\(\int \frac {e^{-3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [855]

Optimal result

Integrand size = 24, antiderivative size = 357 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^4}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^3}+\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)^2}-\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.30 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (64 x+\frac {208+478 a x+74 a^2 x^2-490 a^3 x^3-302 a^4 x^4}{a (-1+a x) (1+a x)^4}+\frac {9 \log (1-a x)}{a}-\frac {201 \log (1+a x)}{a}\right )}{64 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \] Input:

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

Output:

((1 - 1/(a^2*x^2))^(7/2)*(64*x + (208 + 478*a*x + 74*a^2*x^2 - 490*a^3*x^3 
 - 302*a^4*x^4)/(a*(-1 + a*x)*(1 + a*x)^4) + (9*Log[1 - a*x])/a - (201*Log 
[1 + a*x])/a))/(64*(c - c/(a^2*x^2))^(7/2))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.39, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6751, 6747, 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.

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

\(\Big \downarrow \) 6751

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

\(\Big \downarrow \) 6747

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(a^7*Sqrt[1 - 1/(a^2*x^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^3*S 
qrt[c - c/(a^2*x^2)])
 

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 6747

Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_.), x_Symb 
ol] :> Simp[c^p/a^(2*p)   Int[(u/x^(2*p))*(-1 + a*x)^(p - n/2)*(1 + a*x)^(p 
 + n/2), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] &&  !Inte 
gerQ[n/2] && (IntegerQ[p] || GtQ[c, 0]) && IntegersQ[2*p, p + n/2]
 

rule 6751

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.69

method	result	size
default	\(-\frac {\left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x -1\right ) \left (a x +1\right ) \left (-64 x^{6} a^{6}+201 \ln \left (a x +1\right ) x^{5} a^{5}-9 \ln \left (a x -1\right ) x^{5} a^{5}-192 a^{5} x^{5}+603 \ln \left (a x +1\right ) x^{4} a^{4}-27 \ln \left (a x -1\right ) x^{4} a^{4}+174 a^{4} x^{4}+402 \ln \left (a x +1\right ) x^{3} a^{3}-18 a^{3} \ln \left (a x -1\right ) x^{3}+618 a^{3} x^{3}-402 \ln \left (a x +1\right ) x^{2} a^{2}+18 a^{2} \ln \left (a x -1\right ) x^{2}+118 a^{2} x^{2}-603 \ln \left (a x +1\right ) x a +27 a \ln \left (a x -1\right ) x -414 a x -201 \ln \left (a x +1\right )+9 \ln \left (a x -1\right )-208\right )}{64 a^{8} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\)	\(247\)

Input:

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

Output:

-1/64*((a*x-1)/(a*x+1))^(3/2)*(a*x-1)*(a*x+1)*(-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+603*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^8/x^7/(c*(a^2*x^ 
2-1)/a^2/x^2)^(7/2)
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.58 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {{\left (64 \, a^{6} x^{6} + 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} - 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} + 414 \, a x - 201 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x + 1\right ) + 9 \, {\left (a^{5} x^{5} + 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} - 3 \, a x - 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} + 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} - 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x - a^{2} c^{4}\right )}} \] Input:

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

Output:

1/64*(64*a^6*x^6 + 192*a^5*x^5 - 174*a^4*x^4 - 618*a^3*x^3 - 118*a^2*x^2 + 
 414*a*x - 201*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)*l 
og(a*x + 1) + 9*(a^5*x^5 + 3*a^4*x^4 + 2*a^3*x^3 - 2*a^2*x^2 - 3*a*x - 1)* 
log(a*x - 1) + 208)*sqrt(a^2*c)/(a^7*c^4*x^5 + 3*a^6*c^4*x^4 + 2*a^5*c^4*x 
^3 - 2*a^4*c^4*x^2 - 3*a^3*c^4*x - a^2*c^4)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.66 \[ \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (9 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+27 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+18 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-18 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-27 \,\mathrm {log}\left (a x -1\right ) a x -9 \,\mathrm {log}\left (a x -1\right )-201 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-603 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-402 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+402 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+603 \,\mathrm {log}\left (a x +1\right ) a x +201 \,\mathrm {log}\left (a x +1\right )+64 a^{6} x^{6}+250 a^{5} x^{5}-502 a^{3} x^{3}-234 a^{2} x^{2}+240 a x +150\right )}{64 a \,c^{4} \left (a^{5} x^{5}+3 a^{4} x^{4}+2 a^{3} x^{3}-2 a^{2} x^{2}-3 a x -1\right )} \] Input:

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

Output:

(sqrt(c)*(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*log 
(a*x - 1) - 201*log(a*x + 1)*a**5*x**5 - 603*log(a*x + 1)*a**4*x**4 - 402* 
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))