\(\int e^{\coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [553]

Optimal result

Integrand size = 20, antiderivative size = 90 \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{\frac {1}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{2}-p,2,\frac {5}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{3 a} \]

[Out]

-1/3*2^(1/2+p)*(1+1/a/x)^(3/2)*(c-c/a/x)^p*AppellF1(3/2,1/2-p,2,5/2,1/2*(a+1/x)/a,1+1/a/x)/a/((1-1/a/x)^p)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6317, 6314, 141} \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=-\frac {2^{p+\frac {1}{2}} \left (\frac {1}{a x}+1\right )^{3/2} \left (1-\frac {1}{a x}\right )^{-p} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{2}-p,2,\frac {5}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right ) \left (c-\frac {c}{a x}\right )^p}{3 a} \]

[In]

Int[E^ArcCoth[a*x]*(c - c/(a*x))^p,x]

[Out]

-1/3*(2^(1/2 + p)*(1 + 1/(a*x))^(3/2)*(c - c/(a*x))^p*AppellF1[3/2, 1/2 - p, 2, 5/2, (a + x^(-1))/(2*a), 1 + 1
/(a*x)])/(a*(1 - 1/(a*x))^p)

Rule 141

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_), x_Symbol] :> Simp[(b*e - a*f
)^p*((a + b*x)^(m + 1)/(b^(p + 1)*(m + 1)*(b/(b*c - a*d))^n))*AppellF1[m + 1, -n, -p, m + 2, (-d)*((a + b*x)/(
b*c - a*d)), (-f)*((a + b*x)/(b*e - a*f))], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] &&  !IntegerQ[m] &&  !Int
egerQ[n] && IntegerQ[p] && GtQ[b/(b*c - a*d), 0] &&  !(GtQ[d/(d*a - c*b), 0] && SimplerQ[c + d*x, a + b*x])

Rule 6314

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

Rule 6317

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

Rubi steps \begin{align*} \text {integral}& = \left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^p \, dx \\ & = -\left (\left (\left (1-\frac {1}{a x}\right )^{-p} \left (c-\frac {c}{a x}\right )^p\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{-\frac {1}{2}+p} \sqrt {1+\frac {x}{a}}}{x^2} \, dx,x,\frac {1}{x}\right )\right ) \\ & = -\frac {2^{\frac {1}{2}+p} \left (1-\frac {1}{a x}\right )^{-p} \left (1+\frac {1}{a x}\right )^{3/2} \left (c-\frac {c}{a x}\right )^p \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{2}-p,2,\frac {5}{2},\frac {a+\frac {1}{x}}{2 a},1+\frac {1}{a x}\right )}{3 a} \\ \end{align*}

Mathematica [F]

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

[In]

Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^p,x]

[Out]

Integrate[E^ArcCoth[a*x]*(c - c/(a*x))^p, x]

Maple [F]

\[\int \frac {\left (c -\frac {c}{a x}\right )^{p}}{\sqrt {\frac {a x -1}{a x +1}}}d x\]

[In]

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^p,x)

[Out]

int(1/((a*x-1)/(a*x+1))^(1/2)*(c-c/a/x)^p,x)

Fricas [F]

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

[In]

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

[Out]

integral((a*x + 1)*((a*c*x - c)/(a*x))^p*sqrt((a*x - 1)/(a*x + 1))/(a*x - 1), x)

Sympy [F]

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

[In]

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

[Out]

Integral((-c*(-1 + 1/(a*x)))**p/sqrt((a*x - 1)/(a*x + 1)), x)

Maxima [F]

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

[In]

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

[Out]

integrate((c - c/(a*x))^p/sqrt((a*x - 1)/(a*x + 1)), x)

Giac [F]

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

[In]

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

[Out]

integrate((c - c/(a*x))^p/sqrt((a*x - 1)/(a*x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a\,x}\right )}^p}{\sqrt {\frac {a\,x-1}{a\,x+1}}} \,d x \]

[In]

int((c - c/(a*x))^p/((a*x - 1)/(a*x + 1))^(1/2),x)

[Out]

int((c - c/(a*x))^p/((a*x - 1)/(a*x + 1))^(1/2), x)