3.3.78 \(\int \frac {\cosh (x) \sinh (x)}{x^2} \, dx\) [278]

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 = 0.03, 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 = {5556, 12, 3378, 3382} \begin {gather*} \text {Chi}(2 x)-\frac {\sinh (2 x)}{2 x} \end {gather*}

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 3378

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 3382

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 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^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 \begin {gather*} \text {Chi}(2 x)-\frac {\sinh (2 x)}{2 x} \end {gather*}

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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Maple [A]
time = 0.79, size = 15, normalized size = 0.94

method	result	size
default	\(\hyperbolicCosineIntegral \left (2 x \right )-\frac {\sinh \left (2 x \right )}{2 x}\)	\(15\)
risch	\(\frac {{\mathrm e}^{-2 x}}{4 x}-\frac {\expIntegral \left (1, 2 x \right )}{2}-\frac {{\mathrm e}^{2 x}}{4 x}-\frac {\expIntegral \left (1, -2 x \right )}{2}\)	\(34\)
meijerg	\(\frac {\sqrt {\pi }\, \left (\frac {4}{\sqrt {\pi }}-\frac {2 \sinh \left (2 x \right )}{\sqrt {\pi }\, x}+\frac {4 \hyperbolicCosineIntegral \left (2 x \right )-4 \ln \left (2 x \right )-4 \gamma }{\sqrt {\pi }}+\frac {4 \gamma -4+4 \ln \left (2\right )+4 \ln \left (x \right )+2 i \pi }{\sqrt {\pi }}\right )}{4}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(x)/x^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.29, size = 15, normalized size = 0.94 \begin {gather*} \frac {1}{2} \, \Gamma \left (-1, 2 \, x\right ) + \frac {1}{2} \, \Gamma \left (-1, -2 \, x\right ) \end {gather*}

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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Fricas [A]
time = 0.37, size = 24, normalized size = 1.50 \begin {gather*} \frac {x {\rm Ei}\left (2 \, x\right ) + x {\rm Ei}\left (-2 \, x\right ) - 2 \, \cosh \left (x\right ) \sinh \left (x\right )}{2 \, x} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sinh {\left (x \right )} \cosh {\left (x \right )}}{x^{2}}\, dx \end {gather*}

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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (14) = 28\).
time = 0.39, size = 30, normalized size = 1.88 \begin {gather*} \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} \end {gather*}

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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \frac {\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )}{x^2} \,d x \end {gather*}

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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