\(\int \frac {e^{-3 \text {arctanh}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [774]

Optimal result

Integrand size = 24, antiderivative size = 359 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^4}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^3}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)^2}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \] Output:

(-a^2*x^2+1)^(7/2)/a^7/(c-c/a^2/x^2)^(7/2)/x^6+1/32*(-a^2*x^2+1)^(7/2)/a^8 
/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+1)+1/16*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2 
)^(7/2)/x^7/(a*x+1)^4-1/2*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/( 
a*x+1)^3+59/32*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(a*x+1)^2-75 
/16*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(a*x+1)+9/64*(-a^2*x^2+ 
1)^(7/2)*ln(-a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7-201/64*(-a^2*x^2+1)^(7/2)* 
ln(a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.41 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-2 \left (104+207 a x-59 a^2 x^2-309 a^3 x^3-87 a^4 x^4+96 a^5 x^5+32 a^6 x^6\right )-9 (-1+a x) (1+a x)^4 \log (1-a x)+201 (-1+a x) (1+a x)^4 \log (1+a x)\right )}{64 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x) (1+a x)^4} \] Input:

Integrate[1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)),x]
 

Output:

(Sqrt[1 - a^2*x^2]*(-2*(104 + 207*a*x - 59*a^2*x^2 - 309*a^3*x^3 - 87*a^4* 
x^4 + 96*a^5*x^5 + 32*a^6*x^6) - 9*(-1 + a*x)*(1 + a*x)^4*Log[1 - a*x] + 2 
01*(-1 + a*x)*(1 + a*x)^4*Log[1 + a*x]))/(64*a^2*c^3*Sqrt[c - c/(a^2*x^2)] 
*x*(-1 + a*x)*(1 + a*x)^4)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.38, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6710, 6700, 99, 2009}

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.

\(\displaystyle \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx\)

\(\Big \downarrow \) 6710

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {e^{-3 \text {arctanh}(a x)} x^7}{\left (1-a^2 x^2\right )^{7/2}}dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 6700

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

\(\Big \downarrow \) 99

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \int \left (-\frac {201}{64 a^7 (a x+1)}+\frac {75}{16 a^7 (a x+1)^2}-\frac {59}{16 a^7 (a x+1)^3}+\frac {3}{2 a^7 (a x+1)^4}-\frac {1}{4 a^7 (a x+1)^5}+\frac {1}{a^7}+\frac {9}{64 a^7 (a x-1)}+\frac {1}{32 a^7 (a x-1)^2}\right )dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (\frac {1}{32 a^8 (1-a x)}-\frac {75}{16 a^8 (a x+1)}+\frac {59}{32 a^8 (a x+1)^2}-\frac {1}{2 a^8 (a x+1)^3}+\frac {1}{16 a^8 (a x+1)^4}+\frac {9 \log (1-a x)}{64 a^8}-\frac {201 \log (a x+1)}{64 a^8}+\frac {x}{a^7}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\)

Input:

Int[1/(E^(3*ArcTanh[a*x])*(c - c/(a^2*x^2))^(7/2)),x]
 

Output:

((1 - a^2*x^2)^(7/2)*(x/a^7 + 1/(32*a^8*(1 - a*x)) + 1/(16*a^8*(1 + a*x)^4 
) - 1/(2*a^8*(1 + a*x)^3) + 59/(32*a^8*(1 + a*x)^2) - 75/(16*a^8*(1 + a*x) 
) + (9*Log[1 - a*x])/(64*a^8) - (201*Log[1 + a*x])/(64*a^8)))/((c - c/(a^2 
*x^2))^(7/2)*x^7)
 

Defintions of rubi rules used

rule 99

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], 
 x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | 
| (GtQ[m, 0] && GeQ[n, -1]))
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6700

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

rule 6710

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/x^(2*p))*(1 - a 
^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c 
+ a^2*d, 0] &&  !IntegerQ[p] &&  !IntegerQ[n/2]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.67

method	result	size
default	\(-\frac {\left (-64 x^{6} a^{6}+201 \ln \left (a x +1\right ) x^{5} a^{5}-9 \ln \left (a x -1\right ) x^{5} a^{5}-192 a^{5} x^{5}+603 \ln \left (a x +1\right ) x^{4} a^{4}-27 \ln \left (a x -1\right ) x^{4} a^{4}+174 a^{4} x^{4}+402 \ln \left (a x +1\right ) x^{3} a^{3}-18 a^{3} \ln \left (a x -1\right ) x^{3}+618 a^{3} x^{3}-402 \ln \left (a x +1\right ) x^{2} a^{2}+18 a^{2} \ln \left (a x -1\right ) x^{2}+118 a^{2} x^{2}-603 \ln \left (a x +1\right ) x a +27 a \ln \left (a x -1\right ) x -414 a x -201 \ln \left (a x +1\right )+9 \ln \left (a x -1\right )-208\right ) \left (a x -1\right )^{3}}{64 \sqrt {-a^{2} x^{2}+1}\, a^{8} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\)	\(241\)

Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x,method=_RETURNVER 
BOSE)
 

Output:

-1/64*(-64*x^6*a^6+201*ln(a*x+1)*x^5*a^5-9*ln(a*x-1)*x^5*a^5-192*a^5*x^5+6 
03*ln(a*x+1)*x^4*a^4-27*ln(a*x-1)*x^4*a^4+174*a^4*x^4+402*ln(a*x+1)*x^3*a^ 
3-18*a^3*ln(a*x-1)*x^3+618*a^3*x^3-402*ln(a*x+1)*x^2*a^2+18*a^2*ln(a*x-1)* 
x^2+118*a^2*x^2-603*ln(a*x+1)*x*a+27*a*ln(a*x-1)*x-414*a*x-201*ln(a*x+1)+9 
*ln(a*x-1)-208)*(a*x-1)^3/(-a^2*x^2+1)^(1/2)/a^8/x^7/(c*(a^2*x^2-1)/a^2/x^ 
2)^(7/2)
 

Fricas [F]

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

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm= 
"fricas")
 

Output:

integral(-sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9* 
c^4*x^9 + 3*a^8*c^4*x^8 - 8*a^6*c^4*x^6 - 6*a^5*c^4*x^5 + 6*a^4*c^4*x^4 + 
8*a^3*c^4*x^3 - 3*a*c^4*x - c^4), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Timed out} \] Input:

integrate(1/(a*x+1)**3*(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm= 
"maxima")
 

Output:

integrate((-a^2*x^2 + 1)^(3/2)/((a*x + 1)^3*(c - c/(a^2*x^2))^(7/2)), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm= 
"giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {{\left (1-a^2\,x^2\right )}^{3/2}}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (a\,x+1\right )}^3} \,d x \] Input:

int((1 - a^2*x^2)^(3/2)/((c - c/(a^2*x^2))^(7/2)*(a*x + 1)^3),x)
 

Output:

int((1 - a^2*x^2)^(3/2)/((c - c/(a^2*x^2))^(7/2)*(a*x + 1)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, i \left (9 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+27 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+18 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-18 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-27 \,\mathrm {log}\left (a x -1\right ) a x -9 \,\mathrm {log}\left (a x -1\right )-201 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-603 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-402 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+402 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+603 \,\mathrm {log}\left (a x +1\right ) a x +201 \,\mathrm {log}\left (a x +1\right )+64 a^{6} x^{6}+250 a^{5} x^{5}-502 a^{3} x^{3}-234 a^{2} x^{2}+240 a x +150\right )}{64 a \,c^{4} \left (a^{5} x^{5}+3 a^{4} x^{4}+2 a^{3} x^{3}-2 a^{2} x^{2}-3 a x -1\right )} \] Input:

int(1/(a*x+1)^3*(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x)
 

Output:

(sqrt(c)*i*(9*log(a*x - 1)*a**5*x**5 + 27*log(a*x - 1)*a**4*x**4 + 18*log( 
a*x - 1)*a**3*x**3 - 18*log(a*x - 1)*a**2*x**2 - 27*log(a*x - 1)*a*x - 9*l 
og(a*x - 1) - 201*log(a*x + 1)*a**5*x**5 - 603*log(a*x + 1)*a**4*x**4 - 40 
2*log(a*x + 1)*a**3*x**3 + 402*log(a*x + 1)*a**2*x**2 + 603*log(a*x + 1)*a 
*x + 201*log(a*x + 1) + 64*a**6*x**6 + 250*a**5*x**5 - 502*a**3*x**3 - 234 
*a**2*x**2 + 240*a*x + 150))/(64*a*c**4*(a**5*x**5 + 3*a**4*x**4 + 2*a**3* 
x**3 - 2*a**2*x**2 - 3*a*x - 1))