\(\int \frac {e^{3 \coth ^{-1}(a x)}}{(c-\frac {c}{a^2 x^2})^{7/2}} \, dx\) [830]

Optimal result

Integrand size = 24, antiderivative size = 360 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^4}+\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{2 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^3}-\frac {59 \sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)^2}+\frac {75 \sqrt {1-\frac {1}{a^2 x^2}}}{16 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1-a x)}-\frac {\sqrt {1-\frac {1}{a^2 x^2}}}{32 a c^3 \sqrt {c-\frac {c}{a^2 x^2}} (1+a x)}+\frac {201 \sqrt {1-\frac {1}{a^2 x^2}} \log (1-a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}}-\frac {9 \sqrt {1-\frac {1}{a^2 x^2}} \log (1+a x)}{64 a c^3 \sqrt {c-\frac {c}{a^2 x^2}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.32 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\left (1-\frac {1}{a^2 x^2}\right )^{7/2} \left (\frac {2 \left (104-207 a x-59 a^2 x^2+309 a^3 x^3-87 a^4 x^4-96 a^5 x^5+32 a^6 x^6\right )}{(-1+a x)^4 (1+a x)}+201 \log (1-a x)-9 \log (1+a x)\right )}{64 a \left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \] Input:

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

Output:

((1 - 1/(a^2*x^2))^(7/2)*((2*(104 - 207*a*x - 59*a^2*x^2 + 309*a^3*x^3 - 8 
7*a^4*x^4 - 96*a^5*x^5 + 32*a^6*x^6))/((-1 + a*x)^4*(1 + a*x)) + 201*Log[1 
 - a*x] - 9*Log[1 + a*x]))/(64*a*(c - c/(a^2*x^2))^(7/2))
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.40, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {6751, 6747, 25, 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

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 99

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

\(\Big \downarrow \) 2009

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

Input:

Int[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/(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)) + 1/(32*a^ 
8*(1 + a*x)) - (201*Log[1 - a*x])/(64*a^8) + (9*Log[1 + a*x])/(64*a^8)))/( 
c^3*Sqrt[c - c/(a^2*x^2)]))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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

Input:

int(1/((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+9*ln(a*x+1)*x^5 
*a^5-201*ln(a*x-1)*x^5*a^5+192*a^5*x^5-27*ln(a*x+1)*x^4*a^4+603*ln(a*x-1)* 
x^4*a^4+174*a^4*x^4+18*ln(a*x+1)*x^3*a^3-402*a^3*ln(a*x-1)*x^3-618*a^3*x^3 
+18*ln(a*x+1)*x^2*a^2-402*a^2*ln(a*x-1)*x^2+118*a^2*x^2-27*ln(a*x+1)*x*a+6 
03*a*ln(a*x-1)*x+414*a*x+9*ln(a*x+1)-201*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 = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.58 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {{\left (64 \, a^{6} x^{6} - 192 \, a^{5} x^{5} - 174 \, a^{4} x^{4} + 618 \, a^{3} x^{3} - 118 \, a^{2} x^{2} - 414 \, a x - 9 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x + 1\right ) + 201 \, {\left (a^{5} x^{5} - 3 \, a^{4} x^{4} + 2 \, a^{3} x^{3} + 2 \, a^{2} x^{2} - 3 \, a x + 1\right )} \log \left (a x - 1\right ) + 208\right )} \sqrt {a^{2} c}}{64 \, {\left (a^{7} c^{4} x^{5} - 3 \, a^{6} c^{4} x^{4} + 2 \, a^{5} c^{4} x^{3} + 2 \, a^{4} c^{4} x^{2} - 3 \, a^{3} c^{4} x + a^{2} c^{4}\right )}} \] Input:

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

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 - 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) + 201*(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(1/((a*x-1)/(a*x+1))**(3/2)/(c-c/a**2/x**2)**(7/2),x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (201 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}-603 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+402 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+402 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-603 \,\mathrm {log}\left (a x -1\right ) a x +201 \,\mathrm {log}\left (a x -1\right )-9 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}+27 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-18 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}-18 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+27 \,\mathrm {log}\left (a x +1\right ) a x -9 \,\mathrm {log}\left (a x +1\right )+64 a^{6} x^{6}-250 a^{5} x^{5}+502 a^{3} x^{3}-234 a^{2} x^{2}-240 a x +150\right )}{64 a \,c^{4} \left (a^{5} x^{5}-3 a^{4} x^{4}+2 a^{3} x^{3}+2 a^{2} x^{2}-3 a x +1\right )} \] Input:

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

Output:

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