Integrand size = 22, antiderivative size = 339 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {2 a \left (c-\frac {c}{a^2 x^2}\right )^p x^2}{(1-p) (1-a x) (1+a x)}+\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),2-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}+\frac {6 a^2 \left (c-\frac {c}{a^2 x^2}\right )^p x^3 (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (3-2 p),2-p,\frac {1}{2} (5-2 p),a^2 x^2\right )}{3-2 p}+\frac {a^4 \left (c-\frac {c}{a^2 x^2}\right )^p x^5 (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (5-2 p),2-p,\frac {1}{2} (7-2 p),a^2 x^2\right )}{5-2 p}+\frac {2 a^3 \left (c-\frac {c}{a^2 x^2}\right )^p x^4 (1-a x)^{-p} (1+a x)^{-p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p} \]
2*a*(c-c/a^2/x^2)^p*x^2/(1-p)/(-a*x+1)/(a*x+1)+(c-c/a^2/x^2)^p*x*hypergeom ([2-p, 1/2-p],[3/2-p],a^2*x^2)/(1-2*p)/((-a*x+1)^p)/((a*x+1)^p)+6*a^2*(c-c /a^2/x^2)^p*x^3*hypergeom([2-p, 3/2-p],[5/2-p],a^2*x^2)/(3-2*p)/((-a*x+1)^ p)/((a*x+1)^p)+a^4*(c-c/a^2/x^2)^p*x^5*hypergeom([2-p, 5/2-p],[7/2-p],a^2* x^2)/(5-2*p)/((-a*x+1)^p)/((a*x+1)^p)+2*a^3*(c-c/a^2/x^2)^p*x^4*hypergeom( [2-p, 2-p],[3-p],a^2*x^2)/(2-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 = 0.17 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.64 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} \left (-\left (-1+a^2 x^2\right )^2\right )^{-p} \left (-4 (-1+a x)^p (1+a x) \left (1-a^2 x^2\right )^p \operatorname {AppellF1}(1-2 p,1-p,-p,2-2 p,a x,-a x)+4 (-1+a x)^p (1+a x)^{2 p} \left (1-a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1-2 p,2-p,2-2 p,\frac {2 a x}{1+a x}\right )+(1-a x)^p (1+a x) \left (-1+a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,a^2 x^2\right )\right )}{(-1+2 p) (1+a x)} \]
-(((c - c/(a^2*x^2))^p*x*(-4*(-1 + a*x)^p*(1 + a*x)*(1 - a^2*x^2)^p*Appell F1[1 - 2*p, 1 - p, -p, 2 - 2*p, a*x, -(a*x)] + 4*(-1 + a*x)^p*(1 + a*x)^(2 *p)*(1 - a^2*x^2)^p*Hypergeometric2F1[1 - 2*p, 2 - p, 2 - 2*p, (2*a*x)/(1 + a*x)] + (1 - a*x)^p*(1 + a*x)*(-1 + a^2*x^2)^p*Hypergeometric2F1[1/2 - p , -p, 3/2 - p, a^2*x^2]))/((-1 + 2*p)*(1 - a*x)^p*(1 + a*x)*(-(-1 + a^2*x^ 2)^2)^p))
Time = 0.67 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.68, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {6709, 559, 25, 2339, 278, 2340, 27, 557, 242, 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.
| \(\displaystyle \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx\) |
| \(\Big \downarrow \) 6709 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int x^{-2 p} (a x+1)^4 \left (1-a^2 x^2\right )^{p-2}dx\) |
| \(\Big \downarrow \) 559 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (-\frac {\int -x^{-2 p} \left (1-a^2 x^2\right )^{p-2} \left (4 x^3 a^5+(9-2 p) x^2 a^4+4 x a^3+a^2\right )dx}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int x^{-2 p} \left (1-a^2 x^2\right )^{p-2} \left (4 x^3 a^5+(9-2 p) x^2 a^4+4 x a^3+a^2\right )dx}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 2339 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {4 a^5 \int x^{3-2 p} \left (1-a^2 x^2\right )^{p-2}dx+\int x^{-2 p} \left (1-a^2 x^2\right )^{p-2} \left ((9-2 p) x^2 a^4+4 x a^3+a^2\right )dx}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int x^{-2 p} \left (1-a^2 x^2\right )^{p-2} \left ((9-2 p) x^2 a^4+4 x a^3+a^2\right )dx+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 2340 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\frac {\int -4 a^4 x^{-2 p} \left (p^2-5 p-a x+2\right ) \left (1-a^2 x^2\right )^{p-2}dx}{a^2}+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {-4 a^2 \int x^{-2 p} \left (p^2-5 p-a x+2\right ) \left (1-a^2 x^2\right )^{p-2}dx+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 557 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {-4 a^2 \left (\left (p^2-5 p+2\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{p-2}dx-a \int x^{1-2 p} \left (1-a^2 x^2\right )^{p-2}dx\right )+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {-4 a^2 \left (\left (p^2-5 p+2\right ) \int x^{-2 p} \left (1-a^2 x^2\right )^{p-2}dx-\frac {a x^{2-2 p} \left (1-a^2 x^2\right )^{p-1}}{2 (1-p)}\right )+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
| \(\Big \downarrow \) 278 |
| \(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {-4 a^2 \left (\frac {\left (p^2-5 p+2\right ) x^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),2-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a x^{2-2 p} \left (1-a^2 x^2\right )^{p-1}}{2 (1-p)}\right )+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
((c - c/(a^2*x^2))^p*x^(2*p)*(-(a^2*x^(3 - 2*p)*(1 - a^2*x^2)^(-1 + p)) + (a^2*(9 - 2*p)*x^(1 - 2*p)*(1 - a^2*x^2)^(-1 + p) - 4*a^2*(-1/2*(a*x^(2 - 2*p)*(1 - a^2*x^2)^(-1 + p))/(1 - p) + ((2 - 5*p + p^2)*x^(1 - 2*p)*Hyperg eometric2F1[(1 - 2*p)/2, 2 - p, (3 - 2*p)/2, a^2*x^2])/(1 - 2*p)) + (2*a^5 *x^(4 - 2*p)*Hypergeometric2F1[2 - p, 2 - p, 3 - p, a^2*x^2])/(2 - p))/a^2 ))/(1 - a^2*x^2)^p
3.9.1.3.1 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[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
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[(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[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[ {q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1 )*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(b*(m + q + 2*p + 1)) Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1) *Pq - b*f*(m + q + 2*p + 1)*x^q - a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x] && PolyQ [Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -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]
\[\int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}d x\]
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \]
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \]
Timed out. \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \]