\(\int \frac {e^{2 \coth ^{-1}(a x)}}{(c-a^2 c x^2)^{9/2}} \, dx\) [614]

Optimal result

Integrand size = 24, antiderivative size = 120 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=-\frac {2 (1+a x)}{9 a \left (c-a^2 c x^2\right )^{9/2}}-\frac {x}{9 c \left (c-a^2 c x^2\right )^{7/2}}-\frac {2 x}{15 c^2 \left (c-a^2 c x^2\right )^{5/2}}-\frac {8 x}{45 c^3 \left (c-a^2 c x^2\right )^{3/2}}-\frac {16 x}{45 c^4 \sqrt {c-a^2 c x^2}} \] Output:

1/9*(-2*a*x-2)/a/(-a^2*c*x^2+c)^(9/2)-1/9*x/c/(-a^2*c*x^2+c)^(7/2)-2/15*x/ 
c^2/(-a^2*c*x^2+c)^(5/2)-8/45*x/c^3/(-a^2*c*x^2+c)^(3/2)-16/45*x/c^4/(-a^2 
*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.93 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-10-25 a x+60 a^2 x^2+10 a^3 x^3-80 a^4 x^4+24 a^5 x^5+32 a^6 x^6-16 a^7 x^7\right )}{45 a c^4 (1-a x)^{9/2} (1+a x)^{5/2} \sqrt {c-a^2 c x^2}} \] Input:

Integrate[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]
 

Output:

(Sqrt[1 - a^2*x^2]*(-10 - 25*a*x + 60*a^2*x^2 + 10*a^3*x^3 - 80*a^4*x^4 + 
24*a^5*x^5 + 32*a^6*x^6 - 16*a^7*x^7))/(45*a*c^4*(1 - a*x)^(9/2)*(1 + a*x) 
^(5/2)*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.25, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {6717, 6691, 457, 209, 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.

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

\(\Big \downarrow \) 6717

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

\(\Big \downarrow \) 6691

\(\displaystyle -c \int \frac {(a x+1)^2}{\left (c-a^2 c x^2\right )^{11/2}}dx\)

\(\Big \downarrow \) 457

\(\displaystyle -c \left (\frac {7 \int \frac {1}{\left (c-a^2 c x^2\right )^{9/2}}dx}{9 c}+\frac {2 (a x+1)}{9 a c \left (c-a^2 c x^2\right )^{9/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle -c \left (\frac {7 \left (\frac {6 \int \frac {1}{\left (c-a^2 c x^2\right )^{7/2}}dx}{7 c}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{7/2}}\right )}{9 c}+\frac {2 (a x+1)}{9 a c \left (c-a^2 c x^2\right )^{9/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle -c \left (\frac {7 \left (\frac {6 \left (\frac {4 \int \frac {1}{\left (c-a^2 c x^2\right )^{5/2}}dx}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{7/2}}\right )}{9 c}+\frac {2 (a x+1)}{9 a c \left (c-a^2 c x^2\right )^{9/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle -c \left (\frac {7 \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{7/2}}\right )}{9 c}+\frac {2 (a x+1)}{9 a c \left (c-a^2 c x^2\right )^{9/2}}\right )\)

\(\Big \downarrow \) 208

\(\displaystyle -c \left (\frac {7 \left (\frac {6 \left (\frac {4 \left (\frac {2 x}{3 c^2 \sqrt {c-a^2 c x^2}}+\frac {x}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )}{5 c}+\frac {x}{5 c \left (c-a^2 c x^2\right )^{5/2}}\right )}{7 c}+\frac {x}{7 c \left (c-a^2 c x^2\right )^{7/2}}\right )}{9 c}+\frac {2 (a x+1)}{9 a c \left (c-a^2 c x^2\right )^{9/2}}\right )\)

Input:

Int[E^(2*ArcCoth[a*x])/(c - a^2*c*x^2)^(9/2),x]
 

Output:

-(c*((2*(1 + a*x))/(9*a*c*(c - a^2*c*x^2)^(9/2)) + (7*(x/(7*c*(c - a^2*c*x 
^2)^(7/2)) + (6*(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)))/(9*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 6691

