3.3.55 \(\int \frac {\cosh (a+b x) \sinh (a+b x)}{x} \, dx\) [255]

Optimal. Leaf size=27 \[ \frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \]

[Out]

1/2*cosh(2*a)*Shi(2*b*x)+1/2*Chi(2*b*x)*sinh(2*a)

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Rubi [A]
time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {5556, 12, 3384, 3379, 3382} \begin {gather*} \frac {1}{2} \sinh (2 a) \text {Chi}(2 b x)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Cosh[a + b*x]*Sinh[a + b*x])/x,x]

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a])/2 + (Cosh[2*a]*SinhIntegral[2*b*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 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 3384

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) + f*x]
/(c + d*x), x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f},
x] && NeQ[d*e - c*f, 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 (a+b x) \sinh (a+b x)}{x} \, dx &=\int \frac {\sinh (2 a+2 b x)}{2 x} \, dx\\ &=\frac {1}{2} \int \frac {\sinh (2 a+2 b x)}{x} \, dx\\ &=\frac {1}{2} \cosh (2 a) \int \frac {\sinh (2 b x)}{x} \, dx+\frac {1}{2} \sinh (2 a) \int \frac {\cosh (2 b x)}{x} \, dx\\ &=\frac {1}{2} \text {Chi}(2 b x) \sinh (2 a)+\frac {1}{2} \cosh (2 a) \text {Shi}(2 b x)\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 25, normalized size = 0.93 \begin {gather*} \frac {1}{2} (\text {Chi}(2 b x) \sinh (2 a)+\cosh (2 a) \text {Shi}(2 b x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Cosh[a + b*x]*Sinh[a + b*x])/x,x]

[Out]

(CoshIntegral[2*b*x]*Sinh[2*a] + Cosh[2*a]*SinhIntegral[2*b*x])/2

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Maple [A]
time = 2.57, size = 26, normalized size = 0.96

method	result	size
risch	\(\frac {{\mathrm e}^{-2 a} \expIntegral \left (1, 2 b x \right )}{4}-\frac {{\mathrm e}^{2 a} \expIntegral \left (1, -2 b x \right )}{4}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*exp(-2*a)*Ei(1,2*b*x)-1/4*exp(2*a)*Ei(1,-2*b*x)

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Maxima [A]
time = 0.31, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.39, size = 37, normalized size = 1.37 \begin {gather*} \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) - {\rm Ei}\left (-2 \, b x\right )\right )} \cosh \left (2 \, a\right ) + \frac {1}{4} \, {\left ({\rm Ei}\left (2 \, b x\right ) + {\rm Ei}\left (-2 \, b x\right )\right )} \sinh \left (2 \, a\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(Ei(2*b*x) - Ei(-2*b*x))*cosh(2*a) + 1/4*(Ei(2*b*x) + Ei(-2*b*x))*sinh(2*a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(b*x+a)*sinh(b*x+a)/x,x)

[Out]

Integral(sinh(a + b*x)*cosh(a + b*x)/x, x)

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Giac [A]
time = 0.38, size = 23, normalized size = 0.85 \begin {gather*} \frac {1}{4} \, {\rm Ei}\left (2 \, b x\right ) e^{\left (2 \, a\right )} - \frac {1}{4} \, {\rm Ei}\left (-2 \, b x\right ) e^{\left (-2 \, a\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,\mathrm {sinh}\left (a+b\,x\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(a + b*x)*sinh(a + b*x))/x,x)

[Out]

int((cosh(a + b*x)*sinh(a + b*x))/x, x)

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