Integrand size = 27, antiderivative size = 95 \[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\frac {(1-a x)^{1-2 p} \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p} \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a p} \] Output:
(-a*x+1)^(1-2*p)*(-a^2*x^2+1)^p/a/(1-2*p)/((-a^2*c*x^2+c)^p)+(-a^2*x^2+1)^ p/a/p/((-a*x+1)^(2*p))/((-a^2*c*x^2+c)^p)
Time = 0.01 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.61 \[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\frac {(1-a x)^{-2 p} (-1+p+a p x) \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p}}{a p (-1+2 p)} \] Input:
Integrate[E^(2*(1 + p)*ArcTanh[a*x])/(c - a^2*c*x^2)^p,x]
Output:
((-1 + p + a*p*x)*(1 - a^2*x^2)^p)/(a*p*(-1 + 2*p)*(1 - a*x)^(2*p)*(c - a^ 2*c*x^2)^p)
Time = 0.43 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.73, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6693, 6690, 53, 2009}
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.
\(\displaystyle \int e^{2 (p+1) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx\) |
\(\Big \downarrow \) 6693 |
\(\displaystyle \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p} \int e^{2 (p+1) \text {arctanh}(a x)} \left (1-a^2 x^2\right )^{-p}dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p} \int (1-a x)^{-2 p-1} (a x+1)dx\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \left (1-a^2 x^2\right )^p \left (c-a^2 c x^2\right )^{-p} \int \left (2 (1-a x)^{-2 p-1}-(1-a x)^{-2 p}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \left (1-a^2 x^2\right )^p \left (\frac {(1-a x)^{1-2 p}}{a (1-2 p)}+\frac {(1-a x)^{-2 p}}{a p}\right ) \left (c-a^2 c x^2\right )^{-p}\) |
Input:
Int[E^(2*(1 + p)*ArcTanh[a*x])/(c - a^2*c*x^2)^p,x]
Output:
((1 - a^2*x^2)^p*((1 - a*x)^(1 - 2*p)/(a*(1 - 2*p)) + 1/(a*p*(1 - a*x)^(2* p))))/(c - a^2*c*x^2)^p
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p Int[(1 - a*x)^(p - n/2)*(1 + a*x)^(p + n/2), x], x] /; FreeQ[{a , c, d, n, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPart[p]) Int [(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0])
Time = 0.54 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.63
method | result | size |
gosper | \(-\frac {\left (a x -1\right ) \left (a p x +p -1\right ) {\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-p}}{a p \left (2 p -1\right ) \left (a x +1\right )}\) | \(60\) |
orering | \(-\frac {\left (a x -1\right ) \left (a p x +p -1\right ) {\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{-p}}{a p \left (2 p -1\right ) \left (a x +1\right )}\) | \(60\) |
parallelrisch | \(\frac {\left (-{\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )}+{\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )} p -{\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )} p \,a^{2} x^{2}+{\mathrm e}^{2 \left (p +1\right ) \operatorname {arctanh}\left (a x \right )} x a \right ) \left (-a^{2} c \,x^{2}+c \right )^{-p}}{\left (a x +1\right ) a p \left (2 p -1\right )}\) | \(94\) |
Input:
int(exp(2*(p+1)*arctanh(a*x))/((-a^2*c*x^2+c)^p),x,method=_RETURNVERBOSE)
Output:
-(a*x-1)*(a*p*x+p-1)*exp(2*(p+1)*arctanh(a*x))/a/p/(2*p-1)/(a*x+1)/((-a^2* c*x^2+c)^p)
Time = 0.08 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=-\frac {{\left (a^{2} p x^{2} - a x - p + 1\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (2 \, a p^{2} - a p + {\left (2 \, a^{2} p^{2} - a^{2} p\right )} x\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}} \] Input:
integrate(exp(2*(p+1)*arctanh(a*x))/((-a^2*c*x^2+c)^p),x, algorithm="frica s")
Output:
-(a^2*p*x^2 - a*x - p + 1)*(-(a*x + 1)/(a*x - 1))^(p + 1)/((2*a*p^2 - a*p + (2*a^2*p^2 - a^2*p)*x)*(-a^2*c*x^2 + c)^p)
\[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{- p} e^{2 p \operatorname {atanh}{\left (a x \right )}} e^{2 \operatorname {atanh}{\left (a x \right )}}\, dx \] Input:
integrate(exp(2*(p+1)*atanh(a*x))/((-a**2*c*x**2+c)**p),x)
Output:
Integral(exp(2*p*atanh(a*x))*exp(2*atanh(a*x))/(-c*(a*x - 1)*(a*x + 1))**p , x)
Time = 0.04 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.41 \[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\frac {a p x + p - 1}{{\left (2 \, p^{2} - p\right )} {\left (-a x + 1\right )}^{2 \, p} a c^{p}} \] Input:
integrate(exp(2*(p+1)*arctanh(a*x))/((-a^2*c*x^2+c)^p),x, algorithm="maxim a")
Output:
(a*p*x + p - 1)/((2*p^2 - p)*(-a*x + 1)^(2*p)*a*c^p)
\[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\int { \frac {\left (-\frac {a x + 1}{a x - 1}\right )^{p + 1}}{{\left (-a^{2} c x^{2} + c\right )}^{p}} \,d x } \] Input:
integrate(exp(2*(p+1)*arctanh(a*x))/((-a^2*c*x^2+c)^p),x, algorithm="giac" )
Output:
integrate((-(a*x + 1)/(a*x - 1))^(p + 1)/(-a^2*c*x^2 + c)^p, x)
Time = 26.24 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.91 \[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=-\frac {p\,{\left (a\,x+1\right )}^p-{\left (a\,x+1\right )}^p+a\,p\,x\,{\left (a\,x+1\right )}^p}{a\,p\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a\,x\right )}^p-2\,a\,p^2\,{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a\,x\right )}^p} \] Input:
int(exp(2*atanh(a*x)*(p + 1))/(c - a^2*c*x^2)^p,x)
Output:
-(p*(a*x + 1)^p - (a*x + 1)^p + a*p*x*(a*x + 1)^p)/(a*p*(c - a^2*c*x^2)^p* (1 - a*x)^p - 2*a*p^2*(c - a^2*c*x^2)^p*(1 - a*x)^p)
\[ \int e^{2 (1+p) \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^{-p} \, dx=\int \frac {e^{2 \mathit {atanh} \left (a x \right ) p +2 \mathit {atanh} \left (a x \right )}}{\left (-a^{2} c \,x^{2}+c \right )^{p}}d x \] Input:
int(exp(2*(p+1)*atanh(a*x))/((-a^2*c*x^2+c)^p),x)
Output:
int(e**(2*atanh(a*x)*p + 2*atanh(a*x))/( - a**2*c*x**2 + c)**p,x)