Integrand size = 22, antiderivative size = 218 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),1-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac {a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^3 (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),1-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2 (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (1-p,1-p,2-p,a^2 x^2\right )}{1-p} \]
(c-c/a^2/x^2)^p*x*hypergeom([1-p, 1/2-p],[3/2-p],a^2*x^2)/(1-2*p)/((-a*x+1 )^p)/((a*x+1)^p)+a^2*(c-c/a^2/x^2)^p*x^3*hypergeom([1-p, 3/2-p],[5/2-p],a^ 2*x^2)/(3-2*p)/((-a*x+1)^p)/((a*x+1)^p)-a*(c-c/a^2/x^2)^p*x^2*hypergeom([1 -p, 1-p],[2-p],a^2*x^2)/(1-p)/((-a*x+1)^p)/((a*x+1)^p)
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.11 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.65 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} \left (-\left (-1+a^2 x^2\right )^2\right )^{-p} \left (-2 (-1+a x)^p \left (1-a^2 x^2\right )^p \operatorname {AppellF1}(1-2 p,-p,1-p,2-2 p,a x,-a x)+(1-a x)^p \left (-1+a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,a^2 x^2\right )\right )}{-1+2 p} \]
((c - c/(a^2*x^2))^p*x*(-2*(-1 + a*x)^p*(1 - a^2*x^2)^p*AppellF1[1 - 2*p, -p, 1 - p, 2 - 2*p, a*x, -(a*x)] + (1 - a*x)^p*(-1 + a^2*x^2)^p*Hypergeome tric2F1[1/2 - p, -p, 3/2 - p, a^2*x^2]))/((-1 + 2*p)*(1 - a*x)^p*(-(-1 + a ^2*x^2)^2)^p)
Time = 0.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.69, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6709, 570, 559, 27, 557, 278}
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.
\(\displaystyle \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx\) |
\(\Big \downarrow \) 6709 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int \frac {x^{-2 p} \left (1-a^2 x^2\right )^{p+1}}{(a x+1)^2}dx\) |
\(\Big \downarrow \) 570 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int x^{-2 p} (1-a x)^2 \left (1-a^2 x^2\right )^{p-1}dx\) |
\(\Big \downarrow \) 559 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (x^{1-2 p} \left (-\left (1-a^2 x^2\right )^p\right )-\frac {\int -2 a^2 x^{-2 p} (-p-a x+1) \left (1-a^2 x^2\right )^{p-1}dx}{a^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \int x^{-2 p} (-p-a x+1) \left (1-a^2 x^2\right )^{p-1}dx-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\) |
\(\Big \downarrow \) 557 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \left ((1-p) \int x^{-2 p} \left (1-a^2 x^2\right )^{p-1}dx-a \int x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}dx\right )-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (2 \left (\frac {(1-p) x^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),1-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a x^{2-2 p} \operatorname {Hypergeometric2F1}\left (1-p,1-p,2-p,a^2 x^2\right )}{2 (1-p)}\right )-x^{1-2 p} \left (1-a^2 x^2\right )^p\right )\) |
((c - c/(a^2*x^2))^p*x^(2*p)*(-(x^(1 - 2*p)*(1 - a^2*x^2)^p) + 2*(((1 - p) *x^(1 - 2*p)*Hypergeometric2F1[(1 - 2*p)/2, 1 - p, (3 - 2*p)/2, a^2*x^2])/ (1 - 2*p) - (a*x^(2 - 2*p)*Hypergeometric2F1[1 - p, 1 - p, 2 - p, a^2*x^2] )/(2*(1 - p)))))/(1 - a^2*x^2)^p
3.9.6.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( -b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] && !IGtQ[p, 0] && (ILtQ[p, 0 ] || GtQ[a, 0])
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Sym bol] :> Simp[c Int[(e*x)^m*(a + b*x^2)^p, x], x] + Simp[d/e Int[(e*x)^( m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d^n*(e*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*e^(n - 1)*( m + n + 2*p + 1))), x] + Simp[1/(b*(m + n + 2*p + 1)) Int[(e*x)^m*(a + b* x^2)^p*ExpandToSum[b*(m + n + 2*p + 1)*(c + d*x)^n - b*d^n*(m + n + 2*p + 1 )*x^n - a*d^n*(m + n - 1)*x^(n - 2), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && IGtQ[n, 1] && !IntegerQ[m] && NeQ[m + n + 2*p + 1, 0]
Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c^(2*n)/a^n Int[(e*x)^m*((a + b*x^2)^(n + p)/(c - d*x)^ n), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b*c^2 + a*d^2, 0] && I LtQ[n, -1] && !(IGtQ[m, 0] && ILtQ[m + n, 0] && !GtQ[p, 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[u*((1 + a*x)^n/(x^ (2*p)*(1 - a^2*x^2)^(n/2 - p))), x], x] /; FreeQ[{a, c, d, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && IntegerQ[n/2] && !GtQ[c, 0]
\[\int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \left (-a^{2} x^{2}+1\right )}{\left (a x +1\right )^{2}}d x\]
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \]
Result contains complex when optimal does not.
Time = 9.11 (sec) , antiderivative size = 699, normalized size of antiderivative = 3.21 \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\text {Too large to display} \]
-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 + c**p*p*x**(2 - 2*p)*e xp(I*pi*p)*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2 )/(2*a**(2*p)*gamma(2 - p)*gamma(p + 1)) + a**(3 - 2*p)*c**p*p*x**(3 - 2*p )*exp(I*pi*p)*gamma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*a**2*gamma(p - 1/2)*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 + c**p*p*x**(2 - 2*p)*exp(I*pi*p)*gamma (p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*a**(2*p)*ga mma(2 - p)*gamma(p + 1)) + a**(3 - 2*p)*c**p*p*x**(3 - 2*p)*exp(I*pi*p)*ga mma(p)*gamma(p - 3/2)*hyper((1 - p, 3/2 - p), (5/2 - p,), a**2*x**2)/(2*a* *2*gamma(p - 1/2)*gamma(p + 1)), True)) + Piecewise((0**p*log(a**2*x**2 - 1)/(2*a) + 0**p*acoth(a*x)/a - a**(1 - 2*p)*c**p*p*x**(2 - 2*p)*exp(I*pi*p )*gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamm a(2 - p)*gamma(p + 1)) - a**(1 - 2*p)*c**p*p*x**(1 - 2*p)*exp(I*pi*p)*gamm a(p)*gamma(p - 1/2)*hyper((1 - p, 1/2 - p), (3/2 - p,), a**2*x**2)/(2*a*ga mma(p + 1/2)*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**(1 - 2*p)*c**p*p*x**(2 - 2*p)*exp(I*pi*p)* gamma(p)*gamma(1 - p)*hyper((1 - p, 1 - p), (2 - p,), a**2*x**2)/(2*gamma( 2 - p)*gamma(p + 1)) - a**(1 - 2*p)*c**p*p*x**(1 - 2*p)*exp(I*pi*p)*gam...
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \]
\[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { -\frac {{\left (a^{2} x^{2} - 1\right )} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{2}} \,d x } \]
Timed out. \[ \int e^{-2 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,\left (a^2\,x^2-1\right )}{{\left (a\,x+1\right )}^2} \,d x \]