3.278 \(\int \frac {\cosh (x) \sinh (x)}{x^2} \, dx\)

Optimal. Leaf size=16 \[ \text {Chi}(2 x)-\frac {\sinh (2 x)}{2 x} \]

[Out]

Chi(2*x)-1/2*sinh(2*x)/x

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Rubi [A]  time = 0.05, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5448, 12, 3297, 3301} \[ \text {Chi}(2 x)-\frac {\sinh (2 x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]*Sinh[x])/x^2,x]

[Out]

CoshIntegral[2*x] - Sinh[2*x]/(2*x)

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 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 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]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \[ \text {Chi}(2 x)-\frac {\sinh (2 x)}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[x]*Sinh[x])/x^2,x]

[Out]

CoshIntegral[2*x] - Sinh[2*x]/(2*x)

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fricas [A]  time = 0.92, size = 24, normalized size = 1.50 \[ \frac {x {\rm Ei}\left (2 \, x\right ) + x {\rm Ei}\left (-2 \, x\right ) - 2 \, \cosh \relax (x) \sinh \relax (x)}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)/x^2,x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.12, size = 30, normalized size = 1.88 \[ \frac {2 \, x {\rm Ei}\left (2 \, x\right ) + 2 \, x {\rm Ei}\left (-2 \, x\right ) - e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}}{4 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)/x^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.12, size = 15, normalized size = 0.94 \[ \Chi \left (2 x \right )-\frac {\sinh \left (2 x \right )}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)/x^2,x)

[Out]

Chi(2*x)-1/2*sinh(2*x)/x

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maxima [A]  time = 0.36, size = 15, normalized size = 0.94 \[ \frac {1}{2} \, \Gamma \left (-1, 2 \, x\right ) + \frac {1}{2} \, \Gamma \left (-1, -2 \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)/x^2,x, algorithm="maxima")

[Out]

1/2*gamma(-1, 2*x) + 1/2*gamma(-1, -2*x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {\mathrm {cosh}\relax (x)\,\mathrm {sinh}\relax (x)}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(x)*sinh(x))/x^2,x)

[Out]

int((cosh(x)*sinh(x))/x^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh {\relax (x )} \cosh {\relax (x )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(x)/x**2,x)

[Out]

Integral(sinh(x)*cosh(x)/x**2, x)

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