\(\int \frac {e^{-2 \text {arctanh}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\) [1273]

Optimal result

Integrand size = 24, antiderivative size = 111 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {4 x}{21 c^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{7 a c^2 (1+a x)^2 \left (c-a^2 c x^2\right )^{3/2}}-\frac {1}{7 a c^2 (1+a x) \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x}{21 c^3 \sqrt {c-a^2 c x^2}} \] Output:

4/21*x/c^2/(-a^2*c*x^2+c)^(3/2)-1/7/a/c^2/(a*x+1)^2/(-a^2*c*x^2+c)^(3/2)-1 
/7/a/c^2/(a*x+1)/(-a^2*c*x^2+c)^(3/2)+8/21*x/c^3/(-a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.86 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \left (6-9 a x-24 a^2 x^2-4 a^3 x^3+16 a^4 x^4+8 a^5 x^5\right )}{21 a c^3 (1-a x)^{3/2} (1+a x)^{7/2} \sqrt {c-a^2 c x^2}} \] Input:

Integrate[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2)),x]
 

Output:

-1/21*(Sqrt[1 - a^2*x^2]*(6 - 9*a*x - 24*a^2*x^2 - 4*a^3*x^3 + 16*a^4*x^4 
+ 8*a^5*x^5))/(a*c^3*(1 - a*x)^(3/2)*(1 + a*x)^(7/2)*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6692, 457, 209, 209, 208}

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.

\(\displaystyle \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6692

\(\displaystyle c \int \frac {(1-a x)^2}{\left (c-a^2 c x^2\right )^{9/2}}dx\)

\(\Big \downarrow \) 457

\(\displaystyle c \left (\frac {5 \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}}dx}{7 c}-\frac {2 (1-a x)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle c \left (\frac {5 \left (\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}-\frac {2 (1-a x)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle c \left (\frac {5 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}-\frac {2 (1-a x)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\)

\(\Big \downarrow \) 208

\(\displaystyle c \left (\frac {5 \left (\frac {4 \left (\frac {2 x}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}-\frac {2 (1-a x)}{7 a c \left (c-a^2 c x^2\right )^{7/2}}\right )\)

Input:

Int[1/(E^(2*ArcTanh[a*x])*(c - a^2*c*x^2)^(7/2)),x]
 

Output:

c*((-2*(1 - a*x))/(7*a*c*(c - a^2*c*x^2)^(7/2)) + (5*(x/(5*c*(c - a^2*c*x^ 
2)^(5/2)) + (4*(x/(3*c*(c - a^2*c*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c - a^2* 
c*x^2])))/(5*c)))/(7*c))
 

Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 457

Int[((c_) + (d_.)*(x_))^2*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[d*( 
c + d*x)*((a + b*x^2)^(p + 1)/(b*(p + 1))), x] - Simp[d^2*((p + 2)/(b*(p + 
1)))   Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[ 
b*c^2 + a*d^2, 0] && LtQ[p, -1]
 

rule 6692

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[1/c^(n/2)   Int[(c + d*x^2)^(p + n/2)/(1 - a*x)^n, x], x] /; FreeQ[{a, 
 c, d, p}, x] && EqQ[a^2*c + d, 0] &&  !(IntegerQ[p] || GtQ[c, 0]) && ILtQ[ 
n/2, 0]
 

Maple [A] (verified)

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.58

method	result	size
gosper	\(-\frac {\left (a x -1\right )^{2} \left (8 a^{5} x^{5}+16 a^{4} x^{4}-4 a^{3} x^{3}-24 a^{2} x^{2}-9 a x +6\right )}{21 \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}} a}\)	\(64\)
trager	\(-\frac {\left (8 a^{5} x^{5}+16 a^{4} x^{4}-4 a^{3} x^{3}-24 a^{2} x^{2}-9 a x +6\right ) \sqrt {-a^{2} c \,x^{2}+c}}{21 c^{4} \left (a x +1\right )^{4} \left (a x -1\right )^{2} a}\)	\(74\)
orering	\(\frac {\left (8 a^{5} x^{5}+16 a^{4} x^{4}-4 a^{3} x^{3}-24 a^{2} x^{2}-9 a x +6\right ) \left (a x -1\right ) \left (-a^{2} x^{2}+1\right )}{21 a \left (a x +1\right ) \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}\)	\(79\)
default	\(-\frac {x}{5 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}-\frac {4 \left (\frac {x}{3 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {2 x}{3 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}\right )}{5 c}+\frac {-\frac {2}{7 a c \left (x +\frac {1}{a}\right ) \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {12 a \left (-\frac {-2 \left (x +\frac {1}{a}\right ) a^{2} c +2 a c}{10 a^{2} c^{2} \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {-\frac {2 \left (-2 \left (x +\frac {1}{a}\right ) a^{2} c +2 a c \right )}{15 a^{2} c^{2} \left (-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x +\frac {1}{a}\right ) a^{2} c +2 a c \right )}{15 a^{2} c^{3} \sqrt {-\left (x +\frac {1}{a}\right )^{2} a^{2} c +2 \left (x +\frac {1}{a}\right ) a c}}}{c}\right )}{7}}{a}\)	\(268\)

