Integrand size = 19, antiderivative size = 52 \[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=-\frac {\left (1-a^2 x^2\right )^{\frac {1}{2}+p}}{a (1+2 p)}+x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{2}-p,\frac {3}{2},a^2 x^2\right ) \] Output:
-(-a^2*x^2+1)^(1/2+p)/a/(1+2*p)+x*hypergeom([1/2, 1/2-p],[3/2],a^2*x^2)
Time = 0.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=-\frac {2^{\frac {1}{2}+p} (1-a x)^{\frac {1}{2}+p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2}-p,\frac {1}{2}+p,\frac {3}{2}+p,\frac {1}{2} (1-a x)\right )}{a \left (\frac {1}{2}+p\right )} \] Input:
Integrate[E^ArcTanh[a*x]*(1 - a^2*x^2)^p,x]
Output:
-((2^(1/2 + p)*(1 - a*x)^(1/2 + p)*Hypergeometric2F1[-1/2 - p, 1/2 + p, 3/ 2 + p, (1 - a*x)/2])/(a*(1/2 + p)))
Time = 0.39 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6690, 79}
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.
\(\displaystyle \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx\) |
\(\Big \downarrow \) 6690 |
\(\displaystyle \int (1-a x)^{p-\frac {1}{2}} (a x+1)^{p+\frac {1}{2}}dx\) |
\(\Big \downarrow \) 79 |
\(\displaystyle -\frac {2^{p+\frac {3}{2}} (1-a x)^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (-p-\frac {1}{2},p+\frac {1}{2},p+\frac {3}{2},\frac {1}{2} (1-a x)\right )}{a (2 p+1)}\) |
Input:
Int[E^ArcTanh[a*x]*(1 - a^2*x^2)^p,x]
Output:
-((2^(3/2 + p)*(1 - a*x)^(1/2 + p)*Hypergeometric2F1[-1/2 - p, 1/2 + p, 3/ 2 + p, (1 - a*x)/2])/(a*(1 + 2*p)))
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] && !IntegerQ[m] && !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] || !(RationalQ[n] && GtQ[-d/(b*c - a*d), 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])
Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85
method | result | size |
meijerg | \(\frac {a \,x^{2} \operatorname {hypergeom}\left (\left [1, \frac {1}{2}-p \right ], \left [2\right ], a^{2} x^{2}\right )}{2}+x \operatorname {hypergeom}\left (\left [\frac {1}{2}, \frac {1}{2}-p \right ], \left [\frac {3}{2}\right ], a^{2} x^{2}\right )\) | \(44\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x,method=_RETURNVERBOSE)
Output:
1/2*a*x^2*hypergeom([1,1/2-p],[2],a^2*x^2)+x*hypergeom([1/2,1/2-p],[3/2],a ^2*x^2)
\[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x, algorithm="fricas")
Output:
integral(-sqrt(-a^2*x^2 + 1)*(-a^2*x^2 + 1)^p/(a*x - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).
Time = 6.99 (sec) , antiderivative size = 236, normalized size of antiderivative = 4.54 \[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=a \left (\begin {cases} \frac {x^{2}}{2} & \text {for}\: a^{2} = 0 \\\frac {\begin {cases} \frac {\sqrt {- a^{2} x^{2} + 1} \left (- a^{2} x^{2} + 1\right )^{p}}{- 2 p - 1} & \text {for}\: p \neq - \frac {1}{2} \\\sqrt {- a^{2} x^{2} + 1} \left (- a^{2} x^{2} + 1\right )^{p} \log {\left (\frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )} & \text {otherwise} \end {cases}}{a^{2}} & \text {otherwise} \end {cases}\right ) - \frac {a^{2 p} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} 1, - p, - p - \frac {1}{2} \\ \frac {1}{2}, \frac {1}{2} - p \end {matrix}\middle | {\frac {1}{a^{2} x^{2}}} \right )}}{2 \sqrt {\pi } \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} - \frac {a^{2 p + 1} x^{2 p + 1} e^{i \pi p} \Gamma \left (- p - \frac {1}{2}\right ) \Gamma \left (p + \frac {1}{2}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {1}{2}, 1, p + \frac {1}{2} \\ p + 1, p + \frac {3}{2} \end {matrix}\middle | {a^{2} x^{2} e^{2 i \pi }} \right )}}{2 \sqrt {\pi } a \Gamma \left (\frac {1}{2} - p\right ) \Gamma \left (p + 1\right )} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*(-a**2*x**2+1)**p,x)
Output:
a*Piecewise((x**2/2, Eq(a**2, 0)), (Piecewise((sqrt(-a**2*x**2 + 1)*(-a**2 *x**2 + 1)**p/(-2*p - 1), Ne(p, -1/2)), (sqrt(-a**2*x**2 + 1)*(-a**2*x**2 + 1)**p*log(1/sqrt(-a**2*x**2 + 1)), True))/a**2, True)) - a**(2*p)*x**(2* p + 1)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hyper((1, -p, -p - 1/2), (1/2, 1/2 - p), 1/(a**2*x**2))/(2*sqrt(pi)*gamma(1/2 - p)*gamma(p + 1)) - a**(2*p + 1)*x**(2*p + 1)*exp(I*pi*p)*gamma(-p - 1/2)*gamma(p + 1/2)*hype r((1/2, 1, p + 1/2), (p + 1, p + 3/2), a**2*x**2*exp_polar(2*I*pi))/(2*sqr t(pi)*a*gamma(1/2 - p)*gamma(p + 1))
\[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x, algorithm="maxima")
Output:
integrate((a*x + 1)*(-a^2*x^2 + 1)^(p - 1/2), x)
\[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=\int { \frac {{\left (a x + 1\right )} {\left (-a^{2} x^{2} + 1\right )}^{p}}{\sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x, algorithm="giac")
Output:
integrate((a*x + 1)*(-a^2*x^2 + 1)^p/sqrt(-a^2*x^2 + 1), x)
Timed out. \[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^p\,\left (a\,x+1\right )}{\sqrt {1-a^2\,x^2}} \,d x \] Input:
int(((1 - a^2*x^2)^p*(a*x + 1))/(1 - a^2*x^2)^(1/2),x)
Output:
int(((1 - a^2*x^2)^p*(a*x + 1))/(1 - a^2*x^2)^(1/2), x)
\[ \int e^{\text {arctanh}(a x)} \left (1-a^2 x^2\right )^p \, dx=\frac {-\left (-a^{2} x^{2}+1\right )^{p +\frac {1}{2}}+2 \left (\int \frac {\left (-a^{2} x^{2}+1\right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a p +\left (\int \frac {\left (-a^{2} x^{2}+1\right )^{p}}{\sqrt {-a^{2} x^{2}+1}}d x \right ) a}{a \left (2 p +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*(-a^2*x^2+1)^p,x)
Output:
( - ( - a**2*x**2 + 1)**((2*p + 1)/2) + 2*int(( - a**2*x**2 + 1)**p/sqrt( - a**2*x**2 + 1),x)*a*p + int(( - a**2*x**2 + 1)**p/sqrt( - a**2*x**2 + 1) ,x)*a)/(a*(2*p + 1))