Integrand size = 23, antiderivative size = 51 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {(1-a x)^{1+2 p} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (1+2 p)} \] Output:
-(-a*x+1)^(1+2*p)*(-a^2*c*x^2+c)^p/a/(1+2*p)/((-a^2*x^2+1)^p)
Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.71 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {e^{-2 p \text {arctanh}(a x)} (-1+a x) \left (c-a^2 c x^2\right )^p}{a+2 a p} \] Input:
Integrate[(c - a^2*c*x^2)^p/E^(2*p*ArcTanh[a*x]),x]
Output:
((-1 + a*x)*(c - a^2*c*x^2)^p)/(E^(2*p*ArcTanh[a*x])*(a + 2*a*p))
Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6693, 6690, 17}
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.
\(\displaystyle \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx\) |
\(\Big \downarrow \) 6693 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int e^{-2 p \text {arctanh}(a x)} \left (1-a^2 x^2\right )^pdx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int (1-a x)^{2 p}dx\) |
\(\Big \downarrow \) 17 |
\(\displaystyle -\frac {(1-a x)^{2 p+1} \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p}{a (2 p+1)}\) |
Input:
Int[(c - a^2*c*x^2)^p/E^(2*p*ArcTanh[a*x]),x]
Output:
-(((1 - a*x)^(1 + 2*p)*(c - a^2*c*x^2)^p)/(a*(1 + 2*p)*(1 - a^2*x^2)^p))
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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 = 0.53 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78
method | result | size |
gosper | \(\frac {\left (a x -1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 p \,\operatorname {arctanh}\left (a x \right )}}{\left (2 p +1\right ) a}\) | \(40\) |
orering | \(\frac {\left (a x -1\right ) \left (-a^{2} c \,x^{2}+c \right )^{p} {\mathrm e}^{-2 p \,\operatorname {arctanh}\left (a x \right )}}{\left (2 p +1\right ) a}\) | \(40\) |
parallelrisch | \(\frac {\left (\left (-a^{2} c \,x^{2}+c \right )^{p} a x -\left (-a^{2} c \,x^{2}+c \right )^{p}\right ) {\mathrm e}^{-2 p \,\operatorname {arctanh}\left (a x \right )}}{\left (2 p +1\right ) a}\) | \(54\) |
risch | \(\frac {c^{p} \left (\left (a x -1\right )^{p}\right )^{2} \left (a x -1\right ) {\mathrm e}^{-\frac {i p \pi \left (-2 \operatorname {csgn}\left (i \left (a x -1\right )\right )^{3}-\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i \left (a x -1\right )\right )+\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i \left (a x +1\right )\right ) \operatorname {csgn}\left (i \left (a x -1\right )\right )+\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{3}-\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i \left (a x +1\right )\right )-\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}+\operatorname {csgn}\left (i \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right ) \operatorname {csgn}\left (i c \right )-\operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{3}-\operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2} \operatorname {csgn}\left (i c \right )+2 \operatorname {csgn}\left (i \left (a x -1\right )\right )^{2}+2 \operatorname {csgn}\left (i c \left (a x -1\right ) \left (a x +1\right )\right )^{2}-4\right )}{2}}}{a \left (2 p +1\right )}\) | \(305\) |
Input:
int((-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x)),x,method=_RETURNVERBOSE)
Output:
(a*x-1)/(2*p+1)/a*(-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x))
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.88 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {{\left (a x - 1\right )} {\left (-a^{2} c x^{2} + c\right )}^{p}}{{\left (2 \, a p + a\right )} \left (-\frac {a x + 1}{a x - 1}\right )^{p}} \] Input:
integrate((-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="fricas")
Output:
(a*x - 1)*(-a^2*c*x^2 + c)^p/((2*a*p + a)*(-(a*x + 1)/(a*x - 1))^p)
\[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\begin {cases} \frac {x}{\sqrt {c}} & \text {for}\: a = 0 \wedge p = - \frac {1}{2} \\c^{p} x & \text {for}\: a = 0 \\\int \frac {e^{\operatorname {atanh}{\left (a x \right )}}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx & \text {for}\: p = - \frac {1}{2} \\\frac {a x \left (- a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 p \operatorname {atanh}{\left (a x \right )}} + a e^{2 p \operatorname {atanh}{\left (a x \right )}}} - \frac {\left (- a^{2} c x^{2} + c\right )^{p}}{2 a p e^{2 p \operatorname {atanh}{\left (a x \right )}} + a e^{2 p \operatorname {atanh}{\left (a x \right )}}} & \text {otherwise} \end {cases} \] Input:
integrate((-a**2*c*x**2+c)**p/exp(2*p*atanh(a*x)),x)
Output:
Piecewise((x/sqrt(c), Eq(a, 0) & Eq(p, -1/2)), (c**p*x, Eq(a, 0)), (Integr al(exp(atanh(a*x))/sqrt(-c*(a*x - 1)*(a*x + 1)), x), Eq(p, -1/2)), (a*x*(- a**2*c*x**2 + c)**p/(2*a*p*exp(2*p*atanh(a*x)) + a*exp(2*p*atanh(a*x))) - (-a**2*c*x**2 + c)**p/(2*a*p*exp(2*p*atanh(a*x)) + a*exp(2*p*atanh(a*x))), True))
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.65 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {{\left (a c^{p} x - c^{p}\right )} {\left (-a x + 1\right )}^{2 \, p}}{a {\left (2 \, p + 1\right )}} \] Input:
integrate((-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="maxima")
Output:
(a*c^p*x - c^p)*(-a*x + 1)^(2*p)/(a*(2*p + 1))
\[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\int { \frac {{\left (-a^{2} c x^{2} + c\right )}^{p}}{\left (-\frac {a x + 1}{a x - 1}\right )^{p}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^p/exp(2*p*arctanh(a*x)),x, algorithm="giac")
Output:
integrate((-a^2*c*x^2 + c)^p/(-(a*x + 1)/(a*x - 1))^p, x)
Time = 26.11 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.86 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=-\frac {{\left (c-a^2\,c\,x^2\right )}^p\,{\left (1-a\,x\right )}^{p+1}}{a\,\left (2\,p+1\right )\,{\left (a\,x+1\right )}^p} \] Input:
int(exp(-2*p*atanh(a*x))*(c - a^2*c*x^2)^p,x)
Output:
-((c - a^2*c*x^2)^p*(1 - a*x)^(p + 1))/(a*(2*p + 1)*(a*x + 1)^p)
Time = 0.16 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.78 \[ \int e^{-2 p \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p \, dx=\frac {\left (-a^{2} c \,x^{2}+c \right )^{p} \left (a x -1\right )}{e^{2 \mathit {atanh} \left (a x \right ) p} a \left (2 p +1\right )} \] Input:
int((-a^2*c*x^2+c)^p/exp(2*p*atanh(a*x)),x)
Output:
(( - a**2*c*x**2 + c)**p*(a*x - 1))/(e**(2*atanh(a*x)*p)*a*(2*p + 1))