Integrand size = 22, antiderivative size = 184 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {4 c \left (c-\frac {c}{a^2 x^2}\right )^{-1+p}}{a (1-p)}-\frac {c \left (c-\frac {c}{a^2 x^2}\right )^{-1+p}}{a^2 (1-2 p) x}+c \left (c-\frac {c}{a^2 x^2}\right )^{-1+p} x-\frac {4 \left (2-5 p+p^2\right ) \left (1-\frac {1}{a^2 x^2}\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},2-p,\frac {3}{2},\frac {1}{a^2 x^2}\right )}{a^2 (1-2 p) x}+\frac {2 \left (c-\frac {c}{a^2 x^2}\right )^p \operatorname {Hypergeometric2F1}\left (1,p,1+p,1-\frac {1}{a^2 x^2}\right )}{a p} \] Output:
-4*c*(c-c/a^2/x^2)^(-1+p)/a/(1-p)-c*(c-c/a^2/x^2)^(-1+p)/a^2/(1-2*p)/x+c*( c-c/a^2/x^2)^(-1+p)*x-4*(p^2-5*p+2)*(c-c/a^2/x^2)^p*hypergeom([1/2, 2-p],[ 3/2],1/a^2/x^2)/a^2/(1-2*p)/((1-1/a^2/x^2)^p)/x+2*(c-c/a^2/x^2)^p*hypergeo m([1, p],[p+1],1-1/a^2/x^2)/a/p
Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.
Time = 0.16 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.18 \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} \left (-\left (-1+a^2 x^2\right )^2\right )^{-p} \left (-4 (-1+a x)^p (1+a x) \left (1-a^2 x^2\right )^p \operatorname {AppellF1}(1-2 p,1-p,-p,2-2 p,a x,-a x)+4 (-1+a x)^p (1+a x)^{2 p} \left (1-a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1-2 p,2-p,2-2 p,\frac {2 a x}{1+a x}\right )+(1-a x)^p (1+a x) \left (-1+a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,a^2 x^2\right )\right )}{(-1+2 p) (1+a x)} \] Input:
Integrate[E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^p,x]
Output:
-(((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 = 0.70 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.25, 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 |
\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {-4 a^2 \left (\frac {\left (p^2-5 p+2\right ) x^{1-2 p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1-2 p),2-p,\frac {1}{2} (3-2 p),a^2 x^2\right )}{1-2 p}-\frac {a x^{2-2 p} \left (1-a^2 x^2\right )^{p-1}}{2 (1-p)}\right )+a^2 (9-2 p) x^{1-2 p} \left (1-a^2 x^2\right )^{p-1}+\frac {2 a^5 x^{4-2 p} \operatorname {Hypergeometric2F1}\left (2-p,2-p,3-p,a^2 x^2\right )}{2-p}}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\) |
Input:
Int[E^(4*ArcTanh[a*x])*(c - c/(a^2*x^2))^p,x]
Output:
((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
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]
\[\int \frac {\left (a x +1\right )^{4} \left (c -\frac {c}{a^{2} x^{2}}\right )^{p}}{\left (-a^{2} x^{2}+1\right )^{2}}d x\]
Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x)
Output:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="fricas")
Output:
integral((a^2*x^2 + 2*a*x + 1)*((a^2*c*x^2 - c)/(a^2*x^2))^p/(a^2*x^2 - 2* a*x + 1), x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} \left (a x + 1\right )^{2}}{\left (a x - 1\right )^{2}}\, dx \] Input:
integrate((a*x+1)**4/(-a**2*x**2+1)**2*(c-c/a**2/x**2)**p,x)
Output:
Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x + 1)**2/(a*x - 1)**2, x )
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="maxima")
