### 3.279 $$\int \frac{\cosh (x) \sinh (x)}{x^3} \, dx$$

Optimal. Leaf size=27 $\text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x}$

[Out]

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

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Rubi [A]  time = 0.0629465, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 8, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.5, Rules used = {5448, 12, 3297, 3298} $\text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cosh[2*x]/(2*x) - Sinh[2*x]/(4*x^2) + SinhIntegral[2*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 12

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.007349, size = 27, normalized size = 1. $\text{Shi}(2 x)-\frac{\sinh (2 x)}{4 x^2}-\frac{\cosh (2 x)}{2 x}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 24, normalized size = 0.9 \begin{align*} -{\frac{\cosh \left ( 2\,x \right ) }{2\,x}}+{\it Shi} \left ( 2\,x \right ) -{\frac{\sinh \left ( 2\,x \right ) }{4\,{x}^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)/x^3,x)

[Out]

-1/2*cosh(2*x)/x+Shi(2*x)-1/4*sinh(2*x)/x^2

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Maxima [A]  time = 1.20613, size = 18, normalized size = 0.67 \begin{align*} \Gamma \left (-2, 2 \, x\right ) - \Gamma \left (-2, -2 \, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-2, 2*x) - gamma(-2, -2*x)

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Fricas [A]  time = 2.00945, size = 113, normalized size = 4.19 \begin{align*} \frac{x^{2}{\rm Ei}\left (2 \, x\right ) - x^{2}{\rm Ei}\left (-2 \, x\right ) - x \cosh \left (x\right )^{2} - x \sinh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right )}{2 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^2*Ei(2*x) - x^2*Ei(-2*x) - x*cosh(x)^2 - x*sinh(x)^2 - cosh(x)*sinh(x))/x^2

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)/x**3,x)

[Out]

Integral(sinh(x)*cosh(x)/x**3, x)

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Giac [B]  time = 1.18049, size = 65, normalized size = 2.41 \begin{align*} \frac{4 \, x^{2}{\rm Ei}\left (2 \, x\right ) - 4 \, x^{2}{\rm Ei}\left (-2 \, x\right ) - 2 \, x e^{\left (2 \, x\right )} - 2 \, x e^{\left (-2 \, x\right )} - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}}{8 \, x^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(4*x^2*Ei(2*x) - 4*x^2*Ei(-2*x) - 2*x*e^(2*x) - 2*x*e^(-2*x) - e^(2*x) + e^(-2*x))/x^2