Integrand size = 18, antiderivative size = 81 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {32 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {1}{2} (-8+n)} \operatorname {Hypergeometric2F1}\left (5,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)} \] Output:
-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)
Leaf count is larger than twice the leaf count of optimal. \(190\) vs. \(2(81)=162\).
Time = 1.98 (sec) , antiderivative size = 190, normalized size of antiderivative = 2.35 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=-\frac {c^3 e^{n \coth ^{-1}(a x)} \left (e^{2 \coth ^{-1}(a x)} n \left (-48+44 n-12 n^2+n^3\right ) \operatorname {Hypergeometric2F1}\left (1,1+\frac {n}{2},2+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )+(2+n) \left (a n^3 x+n^2 \left (-1-12 a x+a^2 x^2\right )+2 n \left (6+21 a x-6 a^2 x^2+a^3 x^3\right )+6 \left (-7-4 a x+6 a^2 x^2-4 a^3 x^3+a^4 x^4\right )+\left (-48+44 n-12 n^2+n^3\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {n}{2},1+\frac {n}{2},e^{2 \coth ^{-1}(a x)}\right )\right )\right )}{24 a (2+n)} \] Input:
Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^3,x]
Output:
-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))
Time = 0.42 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6725, 141}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (c-a c x)^3 e^{n \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6725 |
| \(\displaystyle a^3 c^3 \int \left (1-\frac {1}{a x}\right )^{3-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{n/2} x^5d\frac {1}{x}\) |
| \(\Big \downarrow \) 141 |
| \(\displaystyle -\frac {32 c^3 \left (1-\frac {1}{a x}\right )^{4-\frac {n}{2}} \left (\frac {1}{a x}+1\right )^{\frac {n-8}{2}} \operatorname {Hypergeometric2F1}\left (5,4-\frac {n}{2},5-\frac {n}{2},\frac {a-\frac {1}{x}}{a+\frac {1}{x}}\right )}{a (8-n)}\) |
Input:
Int[E^(n*ArcCoth[a*x])*(c - a*c*x)^3,x]
Output:
(-32*c^3*(1 - 1/(a*x))^(4 - n/2)*(1 + 1/(a*x))^((-8 + n)/2)*Hypergeometric 2F1[5, 4 - n/2, 5 - n/2, (a - x^(-1))/(a + x^(-1))])/(a*(8 - n))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> 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] || !Su mSimplerQ[p, 1]) && !ILtQ[m, 0]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[-d^p Subst[Int[((1 + c*(x/d))^p*((1 + x/a)^(n/2)/x^(p + 2)))/(1 - x/a )^(n/2), x], x, 1/x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && IntegerQ[p]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{3}d x\]
Input:
int(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x)
Output:
int(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x)
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int { -{\left (a c x - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="fricas")
Output:
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)
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=- c^{3} \left (\int 3 a x e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- 3 a^{2} x^{2} e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx + \int a^{3} x^{3} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {acoth}{\left (a x \right )}}\right )\, dx\right ) \] Input:
integrate(exp(n*acoth(a*x))*(-a*c*x+c)**3,x)
Output:
-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))
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int { -{\left (a c x - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="maxima")
Output:
-integrate((a*c*x - c)^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int { -{\left (a c x - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^3,x, algorithm="giac")
Output:
integrate(-(a*c*x - c)^3*((a*x + 1)/(a*x - 1))^(1/2*n), x)
Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^3 \,d x \] Input:
int(exp(n*acoth(a*x))*(c - a*c*x)^3,x)
Output:
int(exp(n*acoth(a*x))*(c - a*c*x)^3, x)
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^3 \, dx=c^{3} \left (\int e^{\mathit {acoth} \left (a x \right ) n}d x -\left (\int e^{\mathit {acoth} \left (a x \right ) n} x^{3}d x \right ) a^{3}+3 \left (\int e^{\mathit {acoth} \left (a x \right ) n} x^{2}d x \right ) a^{2}-3 \left (\int e^{\mathit {acoth} \left (a x \right ) n} x d x \right ) a \right ) \] Input:
int(exp(n*acoth(a*x))*(-a*c*x+c)^3,x)
Output:
c**3*(int(e**(acoth(a*x)*n),x) - int(e**(acoth(a*x)*n)*x**3,x)*a**3 + 3*in t(e**(acoth(a*x)*n)*x**2,x)*a**2 - 3*int(e**(acoth(a*x)*n)*x,x)*a)