\(\int e^{-2 \coth ^{-1}(a x)} (c-a^2 c x^2)^p \, dx\) [770]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-p,p,1+p,\frac {1}{2} (1+a x)\right )}{a p} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 55, 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, 6277, 692, 71} \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {2^{p+1} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p-1,p,p+1,\frac {1}{2} (a x+1)\right )}{a p} \]

[In]

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

[Out]

-((2^(1 + p)*(c - a^2*c*x^2)^p*Hypergeometric2F1[-1 - p, p, 1 + p, (1 + a*x)/2])/(a*p*(1 - a*x)^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 6277

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/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]) && I
LtQ[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^{-2 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx \\ & = -\left (c \int (1-a x)^2 \left (c-a^2 c x^2\right )^{-1+p} \, dx\right ) \\ & = -\left (\left (c (1-a x)^{-p} (c+a c x)^{-p} \left (c-a^2 c x^2\right )^p\right ) \int (1-a x)^{1+p} (c+a c x)^{-1+p} \, dx\right ) \\ & = -\frac {2^{1+p} (1-a x)^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (-1-p,p,1+p,\frac {1}{2} (1+a x)\right )}{a p} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.33 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {2^{-1+p} (1-a x)^{2+p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1-p,2+p,3+p,\frac {1}{2} (1-a x)\right )}{a (2+p)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 12.31 (sec) , antiderivative size = 648, normalized size of antiderivative = 11.78 \[ \int e^{-2 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*Piecewise((0**p*x/a + 0**p*log(1/(a**2*x**2))/(2*a**2) - 0**p*log(-1 + 1/(a**2*x**2))/(2*a**2) - 0**p*acoth(
1/(a*x))/a**2 - a**(2*p - 1)*c**p*p*x**(2*p + 1)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2),
(1/2 - p,), 1/(a**2*x**2))/(2*gamma(1/2 - p)*gamma(p + 1)) - c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 -
p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)), 1/Abs(a**2*x**2) > 1), (0**p*x/a + 0**p*
log(1/(a**2*x**2))/(2*a**2) - 0**p*log(1 - 1/(a**2*x**2))/(2*a**2) - 0**p*atanh(1/(a*x))/a**2 - a**(2*p - 1)*c
**p*p*x**(2*p + 1)*exp(I*pi*p)*gamma(p)*gamma(-p - 1/2)*hyper((1 - p, -p - 1/2), (1/2 - p,), 1/(a**2*x**2))/(2
*gamma(1/2 - p)*gamma(p + 1)) - c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_pol
ar(2*I*pi))/(2*gamma(-p)*gamma(p + 1)), True)) - Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) + 0**p*acoth(a*x)/a
+ a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi))/(2*gamma(-p)*gam
ma(p + 1)) + a**(2*p - 2)*c**p*p*x**(2*p - 1)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1 - p, 1/2 - p), (3/2
- p,), 1/(a**2*x**2))/(2*gamma(3/2 - p)*gamma(p + 1)), Abs(a**2*x**2) > 1), (0**p*log(-a**2*x**2 + 1)/(2*a) +
0**p*atanh(a*x)/a + a*c**p*x**2*gamma(p)*gamma(1 - p)*hyper((2, 1, 1 - p), (2, 2), a**2*x**2*exp_polar(2*I*pi
))/(2*gamma(-p)*gamma(p + 1)) + a**(2*p - 2)*c**p*p*x**(2*p - 1)*exp(I*pi*p)*gamma(p)*gamma(1/2 - p)*hyper((1
- p, 1/2 - p), (3/2 - p,), 1/(a**2*x**2))/(2*gamma(3/2 - p)*gamma(p + 1)), True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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