\(\int e^{4 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [765]

Optimal result

Integrand size = 22, antiderivative size = 63 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{2+p} c (1+a x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (-2-p,-1+p,p,\frac {1}{2} (1-a x)\right )}{a (1-p)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6302, 6276, 692, 71} \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {c 2^{p+2} (a x+1)^{1-p} \left (c-a^2 c x^2\right )^{p-1} \operatorname {Hypergeometric2F1}\left (-p-2,p-1,p,\frac {1}{2} (1-a x)\right )}{a (1-p)} \]

[In]

Int[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]

[Out]

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

Rule 71

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

Rule 692

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[d^(m - 1)*((a + c*x^2)^(p + 1)/((1
+ e*(x/d))^(p + 1)*(a/d + (c*x)/e)^(p + 1))), Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^p, x], x] /; FreeQ[{a,
c, d, e, m}, x] && EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] && (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (
IntegerQ[3*p] || IntegerQ[4*p]))

Rule 6276

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[c^(n/2), Int[(c + d*x^2)^(p -
n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && IGt
Q[n/2, 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^{4 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx \\ & = c^2 \int (1+a x)^4 \left (c-a^2 c x^2\right )^{-2+p} \, dx \\ & = \left (c^2 (1+a x)^{1-p} (c-a c x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p}\right ) \int (1+a x)^{2+p} (c-a c x)^{-2+p} \, dx \\ & = \frac {2^{2+p} c (1+a x)^{1-p} \left (c-a^2 c x^2\right )^{-1+p} \operatorname {Hypergeometric2F1}\left (-2-p,-1+p,p,\frac {1}{2} (1-a x)\right )}{a (1-p)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.14 \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{2+p} (1-a x)^{-1+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-2-p,-1+p,p,\frac {1}{2} (1-a x)\right )}{a (-1+p)} \]

[In]

Integrate[E^(4*ArcCoth[a*x])*(c - a^2*c*x^2)^p,x]

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

\[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x - 1\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x - 1\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{2} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (a x - 1\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int e^{4 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (a\,x+1\right )}^2}{{\left (a\,x-1\right )}^2} \,d x \]

[In]

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

[Out]

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