Integrand size = 22, antiderivative size = 60 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p x (1-a x)^{-p} \operatorname {AppellF1}\left (1-p,\frac {3}{2}-p,-\frac {3}{2},2-p,a x,-a x\right )}{1-p} \] Output:
(c-c/a/x)^p*x*AppellF1(1-p,3/2-p,-3/2,2-p,a*x,-a*x)/(1-p)/((-a*x+1)^p)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \] Input:
Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^p,x]
Output:
Integrate[E^(3*ArcTanh[a*x])*(c - c/(a*x))^p, x]
Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6684, 6679, 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.
\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx\) |
\(\Big \downarrow \) 6684 |
\(\displaystyle x^p (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \int e^{3 \text {arctanh}(a x)} x^{-p} (1-a x)^pdx\) |
\(\Big \downarrow \) 6679 |
\(\displaystyle x^p (1-a x)^{-p} \left (c-\frac {c}{a x}\right )^p \int x^{-p} (1-a x)^{p-\frac {3}{2}} (a x+1)^{3/2}dx\) |
\(\Big \downarrow \) 150 |
\(\displaystyle \frac {x (1-a x)^{-p} \operatorname {AppellF1}\left (1-p,\frac {3}{2}-p,-\frac {3}{2},2-p,a x,-a x\right ) \left (c-\frac {c}{a x}\right )^p}{1-p}\) |
Input:
Int[E^(3*ArcTanh[a*x])*(c - c/(a*x))^p,x]
Output:
((c - c/(a*x))^p*x*AppellF1[1 - p, 3/2 - p, -3/2, 2 - p, a*x, -(a*x)])/((1 - p)*(1 - a*x)^p)
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^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_))^(p_.), x_Symbol ] :> Simp[c^p Int[u*(1 + d*(x/c))^p*((1 + a*x)^(n/2)/(1 - a*x)^(n/2)), x] , x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c^2 - d^2, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_), x_Symbol] :> Simp[x^p*((c + d/x)^p/(1 + c*(x/d))^p) Int[u*(1 + c*(x/d))^p*(E^(n*Ar cTanh[a*x])/x^p), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c^2 - a^2*d^2, 0] && !IntegerQ[p]
\[\int \frac {\left (a x +1\right )^{3} \left (c -\frac {c}{a x}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x)
Output:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x, algorithm="fricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*((a*c*x - c)/(a*x))^p/(a^2*x^2 - 2*a *x + 1), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(c-c/a/x)**p,x)
Output:
Integral((-c*(-1 + 1/(a*x)))**p*(a*x + 1)**3/(-(a*x - 1)*(a*x + 1))**(3/2) , x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a x}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x, algorithm="maxima")
Output:
integrate((a*x + 1)^3*(c - c/(a*x))^p/(-a^2*x^2 + 1)^(3/2), x)
Exception generated. \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int(((c - c/(a*x))^p*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2),x)
Output:
int(((c - c/(a*x))^p*(a*x + 1)^3)/(1 - a^2*x^2)^(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {-\left (\int \frac {\left (a c x -c \right )^{p}}{x^{p} \sqrt {-a^{2} x^{2}+1}\, a x -x^{p} \sqrt {-a^{2} x^{2}+1}}d x \right )-\left (\int \frac {\left (a c x -c \right )^{p} x^{2}}{x^{p} \sqrt {-a^{2} x^{2}+1}\, a x -x^{p} \sqrt {-a^{2} x^{2}+1}}d x \right ) a^{2}-2 \left (\int \frac {\left (a c x -c \right )^{p} x}{x^{p} \sqrt {-a^{2} x^{2}+1}\, a x -x^{p} \sqrt {-a^{2} x^{2}+1}}d x \right ) a}{a^{p}} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(c-c/a/x)^p,x)
Output:
( - int((a*c*x - c)**p/(x**p*sqrt( - a**2*x**2 + 1)*a*x - x**p*sqrt( - a** 2*x**2 + 1)),x) - int(((a*c*x - c)**p*x**2)/(x**p*sqrt( - a**2*x**2 + 1)*a *x - x**p*sqrt( - a**2*x**2 + 1)),x)*a**2 - 2*int(((a*c*x - c)**p*x)/(x**p *sqrt( - a**2*x**2 + 1)*a*x - x**p*sqrt( - a**2*x**2 + 1)),x)*a)/a**p