Integrand size = 24, antiderivative size = 363 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=-\frac {\left (1-a^2 x^2\right )^{7/2}}{a^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^6}+\frac {\left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{7/2}}{2 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^3}+\frac {59 \left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)^2}-\frac {75 \left (1-a^2 x^2\right )^{7/2}}{16 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{7/2}}{32 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7 (1+a x)}-\frac {201 \left (1-a^2 x^2\right )^{7/2} \log (1-a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7}+\frac {9 \left (1-a^2 x^2\right )^{7/2} \log (1+a x)}{64 a^8 \left (c-\frac {c}{a^2 x^2}\right )^{7/2} x^7} \] Output:
-(-a^2*x^2+1)^(7/2)/a^7/(c-c/a^2/x^2)^(7/2)/x^6+1/16*(-a^2*x^2+1)^(7/2)/a^ 8/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+1)^4-1/2*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x ^2)^(7/2)/x^7/(-a*x+1)^3+59/32*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/ x^7/(-a*x+1)^2-75/16*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(-a*x+ 1)+1/32*(-a^2*x^2+1)^(7/2)/a^8/(c-c/a^2/x^2)^(7/2)/x^7/(a*x+1)-201/64*(-a^ 2*x^2+1)^(7/2)*ln(-a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7+9/64*(-a^2*x^2+1)^(7 /2)*ln(a*x+1)/a^8/(c-c/a^2/x^2)^(7/2)/x^7
Time = 0.07 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.40 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {1-a^2 x^2} \left (2 \left (104-207 a x-59 a^2 x^2+309 a^3 x^3-87 a^4 x^4-96 a^5 x^5+32 a^6 x^6\right )+201 (-1+a x)^4 (1+a x) \log (1-a x)-9 (-1+a x)^4 (1+a x) \log (1+a x)\right )}{64 a^2 c^3 \sqrt {c-\frac {c}{a^2 x^2}} x (-1+a x)^4 (1+a x)} \] Input:
Integrate[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2),x]
Output:
(Sqrt[1 - a^2*x^2]*(2*(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) + 201*(-1 + a*x)^4*(1 + a*x)*Log[1 - a*x] - 9*(-1 + a*x)^4*(1 + a*x)*Log[1 + a*x]))/(64*a^2*c^3*Sqrt[c - c/(a^2*x^2)]* x*(-1 + a*x)^4*(1 + a*x))
Time = 0.54 (sec) , antiderivative size = 141, 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 = {6710, 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.
\(\displaystyle \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 6710 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {e^{3 \text {arctanh}(a x)} x^7}{\left (1-a^2 x^2\right )^{7/2}}dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 6700 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \int \frac {x^7}{(1-a x)^5 (a x+1)^2}dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \int \left (\frac {9}{64 a^7 (a x+1)}-\frac {1}{32 a^7 (a x+1)^2}-\frac {1}{a^7}-\frac {201}{64 a^7 (a x-1)}-\frac {75}{16 a^7 (a x-1)^2}-\frac {59}{16 a^7 (a x-1)^3}-\frac {3}{2 a^7 (a x-1)^4}-\frac {1}{4 a^7 (a x-1)^5}\right )dx}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (1-a^2 x^2\right )^{7/2} \left (-\frac {75}{16 a^8 (1-a x)}+\frac {1}{32 a^8 (a x+1)}+\frac {59}{32 a^8 (1-a x)^2}-\frac {1}{2 a^8 (1-a x)^3}+\frac {1}{16 a^8 (1-a x)^4}-\frac {201 \log (1-a x)}{64 a^8}+\frac {9 \log (a x+1)}{64 a^8}-\frac {x}{a^7}\right )}{x^7 \left (c-\frac {c}{a^2 x^2}\right )^{7/2}}\) |
Input:
Int[E^(3*ArcTanh[a*x])/(c - c/(a^2*x^2))^(7/2),x]
Output:
((1 - a^2*x^2)^(7/2)*(-(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 - c/( a^2*x^2))^(7/2)*x^7)
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_.))*(u_.)*((c_) + (d_.)/(x_)^2)^(p_), x_Symbo l] :> Simp[x^(2*p)*((c + d/x^2)^p/(1 - a^2*x^2)^p) Int[(u/x^(2*p))*(1 - a ^2*x^2)^p*E^(n*ArcTanh[a*x]), x], x] /; FreeQ[{a, c, d, n, p}, x] && EqQ[c + a^2*d, 0] && !IntegerQ[p] && !IntegerQ[n/2]
