\(\int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx\) [360]

Optimal result

Integrand size = 24, antiderivative size = 127 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 (6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {2+n}{2}}}{a (2+n) (4+n)}+\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n} \]

[Out]

-2*(6+n)*(1-1/a/x)^(-1-1/2*n)*(1+1/a/x)^(1+1/2*n)*(-a*c*x+c)^(1+1/2*n)/a/(n^2+6*n+8)+2*(1-1/a/x)^(-1-1/2*n)*(1
+1/a/x)^(1+1/2*n)*x*(-a*c*x+c)^(1+1/2*n)/(4+n)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6311, 6316, 80, 37} \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=\frac {2 x \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{n+4}-\frac {2 (n+6) \left (1-\frac {1}{a x}\right )^{-\frac {n}{2}-1} \left (\frac {1}{a x}+1\right )^{\frac {n+2}{2}} (c-a c x)^{\frac {n+2}{2}}}{a (n+2) (n+4)} \]

[In]

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

[Out]

(-2*(6 + n)*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*(c - a*c*x)^((2 + n)/2))/(a*(2 + n)*(4 + n)) +
(2*(1 - 1/(a*x))^(-1 - n/2)*(1 + 1/(a*x))^((2 + n)/2)*x*(c - a*c*x)^((2 + n)/2))/(4 + n)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 80

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

Rule 6311

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

Rule 6316

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)/(x_))^(p_.)*(x_)^(m_), x_Symbol] :> Dist[(-c^p)*x^m*(1/x)^m, S
ubst[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, m,
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*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} x^{-1-\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \int e^{n \coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^{1+\frac {n}{2}} x^{1+\frac {n}{2}} \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (\frac {1}{x}\right )^{1+\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-3-\frac {n}{2}} \left (1-\frac {x}{a}\right ) \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = \frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n}+\frac {\left ((6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (\frac {1}{x}\right )^{1+\frac {n}{2}} (c-a c x)^{1+\frac {n}{2}}\right ) \text {Subst}\left (\int x^{-2-\frac {n}{2}} \left (1+\frac {x}{a}\right )^{n/2} \, dx,x,\frac {1}{x}\right )}{a (4+n)} \\ & = -\frac {2 (6+n) \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} (c-a c x)^{\frac {2+n}{2}}}{a (2+n) (4+n)}+\frac {2 \left (1-\frac {1}{a x}\right )^{-1-\frac {n}{2}} \left (1+\frac {1}{a x}\right )^{\frac {2+n}{2}} x (c-a c x)^{\frac {2+n}{2}}}{4+n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.61 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 c \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} (1+a x) (c-a c x)^{n/2} (-6+2 a x+n (-1+a x))}{a (2+n) (4+n)} \]

[In]

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

[Out]

(-2*c*(1 + 1/(a*x))^(n/2)*(1 + a*x)*(c - a*c*x)^(n/2)*(-6 + 2*a*x + n*(-1 + a*x)))/(a*(2 + n)*(4 + n)*(1 - 1/(
a*x))^(n/2))

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.48

method	result	size
gosper	\(\frac {2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a n x +2 a x -n -6\right ) \left (a x +1\right )}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )}\)	\(61\)
parallelrisch	\(-\frac {-2 x^{2} \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2} n -4 x^{2} \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} a^{2}+8 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x \left (-a c x +c \right )^{1+\frac {n}{2}} a +2 \left (-a c x +c \right )^{1+\frac {n}{2}} {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} n +12 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{1+\frac {n}{2}}}{\left (a x -1\right ) a \left (n^{2}+6 n +8\right )}\)	\(150\)

[In]

int(exp(n*arccoth(a*x))*(-a*c*x+c)^(1+1/2*n),x,method=_RETURNVERBOSE)

[Out]

2*(-a*c*x+c)^(1+1/2*n)*exp(n*arccoth(a*x))*(a*n*x+2*a*x-n-6)*(a*x+1)/(a*x-1)/a/(n^2+6*n+8)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 \, {\left ({\left (a^{2} n + 2 \, a^{2}\right )} x^{2} - 4 \, a x - n - 6\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n^{2} + 6 \, a n - {\left (a^{2} n^{2} + 6 \, a^{2} n + 8 \, a^{2}\right )} x + 8 \, a} \]

