3.277 \(\int \frac{\cosh (x) \sinh (x)}{x} \, dx\)

Optimal. Leaf size=8 \[ \frac{\text{Shi}(2 x)}{2} \]

[Out]

SinhIntegral[2*x]/2

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Rubi [A]  time = 0.0301795, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {5448, 12, 3298} \[ \frac{\text{Shi}(2 x)}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[2*x]/2

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 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} \, dx &=\int \frac{\sinh (2 x)}{2 x} \, dx\\ &=\frac{1}{2} \int \frac{\sinh (2 x)}{x} \, dx\\ &=\frac{\text{Shi}(2 x)}{2}\\ \end{align*}

Mathematica [A]  time = 0.0057429, size = 8, normalized size = 1. \[ \frac{\text{Shi}(2 x)}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[2*x]/2

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Maple [A]  time = 0.017, size = 7, normalized size = 0.9 \begin{align*}{\frac{{\it Shi} \left ( 2\,x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*Shi(2*x)

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Maxima [B]  time = 1.28468, size = 18, normalized size = 2.25 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, x\right ) - \frac{1}{4} \,{\rm Ei}\left (-2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Ei(2*x) - 1/4*Ei(-2*x)

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Fricas [B]  time = 1.97821, size = 38, normalized size = 4.75 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, x\right ) - \frac{1}{4} \,{\rm Ei}\left (-2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Ei(2*x) - 1/4*Ei(-2*x)

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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}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sinh(x)*cosh(x)/x, x)

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Giac [B]  time = 1.14652, size = 18, normalized size = 2.25 \begin{align*} \frac{1}{4} \,{\rm Ei}\left (2 \, x\right ) - \frac{1}{4} \,{\rm Ei}\left (-2 \, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*Ei(2*x) - 1/4*Ei(-2*x)