3.4.67 \(\int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx\) [367]

Optimal. Leaf size=79 \[ -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \]

[Out]

-8*c*(1-1/a/x)^(2-1/2*n)*(1+1/a/x)^(-2+1/2*n)*hypergeom([3, 2-1/2*n],[3-1/2*n],(a-1/x)/(a+1/x))/a/(4-n)

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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6310, 6315, 133} \begin {gather*} -\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-4}{2}} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcCoth[a*x])*(c - a*c*x),x]

[Out]

(-8*c*(1 - 1/(a*x))^(2 - n/2)*(1 + 1/(a*x))^((-4 + n)/2)*Hypergeometric2F1[3, 2 - n/2, 3 - n/2, (a - x^(-1))/(
a + x^(-1))])/(a*(4 - n))

Rule 133

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*c - a
*d)^n*((a + b*x)^(m + 1)/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)))*Hypergeometric2F1[m + 1, -n, m + 2,
(-(d*e - c*f))*((a + b*x)/((b*c - a*d)*(e + f*x)))], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) &&  !ILtQ[m, 0]

Rule 6310

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> Dist[d^p, Int[u*x^p*(1 + c/(d*
x))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c^2 - d^2, 0] &&  !IntegerQ[n/2] && Inte
gerQ[p]

Rule 6315

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[-c^p, Subst[Int[(1 +
d*(x/c))^p*((1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2))), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ
[c^2 - a^2*d^2, 0] &&  !IntegerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegerQ[m]

Rubi steps

\begin {align*} \int e^{n \coth ^{-1}(a x)} (c-a c x) \, dx &=-\left ((a c) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right ) x \, dx\right )\\ &=(a c) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{1-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {8 c \left (1-\frac {1}{a x}\right )^{2-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-4+n)} \, _2F_1\left (3,2-\frac {n}{2};3-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (4-n)}\\ \end {align*}

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Mathematica [A]
time = 0.35, size = 104, normalized size = 1.32 \begin {gather*} -\frac {c e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} (-2+n) n \, _2F_1\left (1,1+\frac {n}{2};2+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (-1+a (-2+n) x+a^2 x^2+(-2+n) \, _2F_1\left (1,\frac {n}{2};1+\frac {n}{2};e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{2 a (2+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x),x]

[Out]

-1/2*(c*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*(-2 + n)*n*Hypergeometric2F1[1, 1 + n/2, 2 + n/2, E^(2*ArcCoth[
a*x])] + (2 + n)*(-1 + a*(-2 + n)*x + a^2*x^2 + (-2 + n)*Hypergeometric2F1[1, n/2, 1 + n/2, E^(2*ArcCoth[a*x])
])))/(a*(2 + n))

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))*(-a*c*x+c),x)

[Out]

int(exp(n*arccoth(a*x))*(-a*c*x+c),x)

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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*arccoth(a*x))*(-a*c*x+c),x, algorithm="maxima")

[Out]

-integrate((a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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Fricas [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*arccoth(a*x))*(-a*c*x+c),x, algorithm="fricas")

[Out]

integral(-(a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))*(-a*c*x+c),x)

[Out]

-c*(Integral(a*x*exp(n*acoth(a*x)), x) + Integral(-exp(n*acoth(a*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*arccoth(a*x))*(-a*c*x+c),x, algorithm="giac")

[Out]

integrate(-(a*c*x - c)*((a*x + 1)/(a*x - 1))^(1/2*n), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,\left (c-a\,c\,x\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*acoth(a*x))*(c - a*c*x),x)

[Out]

int(exp(n*acoth(a*x))*(c - a*c*x), x)

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