Int[E^(ArcTanh[(a_.)*(x_)]*(n_))*((c_) + (d_.)*(x_)^2)^(p_.), x_Symbol] :> 
Simp[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]) && IGtQ[n/ 
2, 0]
 

rule 6717

Int[E^(ArcCoth[(a_.)*(x_)]*(n_))*(u_.), x_Symbol] :> Simp[(-1)^(n/2)   Int[ 
u*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[a, x] && IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.67

method	result	size
gosper	\(-\frac {\left (a x +1\right )^{2} \left (16 a^{7} x^{7}-32 x^{6} a^{6}-24 a^{5} x^{5}+80 a^{4} x^{4}-10 a^{3} x^{3}-60 a^{2} x^{2}+25 a x +10\right )}{45 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\)	\(80\)
orering	\(-\frac {\left (a x +1\right )^{2} \left (16 a^{7} x^{7}-32 x^{6} a^{6}-24 a^{5} x^{5}+80 a^{4} x^{4}-10 a^{3} x^{3}-60 a^{2} x^{2}+25 a x +10\right )}{45 a \left (-a^{2} c \,x^{2}+c \right )^{\frac {9}{2}}}\)	\(80\)
trager	\(\frac {\left (16 a^{7} x^{7}-32 x^{6} a^{6}-24 a^{5} x^{5}+80 a^{4} x^{4}-10 a^{3} x^{3}-60 a^{2} x^{2}+25 a x +10\right ) \sqrt {-a^{2} c \,x^{2}+c}}{45 c^{5} \left (a x -1\right )^{5} \left (a x +1\right )^{3} a}\)	\(90\)
default	\(\frac {x}{7 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {7}{2}}}+\frac {\frac {6 x}{35 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}+\frac {6 \left (\frac {4 x}{15 c \left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}+\frac {8 x}{15 c^{2} \sqrt {-a^{2} c \,x^{2}+c}}\right )}{7 c}}{c}+\frac {\frac {2}{9 a c \left (x -\frac {1}{a}\right ) \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}-\frac {16 a \left (-\frac {-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c}{14 a^{2} c^{2} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {7}{2}}}+\frac {-\frac {3 \left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right )}{35 a^{2} c^{2} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {5}{2}}}+\frac {6 \left (-\frac {2 \left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right )}{15 a^{2} c^{2} \left (-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c \right )^{\frac {3}{2}}}-\frac {4 \left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right )}{15 a^{2} c^{3} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\right )}{7 c}}{c}\right )}{9}}{a}\)	\(378\)

Input:

int(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^(9/2),x,method=_RETURNVERBOSE)
 

Output:

-1/45*(a*x+1)^2*(16*a^7*x^7-32*a^6*x^6-24*a^5*x^5+80*a^4*x^4-10*a^3*x^3-60 
*a^2*x^2+25*a*x+10)/a/(-a^2*c*x^2+c)^(9/2)
 

Fricas [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.27 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {{\left (16 \, a^{7} x^{7} - 32 \, a^{6} x^{6} - 24 \, a^{5} x^{5} + 80 \, a^{4} x^{4} - 10 \, a^{3} x^{3} - 60 \, a^{2} x^{2} + 25 \, a x + 10\right )} \sqrt {-a^{2} c x^{2} + c}}{45 \, {\left (a^{9} c^{5} x^{8} - 2 \, a^{8} c^{5} x^{7} - 2 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} - 6 \, a^{4} c^{5} x^{3} + 2 \, a^{3} c^{5} x^{2} + 2 \, a^{2} c^{5} x - a c^{5}\right )}} \] Input:

integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^(9/2),x, algorithm="fricas")
 

Output:

1/45*(16*a^7*x^7 - 32*a^6*x^6 - 24*a^5*x^5 + 80*a^4*x^4 - 10*a^3*x^3 - 60* 
a^2*x^2 + 25*a*x + 10)*sqrt(-a^2*c*x^2 + c)/(a^9*c^5*x^8 - 2*a^8*c^5*x^7 - 
 2*a^7*c^5*x^6 + 6*a^6*c^5*x^5 - 6*a^4*c^5*x^3 + 2*a^3*c^5*x^2 + 2*a^2*c^5 
*x - a*c^5)
 

Sympy [F]

\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int \frac {a x + 1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {9}{2}} \left (a x - 1\right )}\, dx \] Input:

integrate(1/(a*x-1)*(a*x+1)/(-a**2*c*x**2+c)**(9/2),x)
 

Output:

Integral((a*x + 1)/((-c*(a*x - 1)*(a*x + 1))**(9/2)*(a*x - 1)), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.98 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {2}{9 \, {\left ({\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a^{2} c x - {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} a c\right )}} - \frac {16 \, x}{45 \, \sqrt {-a^{2} c x^{2} + c} c^{4}} - \frac {8 \, x}{45 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{3}} - \frac {2 \, x}{15 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c^{2}} - \frac {x}{9 \, {\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} c} \] Input:

integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^(9/2),x, algorithm="maxima")
 

