Integrand size = 24, antiderivative size = 278 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2}}{16 a c^3 (1-a x)^4 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{12 a c^3 (1-a x)^3 \sqrt {c-a^2 c x^2}}+\frac {3 \sqrt {1-a^2 x^2}}{32 a c^3 (1-a x)^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a c^3 (1-a x) \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{32 a c^3 (1+a x) \sqrt {c-a^2 c x^2}}+\frac {5 \sqrt {1-a^2 x^2} \text {arctanh}(a x)}{32 a c^3 \sqrt {c-a^2 c x^2}} \] Output:
1/16*(-a^2*x^2+1)^(1/2)/a/c^3/(-a*x+1)^4/(-a^2*c*x^2+c)^(1/2)+1/12*(-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)+1/8*(-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)/(-a^2*c*x^2+c )^(1/2)+5/32*(-a^2*x^2+1)^(1/2)*arctanh(a*x)/a/c^3/(-a^2*c*x^2+c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.36 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (32-15 a x-35 a^2 x^2+45 a^3 x^3-15 a^4 x^4+15 (-1+a x)^4 (1+a x) \text {arctanh}(a x)\right )}{96 a c^3 (-1+a x)^4 (1+a x) \sqrt {c-a^2 c x^2}} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(32 - 15*a*x - 35*a^2*x^2 + 45*a^3*x^3 - 15*a^4*x^4 + 1 5*(-1 + a*x)^4*(1 + a*x)*ArcTanh[a*x]))/(96*a*c^3*(-1 + a*x)^4*(1 + a*x)*S qrt[c - a^2*c*x^2])
Time = 0.70 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.
| \(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6693 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{3 \text {arctanh}(a x)}}{\left (1-a^2 x^2\right )^{7/2}}dx}{c^3 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 6690 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {1}{(1-a x)^5 (a x+1)^2}dx}{c^3 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (\frac {1}{8 (a x-1)^2}+\frac {1}{32 (a x+1)^2}-\frac {3}{16 (a x-1)^3}+\frac {1}{4 (a x-1)^4}-\frac {1}{4 (a x-1)^5}-\frac {5}{32 \left (a^2 x^2-1\right )}\right )dx}{c^3 \sqrt {c-a^2 c x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (\frac {5 \text {arctanh}(a x)}{32 a}+\frac {1}{8 a (1-a x)}-\frac {1}{32 a (a x+1)}+\frac {3}{32 a (1-a x)^2}+\frac {1}{12 a (1-a x)^3}+\frac {1}{16 a (1-a x)^4}\right )}{c^3 \sqrt {c-a^2 c x^2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - a^2*c*x^2)^(7/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(1/(16*a*(1 - a*x)^4) + 1/(12*a*(1 - a*x)^3) + 3/(32*a* (1 - a*x)^2) + 1/(8*a*(1 - a*x)) - 1/(32*a*(1 + a*x)) + (5*ArcTanh[a*x])/( 32*a)))/(c^3*Sqrt[c - a^2*c*x^2])
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])
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])
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])
Time = 0.13 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(\frac {\left (15 \ln \left (a x +1\right ) x^{5} a^{5}-15 \ln \left (a x -1\right ) x^{5} a^{5}-45 \ln \left (a x +1\right ) x^{4} a^{4}+45 \ln \left (a x -1\right ) x^{4} a^{4}-30 a^{4} x^{4}+30 \ln \left (a x +1\right ) x^{3} a^{3}-30 a^{3} \ln \left (a x -1\right ) x^{3}+90 a^{3} x^{3}+30 \ln \left (a x +1\right ) x^{2} a^{2}-30 a^{2} \ln \left (a x -1\right ) x^{2}-70 a^{2} x^{2}-45 \ln \left (a x +1\right ) x a +45 a \ln \left (a x -1\right ) x -30 a x +15 \ln \left (a x +1\right )-15 \ln \left (a x -1\right )+64\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{192 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right ) \left (a x -1\right )^{4} c^{4} a}\) | \(227\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x,method=_RETURNVERB OSE)
Output:
1/192*(15*ln(a*x+1)*x^5*a^5-15*ln(a*x-1)*x^5*a^5-45*ln(a*x+1)*x^4*a^4+45*l n(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+90*a ^3*x^3+30*ln(a*x+1)*x^2*a^2-30*a^2*ln(a*x-1)*x^2-70*a^2*x^2-45*ln(a*x+1)*x *a+45*a*ln(a*x-1)*x-30*a*x+15*ln(a*x+1)-15*ln(a*x-1)+64)/(-a^2*x^2+1)^(1/2 )/(a*x+1)/(a*x-1)^4*(-c*(a^2*x^2-1))^(1/2)/c^4/a