Input:

int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-1/21*(a*x-1)^2*(8*a^5*x^5+16*a^4*x^4-4*a^3*x^3-24*a^2*x^2-9*a*x+6)/(-a^2* 
c*x^2+c)^(7/2)/a
 

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.12 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {{\left (8 \, a^{5} x^{5} + 16 \, a^{4} x^{4} - 4 \, a^{3} x^{3} - 24 \, a^{2} x^{2} - 9 \, a x + 6\right )} \sqrt {-a^{2} c x^{2} + c}}{21 \, {\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \] Input:

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="fric 
as")
 

Output:

-1/21*(8*a^5*x^5 + 16*a^4*x^4 - 4*a^3*x^3 - 24*a^2*x^2 - 9*a*x + 6)*sqrt(- 
a^2*c*x^2 + c)/(a^7*c^4*x^6 + 2*a^6*c^4*x^5 - a^5*c^4*x^4 - 4*a^4*c^4*x^3 
- a^3*c^4*x^2 + 2*a^2*c^4*x + a*c^4)
 

Sympy [F]

\[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=- \int \frac {a x}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \left (- \frac {1}{- a^{7} c^{3} x^{7} \sqrt {- a^{2} c x^{2} + c} - a^{6} c^{3} x^{6} \sqrt {- a^{2} c x^{2} + c} + 3 a^{5} c^{3} x^{5} \sqrt {- a^{2} c x^{2} + c} + 3 a^{4} c^{3} x^{4} \sqrt {- a^{2} c x^{2} + c} - 3 a^{3} c^{3} x^{3} \sqrt {- a^{2} c x^{2} + c} - 3 a^{2} c^{3} x^{2} \sqrt {- a^{2} c x^{2} + c} + a c^{3} x \sqrt {- a^{2} c x^{2} + c} + c^{3} \sqrt {- a^{2} c x^{2} + c}}\right )\, dx \] Input:

integrate(1/(a*x+1)**2*(-a**2*x**2+1)/(-a**2*c*x**2+c)**(7/2),x)
 

Output:

-Integral(a*x/(-a**7*c**3*x**7*sqrt(-a**2*c*x**2 + c) - a**6*c**3*x**6*sqr 
t(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5*sqrt(-a**2*c*x**2 + c) + 3*a**4*c** 
3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3*c**3*x**3*sqrt(-a**2*c*x**2 + c) - 
3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) + a*c**3*x*sqrt(-a**2*c*x**2 + c) 
+ c**3*sqrt(-a**2*c*x**2 + c)), x) - Integral(-1/(-a**7*c**3*x**7*sqrt(-a* 
*2*c*x**2 + c) - a**6*c**3*x**6*sqrt(-a**2*c*x**2 + c) + 3*a**5*c**3*x**5* 
sqrt(-a**2*c*x**2 + c) + 3*a**4*c**3*x**4*sqrt(-a**2*c*x**2 + c) - 3*a**3* 
c**3*x**3*sqrt(-a**2*c*x**2 + c) - 3*a**2*c**3*x**2*sqrt(-a**2*c*x**2 + c) 
 + a*c**3*x*sqrt(-a**2*c*x**2 + c) + c**3*sqrt(-a**2*c*x**2 + c)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=-\frac {2}{7 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a^{2} c x + {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} a c\right )}} + \frac {8 \, x}{21 \, \sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{21 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {x}{7 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c} \] Input:

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="maxi 
ma")
 

Output:

-2/7/((-a^2*c*x^2 + c)^(5/2)*a^2*c*x + (-a^2*c*x^2 + c)^(5/2)*a*c) + 8/21* 
x/(sqrt(-a^2*c*x^2 + c)*c^3) + 4/21*x/((-a^2*c*x^2 + c)^(3/2)*c^2) + 1/7*x 
/((-a^2*c*x^2 + c)^(5/2)*c)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (95) = 190\).