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1+1/2*n),x, algorithm="fricas")

[Out]

-2*((a^2*n + 2*a^2)*x^2 - 4*a*x - n - 6)*(-a*c*x + c)^(1/2*n + 1)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*n^2 + 6*a*n
- (a^2*n^2 + 6*a^2*n + 8*a^2)*x + 8*a)

Sympy [F]

\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=\begin {cases} c^{\frac {n}{2} + 1} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\0^{\frac {n}{2} + 1} x e^{\infty n} & \text {for}\: a = \frac {1}{x} \\- \frac {\int \frac {1}{a x e^{4 \operatorname {acoth}{\left (a x \right )}} - e^{4 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -4 \\\int e^{- 2 \operatorname {acoth}{\left (a x \right )}}\, dx & \text {for}\: n = -2 \\\frac {2 a^{2} n x^{2} \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} + \frac {4 a^{2} x^{2} \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {8 a x \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {2 n \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} - \frac {12 \left (- a c x + c\right )^{\frac {n}{2} + 1} e^{n \operatorname {acoth}{\left (a x \right )}}}{a^{2} n^{2} x + 6 a^{2} n x + 8 a^{2} x - a n^{2} - 6 a n - 8 a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((c**(n/2 + 1)*x*exp(I*pi*n/2), Eq(a, 0)), (0**(n/2 + 1)*x*exp(oo*n), Eq(a, 1/x)), (-Integral(1/(a*x*
exp(4*acoth(a*x)) - exp(4*acoth(a*x))), x)/c, Eq(n, -4)), (Integral(exp(-2*acoth(a*x)), x), Eq(n, -2)), (2*a**
2*n*x**2*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*n - 8*a
) + 4*a**2*x**2*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*
n - 8*a) - 8*a*x*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a
*n - 8*a) - 2*n*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*
n - 8*a) - 12*(-a*c*x + c)**(n/2 + 1)*exp(n*acoth(a*x))/(a**2*n**2*x + 6*a**2*n*x + 8*a**2*x - a*n**2 - 6*a*n
- 8*a), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.54 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {2 \, {\left (a^{2} \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 2\right )} x^{2} - 4 \, a \left (-c\right )^{\frac {1}{2} \, n} c x - \left (-c\right )^{\frac {1}{2} \, n} c {\left (n + 6\right )}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{{\left (n^{2} + 6 \, n + 8\right )} a} \]

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1+1/2*n),x, algorithm="maxima")

[Out]

-2*(a^2*(-c)^(1/2*n)*c*(n + 2)*x^2 - 4*a*(-c)^(1/2*n)*c*x - (-c)^(1/2*n)*c*(n + 6))*(a*x + 1)^(1/2*n)/((n^2 +
6*n + 8)*a)

Giac [F]

\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n + 1} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \]

[In]

integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1+1/2*n),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 4.36 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.10 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{1+\frac {n}{2}} \, dx=-\frac {\left (\frac {\left (2\,n+12\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a^2\,\left (n^2+6\,n+8\right )}-\frac {x^2\,\left (2\,n+4\right )\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{n^2+6\,n+8}+\frac {8\,x\,{\left (c-a\,c\,x\right )}^{\frac {n}{2}+1}}{a\,\left (n^2+6\,n+8\right )}\right )\,{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}}{\left (x-\frac {1}{a}\right )\,{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}} \]

[In]

int(exp(n*acoth(a*x))*(c - a*c*x)^(n/2 + 1),x)

[Out]

-((((2*n + 12)*(c - a*c*x)^(n/2 + 1))/(a^2*(6*n + n^2 + 8)) - (x^2*(2*n + 4)*(c - a*c*x)^(n/2 + 1))/(6*n + n^2
+ 8) + (8*x*(c - a*c*x)^(n/2 + 1))/(a*(6*n + n^2 + 8)))*((a*x + 1)/(a*x))^(n/2))/((x - 1/a)*((a*x - 1)/(a*x))
^(n/2))