Output:
integrate((a*x + 1)^4*(c - c/(a^2*x^2))^p/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a x + 1\right )}^{4} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a^{2} x^{2} - 1\right )}^{2}} \,d x } \] Input:
integrate((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x, algorithm="giac")
Output:
integrate((a*x + 1)^4*(c - c/(a^2*x^2))^p/(a^2*x^2 - 1)^2, x)
Timed out. \[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,{\left (a\,x+1\right )}^4}{{\left (a^2\,x^2-1\right )}^2} \,d x \] Input:
int(((c - c/(a^2*x^2))^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2,x)
Output:
int(((c - c/(a^2*x^2))^p*(a*x + 1)^4)/(a^2*x^2 - 1)^2, x)
\[ \int e^{4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (a^{2} c \,x^{2}-c \right )^{p} a^{2} p \,x^{2}+\left (a^{2} c \,x^{2}-c \right )^{p} a p x -2 \left (a^{2} c \,x^{2}-c \right )^{p} a x +2 \left (a^{2} c \,x^{2}-c \right )^{p} p^{2}-6 \left (a^{2} c \,x^{2}-c \right )^{p} p +2 \left (a^{2} c \,x^{2}-c \right )^{p}-4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) a \,p^{3} x +12 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) a \,p^{2} x -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) a p x +4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) p^{3}-12 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) p^{2}+4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}-x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}+x^{2 p} x}d x \right ) p +4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p} x^{2}}{x^{2 p} a^{3} x^{3}-x^{2 p} a^{2} x^{2}-x^{2 p} a x +x^{2 p}}d x \right ) a^{4} p x -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p} x^{2}}{x^{2 p} a^{3} x^{3}-x^{2 p} a^{2} x^{2}-x^{2 p} a x +x^{2 p}}d x \right ) a^{3} p}{x^{2 p} a^{2 p} a p \left (a x -1\right )} \] Input:
int((a*x+1)^4/(-a^2*x^2+1)^2*(c-c/a^2/x^2)^p,x)
Output:
((a**2*c*x**2 - c)**p*a**2*p*x**2 + (a**2*c*x**2 - c)**p*a*p*x - 2*(a**2*c *x**2 - c)**p*a*x + 2*(a**2*c*x**2 - c)**p*p**2 - 6*(a**2*c*x**2 - c)**p*p + 2*(a**2*c*x**2 - c)**p - 4*x**(2*p)*int((a**2*c*x**2 - c)**p/(x**(2*p)* a**3*x**4 - x**(2*p)*a**2*x**3 - x**(2*p)*a*x**2 + x**(2*p)*x),x)*a*p**3*x + 12*x**(2*p)*int((a**2*c*x**2 - c)**p/(x**(2*p)*a**3*x**4 - x**(2*p)*a** 2*x**3 - x**(2*p)*a*x**2 + x**(2*p)*x),x)*a*p**2*x - 4*x**(2*p)*int((a**2* c*x**2 - c)**p/(x**(2*p)*a**3*x**4 - x**(2*p)*a**2*x**3 - x**(2*p)*a*x**2 + x**(2*p)*x),x)*a*p*x + 4*x**(2*p)*int((a**2*c*x**2 - c)**p/(x**(2*p)*a** 3*x**4 - x**(2*p)*a**2*x**3 - x**(2*p)*a*x**2 + x**(2*p)*x),x)*p**3 - 12*x **(2*p)*int((a**2*c*x**2 - c)**p/(x**(2*p)*a**3*x**4 - x**(2*p)*a**2*x**3 - x**(2*p)*a*x**2 + x**(2*p)*x),x)*p**2 + 4*x**(2*p)*int((a**2*c*x**2 - c) **p/(x**(2*p)*a**3*x**4 - x**(2*p)*a**2*x**3 - x**(2*p)*a*x**2 + x**(2*p)* x),x)*p + 4*x**(2*p)*int(((a**2*c*x**2 - c)**p*x**2)/(x**(2*p)*a**3*x**3 - x**(2*p)*a**2*x**2 - x**(2*p)*a*x + x**(2*p)),x)*a**4*p*x - 4*x**(2*p)*in t(((a**2*c*x**2 - c)**p*x**2)/(x**(2*p)*a**3*x**3 - x**(2*p)*a**2*x**2 - x **(2*p)*a*x + x**(2*p)),x)*a**3*p)/(x**(2*p)*a**(2*p)*a*p*(a*x - 1))