\(\int \frac {e^{3 \text {arctanh}(a x)} (c-a^2 c x^2)^p}{x^2} \, dx\) [1202]

Optimal result

Integrand size = 25, antiderivative size = 187 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\frac {4 a \left (c-a^2 c x^2\right )^p}{(1-2 p) \sqrt {1-a^2 x^2}}-\frac {\left (c-a^2 c x^2\right )^p}{x \sqrt {1-a^2 x^2}}+a^2 (5-2 p) x \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )-\frac {3 a \sqrt {1-a^2 x^2} \left (c-a^2 c x^2\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2}+p,\frac {3}{2}+p,1-a^2 x^2\right )}{1+2 p} \] Output:

4*a*(-a^2*c*x^2+c)^p/(1-2*p)/(-a^2*x^2+1)^(1/2)-(-a^2*c*x^2+c)^p/x/(-a^2*x 
^2+1)^(1/2)+a^2*(5-2*p)*x*(-a^2*c*x^2+c)^p*hypergeom([1/2, 3/2-p],[3/2],a^ 
2*x^2)/((-a^2*x^2+1)^p)-3*a*(-a^2*x^2+1)^(1/2)*(-a^2*c*x^2+c)^p*hypergeom( 
[1, 1/2+p],[3/2+p],-a^2*x^2+1)/(1+2*p)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.71 \[ \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx=\left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{2}-p,\frac {1}{2},a^2 x^2\right )}{x}+a \left (3 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )+\frac {\left (1-a^2 x^2\right )^{-\frac {1}{2}+p} \left (1+3 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2}+p,\frac {1}{2}+p,1-a^2 x^2\right )\right )}{1-2 p}\right )\right ) \] Input:

Integrate[(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p)/x^2,x]
 

Output:

((c - a^2*c*x^2)^p*(-(Hypergeometric2F1[-1/2, 3/2 - p, 1/2, a^2*x^2]/x) + 
a*(3*a*x*Hypergeometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2] + ((1 - a^2*x^2)^(- 
1/2 + p)*(1 + 3*Hypergeometric2F1[1, -1/2 + p, 1/2 + p, 1 - a^2*x^2]))/(1 
- 2*p))))/(1 - a^2*x^2)^p
 

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {6703, 6698, 543, 354, 27, 88, 75, 359, 237}

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.

\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)} \left (c-a^2 c x^2\right )^p}{x^2} \, dx\)

\(\Big \downarrow \) 6703

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \int \frac {e^{3 \text {arctanh}(a x)} \left (1-a^2 x^2\right )^p}{x^2}dx\)

\(\Big \downarrow \) 6698

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

\(\Big \downarrow \) 543

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 88

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\frac {1}{2} a \left (3 \int \frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x^2}dx^2+\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}\right )+\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx\right )\)

\(\Big \downarrow \) 75

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (\int \frac {\left (1-a^2 x^2\right )^{p-\frac {3}{2}} \left (3 a^2 x^2+1\right )}{x^2}dx+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )\right )\)

\(\Big \downarrow \) 359

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a^2 (5-2 p) \int \left (1-a^2 x^2\right )^{p-\frac {3}{2}}dx+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )-\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}\right )\)

\(\Big \downarrow \) 237

\(\displaystyle \left (1-a^2 x^2\right )^{-p} \left (c-a^2 c x^2\right )^p \left (a^2 (5-2 p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{2}-p,\frac {3}{2},a^2 x^2\right )+\frac {1}{2} a \left (\frac {8 \left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{1-2 p}-\frac {6 \left (1-a^2 x^2\right )^{p+\frac {1}{2}} \operatorname {Hypergeometric2F1}\left (1,p+\frac {1}{2},p+\frac {3}{2},1-a^2 x^2\right )}{2 p+1}\right )-\frac {\left (1-a^2 x^2\right )^{p-\frac {1}{2}}}{x}\right )\)

Input:

Int[(E^(3*ArcTanh[a*x])*(c - a^2*c*x^2)^p)/x^2,x]
 

Output:

