Integrand size = 14, antiderivative size = 46 \[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\frac {x (c x)^m \operatorname {AppellF1}\left (-1-m,\frac {n}{2},-\frac {n}{2},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{1+m} \] Output:
x*(c*x)^m*AppellF1(-1-m,1/2*n,-1/2*n,-m,1/a/x,-1/a/x)/(1+m)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int e^{n \coth ^{-1}(a x)} (c x)^m \, dx \] Input:
Integrate[E^(n*ArcCoth[a*x])*(c*x)^m,x]
Output:
Integrate[E^(n*ArcCoth[a*x])*(c*x)^m, x]
Time = 0.39 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6723, 150}
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.
\(\displaystyle \int (c x)^m e^{n \coth ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 6723 |
\(\displaystyle -\left (\frac {1}{x}\right )^m (c x)^m \int \left (1-\frac {1}{a x}\right )^{-n/2} \left (1+\frac {1}{a x}\right )^{n/2} \left (\frac {1}{x}\right )^{-m-2}d\frac {1}{x}\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x (c x)^m \operatorname {AppellF1}\left (-m-1,\frac {n}{2},-\frac {n}{2},-m,\frac {1}{a x},-\frac {1}{a x}\right )}{m+1}\) |
Input:
Int[E^(n*ArcCoth[a*x])*(c*x)^m,x]
Output:
(x*(c*x)^m*AppellF1[-1 - m, n/2, -1/2*n, -m, 1/(a*x), -(1/(a*x))])/(1 + m)
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ ] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 , (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] && !In tegerQ[m] && !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*((c_.)*(x_))^(m_), x_Symbol] :> Simp[(-(c* x)^m)*(1/x)^m Subst[Int[(1 + x/a)^(n/2)/(x^(m + 2)*(1 - x/a)^(n/2)), x], x, 1/x], x] /; FreeQ[{a, c, m, n}, x] && !IntegerQ[n] && !IntegerQ[m]
\[\int {\mathrm e}^{n \,\operatorname {arccoth}\left (a x \right )} \left (x c \right )^{m}d x\]
Input:
int(exp(n*arccoth(a*x))*(x*c)^m,x)
Output:
int(exp(n*arccoth(a*x))*(x*c)^m,x)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int { \left (c x\right )^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(c*x)^m,x, algorithm="fricas")
Output:
integral((c*x)^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int \left (c x\right )^{m} e^{n \operatorname {acoth}{\left (a x \right )}}\, dx \] Input:
integrate(exp(n*acoth(a*x))*(c*x)**m,x)
Output:
Integral((c*x)**m*exp(n*acoth(a*x)), x)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int { \left (c x\right )^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(c*x)^m,x, algorithm="maxima")
Output:
integrate((c*x)^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int { \left (c x\right )^{m} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n} \,d x } \] Input:
integrate(exp(n*arccoth(a*x))*(c*x)^m,x, algorithm="giac")
Output:
integrate((c*x)^m*((a*x + 1)/(a*x - 1))^(1/2*n), x)
Timed out. \[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=\int {\mathrm {e}}^{n\,\mathrm {acoth}\left (a\,x\right )}\,{\left (c\,x\right )}^m \,d x \] Input:
int(exp(n*acoth(a*x))*(c*x)^m,x)
Output:
int(exp(n*acoth(a*x))*(c*x)^m, x)
\[ \int e^{n \coth ^{-1}(a x)} (c x)^m \, dx=c^{m} \left (\int x^{m} e^{\mathit {acoth} \left (a x \right ) n}d x \right ) \] Input:
int(exp(n*acoth(a*x))*(c*x)^m,x)
Output:
c**m*int(x**m*e**(acoth(a*x)*n),x)