3.10.77 \(\int e^{n \sinh (a c+b c x)} \coth (c (a+b x)) \, dx\) [977]

Optimal. Leaf size=18 \[ \frac {\text {Ei}(n \sinh (c (a+b x)))}{b c} \]

[Out]

Ei(n*sinh(c*(b*x+a)))/b/c

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Rubi [A]
time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {4425, 2209} \begin {gather*} \frac {\text {Ei}(n \sinh (c (a+b x)))}{b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*Sinh[a*c + b*c*x])*Coth[c*(a + b*x)],x]

[Out]

ExpIntegralEi[n*Sinh[c*(a + b*x)]]/(b*c)

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 4425

Int[Coth[(c_.)*((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{d = FreeFactors[Sinh[c*(a + b*x)], x]}, Dist[1/(
b*c), Subst[Int[SubstFor[1/x, Sinh[c*(a + b*x)]/d, u, x], x], x, Sinh[c*(a + b*x)]/d], x] /; FunctionOfQ[Sinh[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int e^{n \sinh (a c+b c x)} \coth (c (a+b x)) \, dx &=\frac {\text {Subst}\left (\int \frac {e^{n x}}{x} \, dx,x,\sinh (c (a+b x))\right )}{b c}\\ &=\frac {\text {Ei}(n \sinh (c (a+b x)))}{b c}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} \frac {\text {Ei}(n \sinh (c (a+b x)))}{b c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(n*Sinh[a*c + b*c*x])*Coth[c*(a + b*x)],x]

[Out]

ExpIntegralEi[n*Sinh[c*(a + b*x)]]/(b*c)

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Maple [A]
time = 7.17, size = 31, normalized size = 1.72

method	result	size
default	\(\frac {\hyperbolicCosineIntegral \left (n \sinh \left (c \left (b x +a \right )\right )\right )+\hyperbolicSineIntegral \left (n \sinh \left (c \left (b x +a \right )\right )\right )}{b c}\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*sinh(b*c*x+a*c))*coth(c*(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

1/b/c*(Chi(n*sinh(c*(b*x+a)))+Shi(n*sinh(c*(b*x+a))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*c*x+a*c))*coth(c*(b*x+a)),x, algorithm="maxima")

[Out]

integrate(coth((b*x + a)*c)*e^(n*sinh(b*c*x + a*c)), x)

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Fricas [A]
time = 0.36, size = 19, normalized size = 1.06 \begin {gather*} \frac {{\rm Ei}\left (n \sinh \left (b c x + a c\right )\right )}{b c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*c*x+a*c))*coth(c*(b*x+a)),x, algorithm="fricas")

[Out]

Ei(n*sinh(b*c*x + a*c))/(b*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*c*x+a*c))*coth(c*(b*x+a)),x)

[Out]

Integral(exp(n*sinh(a*c + b*c*x))*coth(a*c + b*c*x), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*sinh(b*c*x+a*c))*coth(c*(b*x+a)),x, algorithm="giac")

[Out]

integrate(coth((b*x + a)*c)*e^(n*sinh(b*c*x + a*c)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int \mathrm {coth}\left (c\,\left (a+b\,x\right )\right )\,{\mathrm {e}}^{n\,\mathrm {sinh}\left (a\,c+b\,c\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(c*(a + b*x))*exp(n*sinh(a*c + b*c*x)),x)

[Out]

int(coth(c*(a + b*x))*exp(n*sinh(a*c + b*c*x)), x)

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