Time = 0.17 (sec) , antiderivative size = 565, normalized size of antiderivative = 2.03 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\left [\frac {15 \, {\left (a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {c} \log \left (-\frac {a^{6} c x^{6} + 5 \, a^{4} c x^{4} - 5 \, a^{2} c x^{2} - 4 \, {\left (a^{3} x^{3} + a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} \sqrt {c} - c}{a^{6} x^{6} - 3 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - 1}\right ) + 4 \, {\left (32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{384 \, {\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}, \frac {15 \, {\left (a^{7} x^{7} - 3 \, a^{6} x^{6} + a^{5} x^{5} + 5 \, a^{4} x^{4} - 5 \, a^{3} x^{3} - a^{2} x^{2} + 3 \, a x - 1\right )} \sqrt {-c} \arctan \left (\frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1} a \sqrt {-c} x}{a^{4} c x^{4} - c}\right ) + 2 \, {\left (32 \, a^{5} x^{5} - 81 \, a^{4} x^{4} + 19 \, a^{3} x^{3} + 99 \, a^{2} x^{2} - 81 \, a x\right )} \sqrt {-a^{2} c x^{2} + c} \sqrt {-a^{2} x^{2} + 1}}{192 \, {\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}}\right ] \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=" fricas")
Output:
[1/384*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a^4*x^4 - 5*a^3*x^3 - a^2*x^ 2 + 3*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*(32*a^5*x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1))/(a^8*c^4*x^ 7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4* x^2 + 3*a^2*c^4*x - a*c^4), 1/192*(15*(a^7*x^7 - 3*a^6*x^6 + a^5*x^5 + 5*a ^4*x^4 - 5*a^3*x^3 - a^2*x^2 + 3*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*(32*a^5*x^5 - 81*a^4*x^4 + 19*a^3*x^3 + 99*a^2*x^2 - 81*a*x)*sqrt(-a^2*c*x^2 + c)*sqrt(- a^2*x^2 + 1))/(a^8*c^4*x^7 - 3*a^7*c^4*x^6 + a^6*c^4*x^5 + 5*a^5*c^4*x^4 - 5*a^4*c^4*x^3 - a^3*c^4*x^2 + 3*a^2*c^4*x - a*c^4)]
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(-a**2*c*x**2+c)**(7/2),x)
Output:
Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(a*x - 1)*(a*x + 1))**(7/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=" maxima")
Output:
integrate((a*x + 1)^3/((-a^2*c*x^2 + c)^(7/2)*(-a^2*x^2 + 1)^(3/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {7}{2}} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x, algorithm=" giac")
Output:
integrate((a*x + 1)^3/((-a^2*c*x^2 + c)^(7/2)*(-a^2*x^2 + 1)^(3/2)), x)
Timed out. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\int \frac {{\left (a\,x+1\right )}^3}{{\left (c-a^2\,c\,x^2\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((a*x + 1)^3/((c - a^2*c*x^2)^(7/2)*(1 - a^2*x^2)^(3/2)),x)
Output:
int((a*x + 1)^3/((c - a^2*c*x^2)^(7/2)*(1 - a^2*x^2)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.80 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{7/2}} \, dx=\frac {\sqrt {c}\, \left (-15 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+45 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}-30 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-30 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+45 \,\mathrm {log}\left (a x -1\right ) a x -15 \,\mathrm {log}\left (a x -1\right )+15 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-45 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}+30 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+30 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-45 \,\mathrm {log}\left (a x +1\right ) a x +15 \,\mathrm {log}\left (a x +1\right )-10 a^{5} x^{5}+70 a^{3} x^{3}-90 a^{2} x^{2}+54\right )}{192 a \,c^{4} \left (a^{5} x^{5}-3 a^{4} x^{4}+2 a^{3} x^{3}+2 a^{2} x^{2}-3 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(-a^2*c*x^2+c)^(7/2),x)
Output:
(sqrt(c)*( - 15*log(a*x - 1)*a**5*x**5 + 45*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 + 45*log(a*x - 1)*a*x - 1 5*log(a*x - 1) + 15*log(a*x + 1)*a**5*x**5 - 45*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 - 45*log(a*x + 1)*a*x + 15*log(a*x + 1) - 10*a**5*x**5 + 70*a**3*x**3 - 90*a**2*x**2 + 54))/(19 2*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) )