Integrand size = 25, antiderivative size = 268 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=-\frac {x \sqrt {1-a^2 x^2}}{a^5 c^2 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1-a x)^2 \sqrt {c-a^2 c x^2}}-\frac {\sqrt {1-a^2 x^2}}{a^6 c^2 (1-a x) \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2}}{8 a^6 c^2 (1+a x) \sqrt {c-a^2 c x^2}}-\frac {23 \sqrt {1-a^2 x^2} \log (1-a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}}+\frac {7 \sqrt {1-a^2 x^2} \log (1+a x)}{16 a^6 c^2 \sqrt {c-a^2 c x^2}} \] Output:
-x*(-a^2*x^2+1)^(1/2)/a^5/c^2/(-a^2*c*x^2+c)^(1/2)+1/8*(-a^2*x^2+1)^(1/2)/ a^6/c^2/(-a*x+1)^2/(-a^2*c*x^2+c)^(1/2)-(-a^2*x^2+1)^(1/2)/a^6/c^2/(-a*x+1 )/(-a^2*c*x^2+c)^(1/2)+1/8*(-a^2*x^2+1)^(1/2)/a^6/c^2/(a*x+1)/(-a^2*c*x^2+ c)^(1/2)-23/16*(-a^2*x^2+1)^(1/2)*ln(-a*x+1)/a^6/c^2/(-a^2*c*x^2+c)^(1/2)+ 7/16*(-a^2*x^2+1)^(1/2)*ln(a*x+1)/a^6/c^2/(-a^2*c*x^2+c)^(1/2)
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.32 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 \left (-8 a x+\frac {1}{(-1+a x)^2}+\frac {8}{-1+a x}+\frac {1}{1+a x}\right )-23 \log (1-a x)+7 \log (1+a x)\right )}{16 a^6 c^2 \sqrt {c-a^2 c x^2}} \] Input:
Integrate[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^(5/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(2*(-8*a*x + (-1 + a*x)^(-2) + 8/(-1 + a*x) + (1 + a*x) ^(-1)) - 23*Log[1 - a*x] + 7*Log[1 + a*x]))/(16*a^6*c^2*Sqrt[c - a^2*c*x^2 ])
Time = 0.52 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.41, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6703, 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.
\(\displaystyle \int \frac {x^5 e^{\text {arctanh}(a x)}}{\left (c-a^2 c x^2\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 6703 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (1-a^2 x^2\right )^{5/2}}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \frac {x^5}{(1-a x)^3 (a x+1)^2}dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \int \left (\frac {7}{16 a^5 (a x+1)}-\frac {1}{8 a^5 (a x+1)^2}-\frac {1}{a^5}-\frac {23}{16 a^5 (a x-1)}-\frac {1}{a^5 (a x-1)^2}-\frac {1}{4 a^5 (a x-1)^3}\right )dx}{c^2 \sqrt {c-a^2 c x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {1-a^2 x^2} \left (-\frac {1}{a^6 (1-a x)}+\frac {1}{8 a^6 (a x+1)}+\frac {1}{8 a^6 (1-a x)^2}-\frac {23 \log (1-a x)}{16 a^6}+\frac {7 \log (a x+1)}{16 a^6}-\frac {x}{a^5}\right )}{c^2 \sqrt {c-a^2 c x^2}}\) |
Input:
Int[(E^ArcTanh[a*x]*x^5)/(c - a^2*c*x^2)^(5/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(-(x/a^5) + 1/(8*a^6*(1 - a*x)^2) - 1/(a^6*(1 - a*x)) + 1/(8*a^6*(1 + a*x)) - (23*Log[1 - a*x])/(16*a^6) + (7*Log[1 + a*x])/(16*a ^6)))/(c^2*Sqrt[c - a^2*c*x^2])
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]))
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])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_), x_ Symbol] :> Simp[c^IntPart[p]*((c + d*x^2)^FracPart[p]/(1 - a^2*x^2)^FracPar t[p]) Int[x^m*(1 - a^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, m, n, p}, x] && EqQ[a^2*c + d, 0] && !(IntegerQ[p] || GtQ[c, 0]) && !I ntegerQ[n/2]
