=-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{2} b \int \frac{\cosh ^2(b x)}{b x^3} \, dx+\frac{1}{2} b \int \frac{\text{Chi}(b x) \sinh (b x)}{x^2} \, dx\\ &=-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}+\frac{1}{2} \int \frac{\cosh ^2(b x)}{x^3} \, dx+\frac{1}{2} b^2 \int \frac{\cosh (b x) \text{Chi}(b x)}{x} \, dx+\frac{1}{2} b^2 \int \frac{\cosh (b x) \sinh (b x)}{b x^2} \, dx\\ &=-\frac{\cosh ^2(b x)}{4 x^2}-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)^2-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}+\frac{1}{2} b \int \frac{\cosh (b x) \sinh (b x)}{x^2} \, dx-\frac{1}{2} b^2 \int \frac{1}{x} \, dx+b^2 \int \frac{\cosh ^2(b x)}{x} \, dx\\ &=-\frac{\cosh ^2(b x)}{4 x^2}-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)^2-\frac{1}{2} b^2 \log (x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}+\frac{1}{2} b \int \frac{\sinh (2 b x)}{2 x^2} \, dx+b^2 \int \left (\frac{1}{2 x}+\frac{\cosh (2 b x)}{2 x}\right ) \, dx\\ &=-\frac{\cosh ^2(b x)}{4 x^2}-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)^2-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}+\frac{1}{4} b \int \frac{\sinh (2 b x)}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{\cosh (2 b x)}{x} \, dx\\ &=-\frac{\cosh ^2(b x)}{4 x^2}-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)^2+\frac{1}{2} b^2 \text{Chi}(2 b x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}-\frac{b \sinh (2 b x)}{4 x}+\frac{1}{2} b^2 \int \frac{\cosh (2 b x)}{x} \, dx\\ &=-\frac{\cosh ^2(b x)}{4 x^2}-\frac{\cosh (b x) \text{Chi}(b x)}{2 x^2}+\frac{1}{4} b^2 \text{Chi}(b x)^2+b^2 \text{Chi}(2 b x)-\frac{b \cosh (b x) \sinh (b x)}{2 x}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}-\frac{b \sinh (2 b x)}{4 x}\\ \end{align*}

Mathematica [A]  time = 0.016309, size = 96, normalized size = 1. \[ \frac{1}{4} b^2 \text{Chi}(b x)^2+b^2 \text{Chi}(2 b x)-\frac{\text{Chi}(b x) \cosh (b x)}{2 x^2}-\frac{b \text{Chi}(b x) \sinh (b x)}{2 x}-\frac{\cosh ^2(b x)}{4 x^2}-\frac{b \sinh (2 b x)}{4 x}-\frac{b \sinh (b x) \cosh (b x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[b*x]*CoshIntegral[b*x])/x^3,x]

[Out]

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

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Maple [F]  time = 0.054, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\it Chi} \left ( bx \right ) \cosh \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(Chi(b*x)*cosh(b*x)/x^3,x)

[Out]

int(Chi(b*x)*cosh(b*x)/x^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*cosh(b*x)/x^3,x, algorithm="maxima")

[Out]

integrate(Chi(b*x)*cosh(b*x)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cosh \left (b x\right ) \operatorname{Chi}\left (b x\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*cosh(b*x)/x^3,x, algorithm="fricas")

[Out]

integral(cosh(b*x)*cosh_integral(b*x)/x^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cosh{\left (b x \right )} \operatorname{Chi}\left (b x\right )}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*cosh(b*x)/x**3,x)

[Out]

Integral(cosh(b*x)*Chi(b*x)/x**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\rm Chi}\left (b x\right ) \cosh \left (b x\right )}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(Chi(b*x)*cosh(b*x)/x^3,x, algorithm="giac")

[Out]

integrate(Chi(b*x)*cosh(b*x)/x^3, x)