Time = 0.19 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.70 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {a^{5} {\left (\frac {14 \, {\left (7 \, c - \frac {15 \, c}{a x + 1}\right )}}{a^{5} {\left (c - \frac {2 \, c}{a x + 1}\right )} c^{3} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )} + \frac {3 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{3} c^{42} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 21 \, a^{30} {\left (c - \frac {2 \, c}{a x + 1}\right )}^{2} c^{43} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 210 \, a^{30} c^{45} \sqrt {-c + \frac {2 \, c}{a x + 1}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6} - 70 \, a^{30} c^{44} {\left (-c + \frac {2 \, c}{a x + 1}\right )}^{\frac {3}{2}} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{6} \mathrm {sgn}\left (a\right )^{6}}{a^{35} c^{49} \mathrm {sgn}\left (\frac {1}{a x + 1}\right )^{7} \mathrm {sgn}\left (a\right )^{7}}\right )} - \frac {256 \, \mathrm {sgn}\left (\frac {1}{a x + 1}\right ) \mathrm {sgn}\left (a\right )}{\sqrt {-c} c^{3}}}{672 \, {\left | a \right |}} \] Input:

integrate(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x, algorithm="giac 
")
 

Output:

1/672*(a^5*(14*(7*c - 15*c/(a*x + 1))/(a^5*(c - 2*c/(a*x + 1))*c^3*sqrt(-c 
 + 2*c/(a*x + 1))*sgn(1/(a*x + 1))*sgn(a)) + (3*a^30*(c - 2*c/(a*x + 1))^3 
*c^42*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 - 21*a^30*(c - 
2*c/(a*x + 1))^2*c^43*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 
 - 210*a^30*c^45*sqrt(-c + 2*c/(a*x + 1))*sgn(1/(a*x + 1))^6*sgn(a)^6 - 70 
*a^30*c^44*(-c + 2*c/(a*x + 1))^(3/2)*sgn(1/(a*x + 1))^6*sgn(a)^6)/(a^35*c 
^49*sgn(1/(a*x + 1))^7*sgn(a)^7)) - 256*sgn(1/(a*x + 1))*sgn(a)/(sqrt(-c)* 
c^3))/abs(a)
 

Mupad [B] (verification not implemented)

Time = 27.10 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {11\,x}{42\,c^4}-\frac {5}{28\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {c-a^2\,c\,x^2}}{28\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {c-a^2\,c\,x^2}}{14\,a\,c^4\,{\left (a\,x+1\right )}^3}-\frac {8\,x\,\sqrt {c-a^2\,c\,x^2}}{21\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \] Input:

int(-(a^2*x^2 - 1)/((c - a^2*c*x^2)^(7/2)*(a*x + 1)^2),x)
 

Output:

((c - a^2*c*x^2)^(1/2)*((11*x)/(42*c^4) - 5/(28*a*c^4)))/((a*x - 1)^2*(a*x 
 + 1)^2) - (c - a^2*c*x^2)^(1/2)/(28*a*c^4*(a*x + 1)^4) - (c - a^2*c*x^2)^ 
(1/2)/(14*a*c^4*(a*x + 1)^3) - (8*x*(c - a^2*c*x^2)^(1/2))/(21*c^4*(a*x - 
1)*(a*x + 1))
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.35 \[ \int \frac {e^{-2 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (9 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+18 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-18 \sqrt {-a^{2} x^{2}+1}\, a x -9 \sqrt {-a^{2} x^{2}+1}+16 a^{5} x^{5}+32 a^{4} x^{4}-8 a^{3} x^{3}-48 a^{2} x^{2}-18 a x +12\right )}{42 \sqrt {-a^{2} x^{2}+1}\, a \,c^{4} \left (a^{4} x^{4}+2 a^{3} x^{3}-2 a x -1\right )} \] Input:

int(1/(a*x+1)^2*(-a^2*x^2+1)/(-a^2*c*x^2+c)^(7/2),x)
 

Output:

(sqrt(c)*(9*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 18*sqrt( - a**2*x**2 + 1)*a 
**3*x**3 - 18*sqrt( - a**2*x**2 + 1)*a*x - 9*sqrt( - a**2*x**2 + 1) + 16*a 
**5*x**5 + 32*a**4*x**4 - 8*a**3*x**3 - 48*a**2*x**2 - 18*a*x + 12))/(42*s 
qrt( - a**2*x**2 + 1)*a*c**4*(a**4*x**4 + 2*a**3*x**3 - 2*a*x - 1))