((c - a^2*c*x^2)^p*(-((1 - a^2*x^2)^(-1/2 + p)/x) + a^2*(5 - 2*p)*x*Hyperg 
eometric2F1[1/2, 3/2 - p, 3/2, a^2*x^2] + (a*((8*(1 - a^2*x^2)^(-1/2 + p)) 
/(1 - 2*p) - (6*(1 - a^2*x^2)^(1/2 + p)*Hypergeometric2F1[1, 1/2 + p, 3/2 
+ p, 1 - a^2*x^2])/(1 + 2*p)))/2))/(1 - a^2*x^2)^p
 

Defintions of rubi rules used

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 75

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x 
)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + 
 d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (IntegerQ[m] 
 || GtQ[-d/(b*c), 0])
 

rule 88

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^Simplify[p + 1], 
 x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] &&  !RationalQ[p] && SumSimpl 
erQ[p, 1]
 

rule 237

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[- 
p, 1/2, 1/2 + 1, (-b)*(x^2/a)], x] /; FreeQ[{a, b, p}, x] &&  !IntegerQ[2*p 
] && GtQ[a, 0]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 359

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + 
Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1))   Int[(e*x)^(m + 2)* 
(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] 
&& LtQ[m, -1] &&  !ILtQ[p, -1]
 

rule 543

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> Module[{k}, Int[x^m*Sum[Binomial[n, 2*k]*c^(n - 2*k)*d^(2*k)*x^(2*k), 
 {k, 0, n/2}]*(a + b*x^2)^p, x] + Int[x^(m + 1)*Sum[Binomial[n, 2*k + 1]*c^ 
(n - 2*k - 1)*d^(2*k + 1)*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2)^p, x]] /; 
 FreeQ[{a, b, c, d, p}, x] && IGtQ[n, 1] && IntegerQ[m] &&  !IntegerQ[2*p] 
&&  !(EqQ[m, 1] && EqQ[b*c^2 + a*d^2, 0])
 

rule 6698

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]
 

rule 6703

Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ 
Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar 
t[p])   Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, 
d, m, n, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) &&  !I 
ntegerQ[n/2]
 

Maple [F]

\[\int \frac {\left (a x +1\right )^{3} \left (-a^{2} c \,x^{2}+c \right )^{p}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} x^{2}}d x\]

Input:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x)
 

Output:

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x)
 

Fricas [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x, algorithm=" 
fricas")
 

Output:

integral(sqrt(-a^2*x^2 + 1)*(a*x + 1)*(-a^2*c*x^2 + c)^p/(a^2*x^4 - 2*a*x^ 
3 + x^2), x)
 

Sympy [F]

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

integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)*(-a**2*c*x**2+c)**p/x**2,x)
 

Output:

Integral((-c*(a*x - 1)*(a*x + 1))**p*(a*x + 1)**3/(x**2*(-(a*x - 1)*(a*x + 
 1))**(3/2)), x)
 

Maxima [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x, algorithm=" 
maxima")
                                                                                    
                                                                                    
 

Output:

integrate((a*x + 1)^3*(-a^2*c*x^2 + c)^p/((-a^2*x^2 + 1)^(3/2)*x^2), x)
 

Giac [F]

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

integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x, algorithm=" 
giac")
 

Output:

integrate((a*x + 1)^3*(-a^2*c*x^2 + c)^p/((-a^2*x^2 + 1)^(3/2)*x^2), x)
 

Mupad [F(-1)]

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

int(((c - a^2*c*x^2)^p*(a*x + 1)^3)/(x^2*(1 - a^2*x^2)^(3/2)),x)
 

Output:

int(((c - a^2*c*x^2)^p*(a*x + 1)^3)/(x^2*(1 - a^2*x^2)^(3/2)), x)
 

Reduce [F]

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

int((a*x+1)^3/(-a^2*x^2+1)^(3/2)*(-a^2*c*x^2+c)^p/x^2,x)
 

Output:

 - int(( - a**2*c*x**2 + c)**p/(sqrt( - a**2*x**2 + 1)*a*x**3 - sqrt( - a* 
*2*x**2 + 1)*x**2),x) - 2*int(( - a**2*c*x**2 + c)**p/(sqrt( - a**2*x**2 + 
 1)*a*x**2 - sqrt( - a**2*x**2 + 1)*x),x)*a - int(( - a**2*c*x**2 + c)**p/ 
(sqrt( - a**2*x**2 + 1)*a*x - sqrt( - a**2*x**2 + 1)),x)*a**2