3.3.77 \(\int \frac {\cosh (x) \sinh (x)}{x} \, dx\) [277]

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

[Out]

1/2*Shi(2*x)

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Rubi [A]
time = 0.02, antiderivative size = 8, normalized size of antiderivative = 1.00, 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 = {5556, 12, 3379} \begin {gather*} \frac {\text {Shi}(2 x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[2*x]/2

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]

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*}

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Mathematica [A]
time = 0.00, size = 8, normalized size = 1.00 \begin {gather*} \frac {\text {Shi}(2 x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

SinhIntegral[2*x]/2

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Maple [A]
time = 0.71, size = 7, normalized size = 0.88

method	result	size
default	\(\frac {\hyperbolicSineIntegral \left (2 x \right )}{2}\)	\(7\)
meijerg	\(-\frac {i \sinIntegral \left (2 i x \right )}{2}\)	\(9\)
risch	\(\frac {\expIntegral \left (1, 2 x \right )}{4}-\frac {\expIntegral \left (1, -2 x \right )}{4}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)/x,x,method=_RETURNVERBOSE)

[Out]

1/2*Shi(2*x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (6) = 12\).
time = 0.29, size = 13, normalized size = 1.62 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, x\right ) - \frac {1}{4} \, {\rm Ei}\left (-2 \, x\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (6) = 12\).
time = 0.36, size = 13, normalized size = 1.62 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, x\right ) - \frac {1}{4} \, {\rm Ei}\left (-2 \, x\right ) \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{x}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (6) = 12\).
time = 0.39, size = 13, normalized size = 1.62 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, x\right ) - \frac {1}{4} \, {\rm Ei}\left (-2 \, x\right ) \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.12 \begin {gather*} \int \frac {\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((cosh(x)*sinh(x))/x, x)

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