\(\int e^{-4 \text {arctanh}(a x)} (c-\frac {c}{a^2 x^2})^p \, dx\) [790]

Optimal result

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*hypergeom 
([1, p],[p+1],1-1/a^2/x^2)/a/p
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.17 \[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=-\frac {\left (c-\frac {c}{a^2 x^2}\right )^p x (1-a x)^{-p} \left (-\left (-1+a^2 x^2\right )^2\right )^{-p} \left (-4 (-1+a x)^p (1+a x) \left (1-a^2 x^2\right )^p \operatorname {AppellF1}(1-2 p,-p,1-p,2-2 p,a x,-a x)+4 (-1+a x)^p (1+a x)^{2 p} \left (1-a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (1-2 p,-p,2-2 p,\frac {2 a x}{1+a x}\right )+(1-a x)^p (1+a x) \left (-1+a^2 x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-p,-p,\frac {3}{2}-p,a^2 x^2\right )\right )}{(-1+2 p) (1+a x)} \] Input:

Integrate[(c - c/(a^2*x^2))^p/E^(4*ArcTanh[a*x]),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, -p, 1 - 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, -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))
 

Rubi [A] (verified)

Time = 1.10 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.25, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6709, 570, 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

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int \frac {x^{-2 p} \left (1-a^2 x^2\right )^{p+2}}{(a x+1)^4}dx\)

\(\Big \downarrow \) 570

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \int x^{-2 p} (1-a x)^4 \left (1-a^2 x^2\right )^{p-2}dx\)

\(\Big \downarrow \) 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

\(\displaystyle x^{2 p} \left (1-a^2 x^2\right )^{-p} \left (c-\frac {c}{a^2 x^2}\right )^p \left (\frac {\int x^{-2 p} \left (1-a^2 x^2\right )^{p-2} \left ((9-2 p) x^2 a^4-4 x a^3+a^2\right )dx-4 a^5 \int x^{3-2 p} \left (1-a^2 x^2\right )^{p-2}dx}{a^2}-a^2 x^{3-2 p} \left (1-a^2 x^2\right )^{p-1}\right )\)

\(\Big \downarrow \) 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[(c - c/(a^2*x^2))^p/E^(4*ArcTanh[a*x]),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*((a*x^(2 - 2*p)* 
(1 - a^2*x^2)^(-1 + p))/(2*(1 - p)) + ((2 - 5*p + p^2)*x^(1 - 2*p)*Hyperge 
ometric2F1[(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
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 242

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]
 

rule 278

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])
 

rule 557

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]
 

rule 559

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]
 

rule 570

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])
 

rule 2339

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])
 

rule 2340

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])
 

rule 6709

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]
 

Maple [F]

\[\int \frac {\left (c -\frac {c}{a^{2} x^{2}}\right )^{p} \left (-a^{2} x^{2}+1\right )^{2}}{\left (a x +1\right )^{4}}d x\]

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)
 

Fricas [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^4*(-a^2*x^2+1)^2,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)
 

Sympy [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {\left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{p} \left (a x - 1\right )^{2}}{\left (a x + 1\right )^{2}}\, dx \] Input:

integrate((c-c/a**2/x**2)**p/(a*x+1)**4*(-a**2*x**2+1)**2,x)
 

Output:

Integral((-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**p*(a*x - 1)**2/(a*x + 1)**2, x 
)
 

Maxima [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="maxima")
 

Output:

integrate((a^2*x^2 - 1)^2*(c - c/(a^2*x^2))^p/(a*x + 1)^4, x)
 

Giac [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int { \frac {{\left (a^{2} x^{2} - 1\right )}^{2} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{p}}{{\left (a x + 1\right )}^{4}} \,d x } \] Input:

integrate((c-c/a^2/x^2)^p/(a*x+1)^4*(-a^2*x^2+1)^2,x, algorithm="giac")
 

Output:

integrate((a^2*x^2 - 1)^2*(c - c/(a^2*x^2))^p/(a*x + 1)^4, x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\int \frac {{\left (c-\frac {c}{a^2\,x^2}\right )}^p\,{\left (a^2\,x^2-1\right )}^2}{{\left (a\,x+1\right )}^4} \,d x \] Input:

int(((c - c/(a^2*x^2))^p*(a^2*x^2 - 1)^2)/(a*x + 1)^4,x)
                                                                                    
                                                                                    
 

Output:

int(((c - c/(a^2*x^2))^p*(a^2*x^2 - 1)^2)/(a*x + 1)^4, x)
 

Reduce [F]

\[ \int e^{-4 \text {arctanh}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^p \, dx=\frac {\left (a^{2} c \,x^{2}-c \right )^{p} a^{2} p \,x^{2}-\left (a^{2} c \,x^{2}-c \right )^{p} a p x +2 \left (a^{2} c \,x^{2}-c \right )^{p} a x +2 \left (a^{2} c \,x^{2}-c \right )^{p} p^{2}-6 \left (a^{2} c \,x^{2}-c \right )^{p} p +2 \left (a^{2} c \,x^{2}-c \right )^{p}-4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) a \,p^{3} x +12 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) a \,p^{2} x -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) a p x -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) p^{3}+12 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) p^{2}-4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p}}{x^{2 p} a^{3} x^{4}+x^{2 p} a^{2} x^{3}-x^{2 p} a \,x^{2}-x^{2 p} x}d x \right ) p -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p} x^{2}}{x^{2 p} a^{3} x^{3}+x^{2 p} a^{2} x^{2}-x^{2 p} a x -x^{2 p}}d x \right ) a^{4} p x -4 x^{2 p} \left (\int \frac {\left (a^{2} c \,x^{2}-c \right )^{p} x^{2}}{x^{2 p} a^{3} x^{3}+x^{2 p} a^{2} x^{2}-x^{2 p} a x -x^{2 p}}d x \right ) a^{3} p}{x^{2 p} a^{2 p} a p \left (a x +1\right )} \] Input:

int((c-c/a^2/x^2)^p/(a*x+1)^4*(-a^2*x^2+1)^2,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))