Integrand size = 22, antiderivative size = 217 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {a \left (c-\frac {c}{a^2 x^2}\right )^p x^2}{\sqrt {1-a^2 x^2}}+\frac {3 a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^3 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}+\frac {a (5-2 p) \left (c-\frac {c}{a^2 x^2}\right )^p x^2 \left (1-a^2 x^2\right )^{-p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)} \]
3*a^2*(c-c/a^2/x^2)^p*x^3*hypergeom([3/2-p, 3/2-p],[5/2-p],a^2*x^2)/(3-2*p )/((-a^2*x^2+1)^p)+1/2*a*(5-2*p)*(c-c/a^2/x^2)^p*x^2*hypergeom([1-p, 3/2-p ],[2-p],a^2*x^2)/(1-p)/((-a^2*x^2+1)^p)+(c-c/a^2/x^2)^p*x/(1-2*p)/(-a^2*x^ 2+1)^(1/2)-a*(c-c/a^2/x^2)^p*x^2/(-a^2*x^2+1)^(1/2)
Time = 0.14 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.81 \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {1}{2} \left (c-\frac {c}{a^2 x^2}\right )^p x \left (1-a^2 x^2\right )^{-p} \left (\frac {2 \left (1-a^2 x^2\right )^{-\frac {1}{2}+p}}{1-2 p}-\frac {3 a x \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{-1+p}+\frac {6 a^2 x^2 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,\frac {3}{2}-p,\frac {5}{2}-p,a^2 x^2\right )}{3-2 p}+\frac {a^3 x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2}-p,2-p,3-p,a^2 x^2\right )}{2-p}\right ) \]
((c - c/(a^2*x^2))^p*x*((2*(1 - a^2*x^2)^(-1/2 + p))/(1 - 2*p) - (3*a*x*Hy pergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x^2])/(-1 + p) + (6*a^2*x^2*Hyp ergeometric2F1[3/2 - p, 3/2 - p, 5/2 - p, a^2*x^2])/(3 - 2*p) + (a^3*x^3*H ypergeometric2F1[3/2 - p, 2 - p, 3 - p, a^2*x^2])/(2 - p)))/(2*(1 - a^2*x^ 2)^p)
Time = 0.65 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {6710, 6698, 559, 25, 2339, 27, 278, 545, 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.
\(\displaystyle \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int e^{3 \text {arctanh}(a x)} x^{-2 p} \left (1-a^2 x^2\right )^pdx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int x^{-2 p} (a x+1)^3 \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx\) |
\(\Big \downarrow \) 559 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (-\frac {\int -x^{-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 x^2 a^4+(5-2 p) x a^3+a^2\right )dx}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int x^{-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 x^2 a^4+(5-2 p) x a^3+a^2\right )dx}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 2339 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int a^2 x^{-2 p} (a (5-2 p) x+1) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+3 a^4 \int x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \int x^{-2 p} (a (5-2 p) x+1) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+3 a^4 \int x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \int x^{-2 p} (a (5-2 p) x+1) \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 545 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \left (a (5-2 p) \int x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+\frac {x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}\right )+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
\(\Big \downarrow \) 278 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {a^2 \left (\frac {a (5-2 p) x^{2-2 p} \operatorname {Hypergeometric2F1}\left (1-p,\frac {3}{2}-p,2-p,a^2 x^2\right )}{2 (1-p)}+\frac {x^{1-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}\right )+\frac {3 a^4 x^{3-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),\frac {3}{2}-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}}{a^2}-a x^{2-2 p} \left (1-a^2 x^2\right )^{p-\frac {1}{2}}\right )\) |
((c - c/(a^2*x^2))^p*x^(2*p)*(-(a*x^(2 - 2*p)*(1 - a^2*x^2)^(-1/2 + p)) + ((3*a^4*x^(3 - 2*p)*Hypergeometric2F1[(3 - 2*p)/2, 3/2 - p, (5 - 2*p)/2, a ^2*x^2])/(3 - 2*p) + a^2*((x^(1 - 2*p)*(1 - a^2*x^2)^(-1/2 + p))/(1 - 2*p) + (a*(5 - 2*p)*x^(2 - 2*p)*Hypergeometric2F1[1 - p, 3/2 - p, 2 - p, a^2*x ^2])/(2*(1 - p))))/a^2))/(1 - a^2*x^2)^p
3.9.2.3.1 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)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*e*(p + 1))), 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] && EqQ[Simplify[m + 2*p + 3], 0]
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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> With [{q = Expon[Pq, x]}, Simp[Coeff[Pq, x, q]/c^q Int[(c*x)^(m + q)*(a + b*x^ 2)^p, x], x] + Simp[1/c^q Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[c^q*Pq - Coeff[Pq, x, q]*(c*x)^q, x], x], x] /; EqQ[q, 1] || EqQ[m + q + 2*p + 1, 0] ] /; FreeQ[{a, b, c, m, p}, x] && PolyQ[Pq, x] && !(IGtQ[m, 0] && ILtQ[p + 1/2, 0])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
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/x^(2*p))*(1 - a ^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && !IntegerQ[n/2]
\[\int \frac {\left (a x +1\right )^{3} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a^2*x ^2 - 2*a*x + 1), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x + 1)**3/(-(a*x - 1)*(a* x + 1))**(3/2), x)
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{3} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int e^{3 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,{\left (a\,x+1\right )}^3}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \]