\(\int e^{-2 \coth ^{-1}(a x)} (c-\frac {c}{a x})^p \, dx\) [555]

Optimal result

Integrand size = 22, antiderivative size = 114 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}+\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (2+p)}-\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,1-\frac {1}{a x}\right )}{a c^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6302, 6268, 25, 528, 382, 105, 162, 67, 70} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^{p+2} \operatorname {Hypergeometric2F1}\left (1,p+2,p+3,\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (p+2)}-\frac {\left (c-\frac {c}{a x}\right )^{p+2} \operatorname {Hypergeometric2F1}\left (1,p+2,p+3,1-\frac {1}{a x}\right )}{a c^2}+\frac {x \left (c-\frac {c}{a x}\right )^{p+2}}{c^2} \]

[In]

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

[Out]

((c - c/(a*x))^(2 + p)*x)/c^2 + ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, (a - x^(-1))/(2*a)])
/(2*a*c^2*(2 + p)) - ((c - c/(a*x))^(2 + p)*Hypergeometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])/(a*c^2)

Rule 25

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(q_.))^(p_.), x_Symbol] :> Dist[(d/a)^p, Int[u*((
a + b*x^n)^(m + p)/x^(n*p)), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[q, -n] && IntegerQ[p] && EqQ[a*c -
b*d, 0] &&  !(IntegerQ[m] && NegQ[n])

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 70

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b*c - a*d)^n*((a + b*x)^(m + 1)/(b^(
n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rule 105

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] &
& (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])

Rule 162

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 382

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[(a + b/x^n)^p*((c +
d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 6268

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

Rule 6302

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Dist[(-1)^(n/2), Int[u*E^(n*ArcTanh[a*x]), x], x] /; Free
Q[a, x] && IntegerQ[n/2]

Rubi steps \begin{align*} \text {integral}& = -\int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx \\ & = -\int \frac {\left (c-\frac {c}{a x}\right )^p (1-a x)}{1+a x} \, dx \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{1+p} x}{1+a x} \, dx}{c} \\ & = \frac {a \int \frac {\left (c-\frac {c}{a x}\right )^{1+p}}{a+\frac {1}{x}} \, dx}{c} \\ & = -\frac {a \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{x^2 (a+x)} \, dx,x,\frac {1}{x}\right )}{c} \\ & = \frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}+\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p} \left (c (2+p)+\frac {c (1+p) x}{a}\right )}{x (a+x)} \, dx,x,\frac {1}{x}\right )}{c^2} \\ & = \frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}-\frac {\text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{a+x} \, dx,x,\frac {1}{x}\right )}{a c}+\frac {(2+p) \text {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^{1+p}}{x} \, dx,x,\frac {1}{x}\right )}{a c} \\ & = \frac {\left (c-\frac {c}{a x}\right )^{2+p} x}{c^2}+\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {a-\frac {1}{x}}{2 a}\right )}{2 a c^2 (2+p)}-\frac {\left (c-\frac {c}{a x}\right )^{2+p} \operatorname {Hypergeometric2F1}\left (1,2+p,3+p,1-\frac {1}{a x}\right )}{a c^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.76 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^p \, dx=\frac {\left (c-\frac {c}{a x}\right )^p (-1+a x)^2 \left (\operatorname {Hypergeometric2F1}\left (1,2+p,3+p,\frac {a-\frac {1}{x}}{2 a}\right )+2 (2+p) \left (a x-\operatorname {Hypergeometric2F1}\left (1,2+p,3+p,1-\frac {1}{a x}\right )\right )\right )}{2 a^3 (2+p) x^2} \]

[In]

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

[Out]

((c - c/(a*x))^p*(-1 + a*x)^2*(Hypergeometric2F1[1, 2 + p, 3 + p, (a - x^(-1))/(2*a)] + 2*(2 + p)*(a*x - Hyper
geometric2F1[1, 2 + p, 3 + p, 1 - 1/(a*x)])))/(2*a^3*(2 + p)*x^2)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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