Output:

2/9/((-a^2*c*x^2 + c)^(7/2)*a^2*c*x - (-a^2*c*x^2 + c)^(7/2)*a*c) - 16/45* 
x/(sqrt(-a^2*c*x^2 + c)*c^4) - 8/45*x/((-a^2*c*x^2 + c)^(3/2)*c^3) - 2/15* 
x/((-a^2*c*x^2 + c)^(5/2)*c^2) - 1/9*x/((-a^2*c*x^2 + c)^(7/2)*c)
 

Giac [F]

\[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\int { \frac {a x + 1}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {9}{2}} {\left (a x - 1\right )}} \,d x } \] Input:

integrate(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^(9/2),x, algorithm="giac")
 

Output:

integrate((a*x + 1)/((-a^2*c*x^2 + c)^(9/2)*(a*x - 1)), x)
 

Mupad [B] (verification not implemented)

Time = 13.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.48 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {\sqrt {c-a^2\,c\,x^2}}{72\,a\,c^5\,{\left (a\,x-1\right )}^5}-\frac {5\,\sqrt {c-a^2\,c\,x^2}}{144\,a\,c^5\,{\left (a\,x-1\right )}^4}+\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {31\,x}{120\,c^5}+\frac {5}{24\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {c-a^2\,c\,x^2}\,\left (\frac {8\,x}{45\,c^5}-\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}+\frac {16\,x\,\sqrt {c-a^2\,c\,x^2}}{45\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \] Input:

int((a*x + 1)/((c - a^2*c*x^2)^(9/2)*(a*x - 1)),x)
 

Output:

(c - a^2*c*x^2)^(1/2)/(72*a*c^5*(a*x - 1)^5) - (5*(c - a^2*c*x^2)^(1/2))/( 
144*a*c^5*(a*x - 1)^4) + ((c - a^2*c*x^2)^(1/2)*((31*x)/(120*c^5) + 5/(24* 
a*c^5)))/((a*x - 1)^3*(a*x + 1)^3) - ((c - a^2*c*x^2)^(1/2)*((8*x)/(45*c^5 
) - 5/(144*a*c^5)))/((a*x - 1)^2*(a*x + 1)^2) + (16*x*(c - a^2*c*x^2)^(1/2 
))/(45*c^5*(a*x - 1)*(a*x + 1))
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.06 \[ \int \frac {e^{2 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^{9/2}} \, dx=\frac {\sqrt {c}\, \left (25 \sqrt {-a^{2} x^{2}+1}\, a^{6} x^{6}-50 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-25 \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+100 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-25 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-50 \sqrt {-a^{2} x^{2}+1}\, a x +25 \sqrt {-a^{2} x^{2}+1}-32 a^{7} x^{7}+64 a^{6} x^{6}+48 a^{5} x^{5}-160 a^{4} x^{4}+20 a^{3} x^{3}+120 a^{2} x^{2}-50 a x -20\right )}{90 \sqrt {-a^{2} x^{2}+1}\, a \,c^{5} \left (a^{6} x^{6}-2 a^{5} x^{5}-a^{4} x^{4}+4 a^{3} x^{3}-a^{2} x^{2}-2 a x +1\right )} \] Input:

int(1/(a*x-1)*(a*x+1)/(-a^2*c*x^2+c)^(9/2),x)
 

Output:

(sqrt(c)*(25*sqrt( - a**2*x**2 + 1)*a**6*x**6 - 50*sqrt( - a**2*x**2 + 1)* 
a**5*x**5 - 25*sqrt( - a**2*x**2 + 1)*a**4*x**4 + 100*sqrt( - a**2*x**2 + 
1)*a**3*x**3 - 25*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 50*sqrt( - a**2*x**2 
+ 1)*a*x + 25*sqrt( - a**2*x**2 + 1) - 32*a**7*x**7 + 64*a**6*x**6 + 48*a* 
*5*x**5 - 160*a**4*x**4 + 20*a**3*x**3 + 120*a**2*x**2 - 50*a*x - 20))/(90 
*sqrt( - a**2*x**2 + 1)*a*c**5*(a**6*x**6 - 2*a**5*x**5 - a**4*x**4 + 4*a* 
*3*x**3 - a**2*x**2 - 2*a*x + 1))