Time = 0.91 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {\left (-16 a^{4} x^{4}+7 \ln \left (a x +1\right ) x^{3} a^{3}-23 a^{3} \ln \left (a x -1\right ) x^{3}+16 a^{3} x^{3}-7 \ln \left (a x +1\right ) x^{2} a^{2}+23 a^{2} \ln \left (a x -1\right ) x^{2}+34 a^{2} x^{2}-7 \ln \left (a x +1\right ) x a +23 a \ln \left (a x -1\right ) x -18 a x +7 \ln \left (a x +1\right )-23 \ln \left (a x -1\right )-12\right ) \sqrt {-c \left (a^{2} x^{2}-1\right )}}{16 \sqrt {-a^{2} x^{2}+1}\, \left (a x +1\right ) \left (a x -1\right )^{2} c^{3} a^{6}}\) | \(171\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),x,method=_RETURNVE RBOSE)
Output:
1/16*(-16*a^4*x^4+7*ln(a*x+1)*x^3*a^3-23*a^3*ln(a*x-1)*x^3+16*a^3*x^3-7*ln (a*x+1)*x^2*a^2+23*a^2*ln(a*x-1)*x^2+34*a^2*x^2-7*ln(a*x+1)*x*a+23*a*ln(a* x-1)*x-18*a*x+7*ln(a*x+1)-23*ln(a*x-1)-12)/(-a^2*x^2+1)^(1/2)/(a*x+1)/(a*x -1)^2*(-c*(a^2*x^2-1))^(1/2)/c^3/a^6
\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (a x + 1\right )} x^{5}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),x, algorithm ="fricas")
Output:
integral(sqrt(-a^2*c*x^2 + c)*sqrt(-a^2*x^2 + 1)*x^5/(a^7*c^3*x^7 - a^6*c^ 3*x^6 - 3*a^5*c^3*x^5 + 3*a^4*c^3*x^4 + 3*a^3*c^3*x^3 - 3*a^2*c^3*x^2 - a* c^3*x + c^3), x)
\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a x + 1\right )}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)*x**5/(-a**2*c*x**2+c)**(5/2),x)
Output:
Integral(x**5*(a*x + 1)/(sqrt(-(a*x - 1)*(a*x + 1))*(-c*(a*x - 1)*(a*x + 1 ))**(5/2)), x)
\[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (a x + 1\right )} x^{5}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \sqrt {-a^{2} x^{2} + 1}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),x, algorithm ="maxima")
Output:
a*integrate(-x^6/((a^4*c^(5/2)*x^4 - 2*a^2*c^(5/2)*x^2 + c^(5/2))*(a*x + 1 )*(a*x - 1)), x) + 1/4/(a^10*c^(5/2)*x^4 - 2*a^8*c^(5/2)*x^2 + a^6*c^(5/2) ) + 1/(a^8*c^(5/2)*x^2 - a^6*c^(5/2)) - 1/2*log(-a^2*c*x^2 + c)/(a^6*c^(5/ 2))
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),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
Timed out. \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^5\,\left (a\,x+1\right )}{{\left (c-a^2\,c\,x^2\right )}^{5/2}\,\sqrt {1-a^2\,x^2}} \,d x \] Input:
int((x^5*(a*x + 1))/((c - a^2*c*x^2)^(5/2)*(1 - a^2*x^2)^(1/2)),x)
Output:
int((x^5*(a*x + 1))/((c - a^2*c*x^2)^(5/2)*(1 - a^2*x^2)^(1/2)), x)
Time = 0.16 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.55 \[ \int \frac {e^{\text {arctanh}(a x)} x^5}{\left (c-a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {c}\, \left (-23 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}+23 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+23 \,\mathrm {log}\left (a x -1\right ) a x -23 \,\mathrm {log}\left (a x -1\right )+7 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}-7 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-7 \,\mathrm {log}\left (a x +1\right ) a x +7 \,\mathrm {log}\left (a x +1\right )-16 a^{4} x^{4}+50 a^{3} x^{3}-52 a x +22\right )}{16 a^{6} c^{3} \left (a^{3} x^{3}-a^{2} x^{2}-a x +1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)*x^5/(-a^2*c*x^2+c)^(5/2),x)
Output:
(sqrt(c)*( - 23*log(a*x - 1)*a**3*x**3 + 23*log(a*x - 1)*a**2*x**2 + 23*lo g(a*x - 1)*a*x - 23*log(a*x - 1) + 7*log(a*x + 1)*a**3*x**3 - 7*log(a*x + 1)*a**2*x**2 - 7*log(a*x + 1)*a*x + 7*log(a*x + 1) - 16*a**4*x**4 + 50*a** 3*x**3 - 52*a*x + 22))/(16*a**6*c**3*(a**3*x**3 - a**2*x**2 - a*x + 1))