3.365 \(\int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx\)

Optimal. Leaf size=81 \[ -\frac {32 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \, _2F_1\left (5,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]

[Out]

-32*c^3*(1-1/a/x)^(4-1/2*n)*(1+1/a/x)^(-4+1/2*n)*hypergeom([5, 4-1/2*n],[5-1/2*n],(a-1/x)/(a+1/x))/a/(8-n)

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Rubi [A]  time = 0.13, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6175, 6180, 131} \[ -\frac {32 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \, _2F_1\left (5,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \]

Antiderivative was successfully verified.

[In]

Int[E^(n*ArcCoth[a*x])*(c - a*c*x)^3,x]

[Out]

(-32*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((-8 + n)/2)*Hypergeometric2F1[5, 4 - n/2, 5 - n/2, (a - x^(-1)
)/(a + x^(-1))])/(a*(8 - n))

Rule 131

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)*Hypergeometric2F1[m + 1, -n, m + 2, -(((d*e - c*f)*(a + b*x))/((b*c - a*d)*(e + f*x))
)])/((m + 1)*(b*e - a*f)^(n + 1)*(e + f*x)^(m + 1)), x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p
+ 2, 0] && ILtQ[n, 0]

Rule 6175

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 6180

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)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^3 x^3 \, dx\right )\\ &=\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {32 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \, _2F_1\left (5,4-\frac {n}{2};5-\frac {n}{2};\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)}\\ \end {align*}

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Mathematica [B]  time = 1.98, size = 190, normalized size = 2.35 \[ -\frac {c^3 e^{n \coth ^{-1}(a x)} \left ((n+2) \left (n^2 \left (a^2 x^2-12 a x-1\right )+2 n \left (a^3 x^3-6 a^2 x^2+21 a x+6\right )+6 \left (a^4 x^4-4 a^3 x^3+6 a^2 x^2-4 a x-7\right )+\left (n^3-12 n^2+44 n-48\right ) \, _2F_1\left (1,\frac {n}{2};\frac {n}{2}+1;e^{2 \coth ^{-1}(a x)}\right )+a n^3 x\right )+n \left (n^3-12 n^2+44 n-48\right ) e^{2 \coth ^{-1}(a x)} \, _2F_1\left (1,\frac {n}{2}+1;\frac {n}{2}+2;e^{2 \coth ^{-1}(a x)}\right )\right )}{24 a (n+2)} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^3,x]

[Out]

-1/24*(c^3*E^(n*ArcCoth[a*x])*(E^(2*ArcCoth[a*x])*n*(-48 + 44*n - 12*n^2 + n^3)*Hypergeometric2F1[1, 1 + n/2,
2 + n/2, E^(2*ArcCoth[a*x])] + (2 + n)*(a*n^3*x + n^2*(-1 - 12*a*x + a^2*x^2) + 2*n*(6 + 21*a*x - 6*a^2*x^2 +
a^3*x^3) + 6*(-7 - 4*a*x + 6*a^2*x^2 - 4*a^3*x^3 + a^4*x^4) + (-48 + 44*n - 12*n^2 + n^3)*Hypergeometric2F1[1,
n/2, 1 + n/2, E^(2*ArcCoth[a*x])])))/(a*(2 + n))

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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="fricas")

[Out]

integral(-(a^3*c^3*x^3 - 3*a^2*c^3*x^2 + 3*a*c^3*x - c^3)*((a*x - 1)/(a*x + 1))^(1/2*n), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int -{\left (a c x - c\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="giac")

[Out]

integrate(-(a*c*x - c)^3*((a*x - 1)/(a*x + 1))^(1/2*n), x)

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maple [F]  time = 0.36, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )} \left (-a c x +c \right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x)

[Out]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\int {\left (a c x - c\right )}^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="maxima")

[Out]

-integrate((a*c*x - c)^3*((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 \[ \int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(n*acoth(a*x))*(c - a*c*x)^3,x)

[Out]

int(exp(n*acoth(a*x))*(c - a*c*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ - c^{3} \left (\int 3 a x e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- 3 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{3} x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(n*acoth(a*x))*(-a*c*x+c)**3,x)

[Out]

-c**3*(Integral(3*a*x*exp(n*acoth(a*x)), x) + Integral(-3*a**2*x**2*exp(n*acoth(a*x)), x) + Integral(a**3*x**3
*exp(n*acoth(a*x)), x) + Integral(-exp(n*acoth(a*x)), x))

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