\(\int \frac {e^{\text {arctanh}(a x)}}{(c-a^2 c x^2)^{7/2}} \, dx\) [1023]

Optimal result

Integrand size = 22, antiderivative size = 277 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2}}{24 a c^3 (1-a x)^3 \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{16 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1+a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{16 a c^3 \sqrt {c-a^2 c x^2}} \] Output:

1/24*(-a^2*x^2+1)^(1/2)/a/c^3/(-a*x+1)^3/(-a^2*c*x^2+c)^(1/2)+3/32*(-a^2*x 
^2+1)^(1/2)/a/c^3/(-a*x+1)^2/(-a^2*c*x^2+c)^(1/2)+3/16*(-a^2*x^2+1)^(1/2)/ 
a/c^3/(-a*x+1)/(-a^2*c*x^2+c)^(1/2)-1/32*(-a^2*x^2+1)^(1/2)/a/c^3/(a*x+1)^ 
2/(-a^2*c*x^2+c)^(1/2)-1/8*(-a^2*x^2+1)^(1/2)/a/c^3/(a*x+1)/(-a^2*c*x^2+c) 
^(1/2)+5/16*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a/c^3/(-a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.37 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (-8-25 a x+25 a^2 x^2+15 a^3 x^3-15 a^4 x^4+15 (-1+a x)^3 (1+a x)^2 \text {arctanh}(a x)\right )}{48 a c^3 (-1+a x)^3 (1+a x)^2 \sqrt {c-a^2 c x^2}} \] Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(-8 - 25*a*x + 25*a^2*x^2 + 15*a^3*x^3 - 15*a^4*x^4 + 1 
5*(-1 + a*x)^3*(1 + a*x)^2*ArcTanh[a*x]))/(48*a*c^3*(-1 + a*x)^3*(1 + a*x) 
^2*Sqrt[c - a^2*c*x^2])
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6693, 6690, 54, 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.

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

\(\Big \downarrow \) 6693

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

\(\Big \downarrow \) 6690

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1-a x)^4 (a x+1)^3}dx}{c^3 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (\frac {3}{16 (a x-1)^2}+\frac {1}{8 (a x+1)^2}-\frac {3}{16 (a x-1)^3}+\frac {1}{16 (a x+1)^3}+\frac {1}{8 (a x-1)^4}-\frac {5}{16 \left (a^2 x^2-1\right )}\right )dx}{c^3 \sqrt {c-a^2 c x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {5 \text {arctanh}(a x)}{16 a}+\frac {3}{16 a (1-a x)}-\frac {1}{8 a (a x+1)}+\frac {3}{32 a (1-a x)^2}-\frac {1}{32 a (a x+1)^2}+\frac {1}{24 a (1-a x)^3}\right )}{c^3 \sqrt {c-a^2 c x^2}}\)

Input:

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

Output:

(Sqrt[1 - a^2*x^2]*(1/(24*a*(1 - a*x)^3) + 3/(32*a*(1 - a*x)^2) + 3/(16*a* 
(1 - a*x)) - 1/(32*a*(1 + a*x)^2) - 1/(8*a*(1 + a*x)) + (5*ArcTanh[a*x])/( 
16*a)))/(c^3*Sqrt[c - a^2*c*x^2])
 

Defintions of rubi rules used

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

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

rule 6690

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

rule 6693

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

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.82

method	result	size
default	\(\frac {\left (15 \ln \left (a x +1\right ) x^{5} a^{5}-15 \ln \left (a x -1\right ) x^{5} a^{5}-15 \ln \left (a x +1\right ) x^{4} a^{4}+15 \ln \left (a x -1\right ) x^{4} a^{4}-30 a^{4} x^{4}-30 \ln \left (a x +1\right ) x^{3} a^{3}+30 a^{3} \ln \left (a x -1\right ) x^{3}+30 a^{3} x^{3}+30 \ln \left (a x +1\right ) x^{2} a^{2}-30 a^{2} \ln \left (a x -1\right ) x^{2}+50 a^{2} x^{2}+15 \ln \left (a x +1\right ) x a -15 a \ln \left (a x -1\right ) x -50 a x -15 \ln \left (a x +1\right )+15 \ln \left (a x -1\right )-16\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{96 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right )^{2} \left (a x -1\right )^{3} c^{4} a}\)	\(227\)

Input:

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

Output:

1/96*(15*ln(a*x+1)*x^5*a^5-15*ln(a*x-1)*x^5*a^5-15*ln(a*x+1)*x^4*a^4+15*ln 
(a*x-1)*x^4*a^4-30*a^4*x^4-30*ln(a*x+1)*x^3*a^3+30*a^3*ln(a*x-1)*x^3+30*a^ 
3*x^3+30*ln(a*x+1)*x^2*a^2-30*a^2*ln(a*x-1)*x^2+50*a^2*x^2+15*ln(a*x+1)*x* 
a-15*a*ln(a*x-1)*x-50*a*x-15*ln(a*x+1)+15*ln(a*x-1)-16)/(-a^2*x^2+1)^(1/2) 
/(a*x+1)^2/(a*x-1)^3*(-c*(a^2*x^2-1))^(1/2)/c^4/a
 

Fricas [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 567, normalized size of antiderivative = 2.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\left [\frac {15 \, {\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) + 4 \, {\left (8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{192 \, {\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}, \frac {15 \, {\left (a^{7} x^{7} - a^{6} x^{6} - 3 \, a^{5} x^{5} + 3 \, a^{4} x^{4} + 3 \, a^{3} x^{3} - 3 \, a^{2} x^{2} - a x + 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) + 2 \, {\left (8 \, a^{5} x^{5} + 7 \, a^{4} x^{4} - 31 \, a^{3} x^{3} - 9 \, a^{2} x^{2} + 33 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{96 \, {\left (a^{8} c^{4} x^{7} - a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} + 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x + a c^{4}\right )}}\right ] \] Input:

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

Output:

[1/192*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x^4 + 3*a^3*x^3 - 3*a^2* 
x^2 - a*x + 1)*sqrt(c)*log(-(a^6*c*x^6 + 5*a^4*c*x^4 - 5*a^2*c*x^2 - 4*(a^ 
3*x^3 + a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*sqrt(c) - c)/(a^6*x^6 
 - 3*a^4*x^4 + 3*a^2*x^2 - 1)) + 4*(8*a^5*x^5 + 7*a^4*x^4 - 31*a^3*x^3 - 9 
*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^7 - 
 a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4*c^4*x^3 - 3*a^3*c^4*x 
^2 - a^2*c^4*x + a*c^4), 1/96*(15*(a^7*x^7 - a^6*x^6 - 3*a^5*x^5 + 3*a^4*x 
^4 + 3*a^3*x^3 - 3*a^2*x^2 - a*x + 1)*sqrt(-c)*arctan(2*sqrt(-a^2*c*x^2 + 
c)*sqrt(-a^2*x^2 + 1)*a*sqrt(-c)*x/(a^4*c*x^4 - c)) + 2*(8*a^5*x^5 + 7*a^4 
*x^4 - 31*a^3*x^3 - 9*a^2*x^2 + 33*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 
 + 1))/(a^8*c^4*x^7 - a^7*c^4*x^6 - 3*a^6*c^4*x^5 + 3*a^5*c^4*x^4 + 3*a^4* 
c^4*x^3 - 3*a^3*c^4*x^2 - a^2*c^4*x + a*c^4)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.82 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-15 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+15 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}+30 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-30 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}-15 \,\mathrm {log}\left (a x -1\right ) a x +15 \,\mathrm {log}\left (a x -1\right )+15 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-15 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}-30 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+30 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}+15 \,\mathrm {log}\left (a x +1\right ) a x -15 \,\mathrm {log}\left (a x +1\right )-30 a^{5} x^{5}+90 a^{3} x^{3}-10 a^{2} x^{2}-80 a x +14\right )}{96 a \,c^{4} \left (a^{5} x^{5}-a^{4} x^{4}-2 a^{3} x^{3}+2 a^{2} x^{2}+a x -1\right )} \] Input:

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

Output:

(sqrt(c)*( - 15*log(a*x - 1)*a**5*x**5 + 15*log(a*x - 1)*a**4*x**4 + 30*lo 
g(a*x - 1)*a**3*x**3 - 30*log(a*x - 1)*a**2*x**2 - 15*log(a*x - 1)*a*x + 1 
5*log(a*x - 1) + 15*log(a*x + 1)*a**5*x**5 - 15*log(a*x + 1)*a**4*x**4 - 3 
0*log(a*x + 1)*a**3*x**3 + 30*log(a*x + 1)*a**2*x**2 + 15*log(a*x + 1)*a*x 
 - 15*log(a*x + 1) - 30*a**5*x**5 + 90*a**3*x**3 - 10*a**2*x**2 - 80*a*x + 
 14))/(96*a*c**4*(a**5*x**5 - a**4*x**4 - 2*a**3*x**3 + 2*a**2*x**2 + a*x 
- 1))