Integrand size = 22, antiderivative size = 36 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 e^{n \coth ^{-1}(a x)} (1+a x) (c-a c x)^{n/2}}{a (2+n)} \] Output:
2*exp(n*arccoth(a*x))*(a*x+1)*(-a*c*x+c)^(1/2*n)/a/(2+n)
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.61 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=-\frac {\left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{1+\frac {n}{2}} x (c-a c x)^{n/2}}{-1-\frac {n}{2}} \] Input:
Integrate[E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2),x]
Output:
-(((1 + 1/(a*x))^(1 + n/2)*x*(c - a*c*x)^(n/2))/((-1 - n/2)*(1 - 1/(a*x))^ (n/2)))
Time = 0.36 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6726}
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.
\(\displaystyle \int (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6726 |
\(\displaystyle \frac {2 (a x+1) (c-a c x)^{n/2} e^{n \coth ^{-1}(a x)}}{a (n+2)}\) |
Input:
Int[E^(n*ArcCoth[a*x])*(c - a*c*x)^(n/2),x]
Output:
(2*E^(n*ArcCoth[a*x])*(1 + a*x)*(c - a*c*x)^(n/2))/(a*(2 + n))
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.), x_Symbol] :> S imp[(1 + a*x)*(c + d*x)^p*(E^(n*ArcCoth[a*x])/(a*(p + 1))), x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a*c + d, 0] && !IntegerQ[p] && EqQ[p, n/2]
Time = 0.97 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94
method | result | size |
gosper | \(\frac {2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a x +1\right ) \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )}\) | \(34\) |
orering | \(\frac {2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (a x +1\right ) \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )}\) | \(34\) |
parallelrisch | \(-\frac {-2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} x \left (-a c x +c \right )^{\frac {n}{2}} a -2 \,{\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (-a c x +c \right )^{\frac {n}{2}}}{a \left (2+n \right )}\) | \(54\) |
risch | \(\frac {2 \left (a x +1\right ) \left (a x +1\right )^{\frac {n}{2}} \left (a x -1\right )^{-\frac {n}{2}} \left (a x -1\right )^{\frac {n}{2}} c^{\frac {n}{2}} {\mathrm e}^{-\frac {i n \pi \left (-\operatorname {csgn}\left (i c \left (a x -1\right )\right )^{3}-\operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2} \operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right )\right ) \operatorname {csgn}\left (i c \right )+2 \operatorname {csgn}\left (i c \left (a x -1\right )\right )^{2}-2\right )}{4}}}{a \left (2+n \right )}\) | \(151\) |
Input:
int(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2*n),x,method=_RETURNVERBOSE)
Output:
2*exp(n*arccoth(a*x))*(a*x+1)*(-a*c*x+c)^(1/2*n)/a/(2+n)
Time = 0.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.22 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 \, {\left (a x + 1\right )} {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a n + 2 \, a} \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2*n),x, algorithm="fricas")
Output:
2*(a*x + 1)*(-a*c*x + c)^(1/2*n)*((a*x + 1)/(a*x - 1))^(1/2*n)/(a*n + 2*a)
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\begin {cases} - \frac {x}{c} & \text {for}\: a = 0 \wedge n = -2 \\c^{\frac {n}{2}} x e^{\frac {i \pi n}{2}} & \text {for}\: a = 0 \\- \frac {\int \frac {1}{a x e^{2 \operatorname {acoth}{\left (a x \right )}} - e^{2 \operatorname {acoth}{\left (a x \right )}}}\, dx}{c} & \text {for}\: n = -2 \\\frac {2 a x \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} + \frac {2 \left (- a c x + c\right )^{\frac {n}{2}} e^{n \operatorname {acoth}{\left (a x \right )}}}{a n + 2 a} & \text {otherwise} \end {cases} \] Input:
integrate(exp(n*acoth(a*x))*(-a*c*x+c)**(1/2*n),x)
Output:
Piecewise((-x/c, Eq(a, 0) & Eq(n, -2)), (c**(n/2)*x*exp(I*pi*n/2), Eq(a, 0 )), (-Integral(1/(a*x*exp(2*acoth(a*x)) - exp(2*acoth(a*x))), x)/c, Eq(n, -2)), (2*a*x*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n + 2*a) + 2*(-a*c*x + c)**(n/2)*exp(n*acoth(a*x))/(a*n + 2*a), True))
Time = 0.04 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2 \, {\left (a \left (-c\right )^{\frac {1}{2} \, n} x + \left (-c\right )^{\frac {1}{2} \, n}\right )} {\left (a x + 1\right )}^{\frac {1}{2} \, n}}{a {\left (n + 2\right )}} \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2*n),x, algorithm="maxima")
Output:
2*(a*(-c)^(1/2*n)*x + (-c)^(1/2*n))*(a*x + 1)^(1/2*n)/(a*(n + 2))
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\int { {\left (-a c x + c\right )}^{\frac {1}{2} \, n} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(-a*c*x+c)^(1/2*n),x, algorithm="giac")
Output:
integrate((-a*c*x + c)^(1/2*n)*((a*x + 1)/(a*x - 1))^(1/2*n), x)
Time = 13.65 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.53 \[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\frac {2\,{\left (\frac {1}{a\,x}+1\right )}^{n/2}\,{\left (c-a\,c\,x\right )}^{n/2}\,\left (a\,x+1\right )}{a\,{\left (1-\frac {1}{a\,x}\right )}^{n/2}\,\left (n+2\right )} \] Input:
int(exp(n*acoth(a*x))*(c - a*c*x)^(n/2),x)
Output:
(2*(1/(a*x) + 1)^(n/2)*(c - a*c*x)^(n/2)*(a*x + 1))/(a*(1 - 1/(a*x))^(n/2) *(n + 2))
\[ \int e^{n \coth ^{-1}(a x)} (c-a c x)^{n/2} \, dx=\int e^{\mathit {acoth} \left (a x \right ) n} \left (-a c x +c \right )^{\frac {n}{2}}d x \] Input:
int(exp(n*acoth(a*x))*(-a*c*x+c)^(1/2*n),x)
Output:
int(e**(acoth(a*x)*n)*( - a*c*x + c)**(n/2),x)