Time = 0.22 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\left (-64 x^{6} a^{6}+9 \ln \left (a x +1\right ) x^{5} a^{5}-201 \ln \left (a x -1\right ) x^{5} a^{5}+192 a^{5} x^{5}-27 \ln \left (a x +1\right ) x^{4} a^{4}+603 \ln \left (a x -1\right ) x^{4} a^{4}+174 a^{4} x^{4}+18 \ln \left (a x +1\right ) x^{3} a^{3}-402 a^{3} \ln \left (a x -1\right ) x^{3}-618 a^{3} x^{3}+18 \ln \left (a x +1\right ) x^{2} a^{2}-402 a^{2} \ln \left (a x -1\right ) x^{2}+118 a^{2} x^{2}-27 \ln \left (a x +1\right ) x a +603 a \ln \left (a x -1\right ) x +414 a x +9 \ln \left (a x +1\right )-201 \ln \left (a x -1\right )-208\right ) \left (a x +1\right )^{3}}{64 \sqrt {-a^{2} x^{2}+1}\, a^{8} x^{7} {\left (\frac {c \left (a^{2} x^{2}-1\right )}{a^{2} x^{2}}\right )}^{\frac {7}{2}}}\) | \(241\) |
Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x,method=_RETURNVERBO SE)
Output:
1/64*(-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+603*a*ln(a*x-1)*x+414*a*x+9*ln(a*x+1)-201* ln(a*x-1)-208)*(a*x+1)^3/(-a^2*x^2+1)^(1/2)/a^8/x^7/(c*(a^2*x^2-1)/a^2/x^2 )^(7/2)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="f ricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*a^8*x^8*sqrt((a^2*c*x^2 - c)/(a^2*x^2))/(a^9*c ^4*x^9 - 3*a^8*c^4*x^8 + 8*a^6*c^4*x^6 - 6*a^5*c^4*x^5 - 6*a^4*c^4*x^4 + 8 *a^3*c^4*x^3 - 3*a*c^4*x + c^4), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {\left (a x + 1\right )^{3}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \left (- c \left (-1 + \frac {1}{a x}\right ) \left (1 + \frac {1}{a x}\right )\right )^{\frac {7}{2}}}\, dx \] Input:
integrate((a*x+1)**3/(-a**2*x**2+1)**(3/2)/(c-c/a**2/x**2)**(7/2),x)
Output:
Integral((a*x + 1)**3/((-(a*x - 1)*(a*x + 1))**(3/2)*(-c*(-1 + 1/(a*x))*(1 + 1/(a*x)))**(7/2)), x)
\[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int { \frac {{\left (a x + 1\right )}^{3}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c - \frac {c}{a^{2} x^{2}}\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="m axima")
Output:
integrate((a*x + 1)^3/((-a^2*x^2 + 1)^(3/2)*(c - c/(a^2*x^2))^(7/2)), x)
Exception generated. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x, algorithm="g iac")
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. \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\int \frac {{\left (a\,x+1\right )}^3}{{\left (c-\frac {c}{a^2\,x^2}\right )}^{7/2}\,{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] Input:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(7/2)*(1 - a^2*x^2)^(3/2)),x)
Output:
int((a*x + 1)^3/((c - c/(a^2*x^2))^(7/2)*(1 - a^2*x^2)^(3/2)), x)
Time = 0.15 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.65 \[ \int \frac {e^{3 \text {arctanh}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^{7/2}} \, dx=\frac {\sqrt {c}\, i \left (-201 \,\mathrm {log}\left (a x -1\right ) a^{5} x^{5}+603 \,\mathrm {log}\left (a x -1\right ) a^{4} x^{4}-402 \,\mathrm {log}\left (a x -1\right ) a^{3} x^{3}-402 \,\mathrm {log}\left (a x -1\right ) a^{2} x^{2}+603 \,\mathrm {log}\left (a x -1\right ) a x -201 \,\mathrm {log}\left (a x -1\right )+9 \,\mathrm {log}\left (a x +1\right ) a^{5} x^{5}-27 \,\mathrm {log}\left (a x +1\right ) a^{4} x^{4}+18 \,\mathrm {log}\left (a x +1\right ) a^{3} x^{3}+18 \,\mathrm {log}\left (a x +1\right ) a^{2} x^{2}-27 \,\mathrm {log}\left (a x +1\right ) a x +9 \,\mathrm {log}\left (a x +1\right )-64 a^{6} x^{6}+250 a^{5} x^{5}-502 a^{3} x^{3}+234 a^{2} x^{2}+240 a x -150\right )}{64 a \,c^{4} \left (a^{5} x^{5}-3 a^{4} x^{4}+2 a^{3} x^{3}+2 a^{2} x^{2}-3 a x +1\right )} \] Input:
int((a*x+1)^3/(-a^2*x^2+1)^(3/2)/(c-c/a^2/x^2)^(7/2),x)
Output:
(sqrt(c)*i*( - 201*log(a*x - 1)*a**5*x**5 + 603*log(a*x - 1)*a**4*x**4 - 4 02*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) + 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))