Integrand size = 20, antiderivative size = 136 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {8 \sqrt {1-a^2 x^2}}{a c^3 (1-a x)}-\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {14 \left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^3 (1-a x)^2}+\frac {4 \arcsin (a x)}{a c^3} \] Output:
-8*(-a^2*x^2+1)^(1/2)/a/c^3/(-a*x+1)-1/5*(-a^2*x^2+1)^(3/2)/a/c^3/(-a*x+1) ^4+14/15*(-a^2*x^2+1)^(3/2)/a/c^3/(-a*x+1)^3+(-a^2*x^2+1)^(3/2)/a/c^3/(-a* x+1)^2+4*arcsin(a*x)/a/c^3
Time = 0.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.53 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {\frac {\sqrt {1+a x} \left (-94+222 a x-149 a^2 x^2+15 a^3 x^3\right )}{(1-a x)^{5/2}}-120 \arcsin \left (\frac {\sqrt {1-a x}}{\sqrt {2}}\right )}{15 a c^3} \] Input:
Integrate[E^ArcTanh[a*x]/(c - c/(a*x))^3,x]
Output:
((Sqrt[1 + a*x]*(-94 + 222*a*x - 149*a^2*x^2 + 15*a^3*x^3))/(1 - a*x)^(5/2 ) - 120*ArcSin[Sqrt[1 - a*x]/Sqrt[2]])/(15*a*c^3)
Time = 0.90 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6681, 6678, 581, 25, 2168, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle -\frac {a^3 \int \frac {e^{\text {arctanh}(a x)} x^3}{(1-a x)^3}dx}{c^3}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle -\frac {a^3 \int \frac {x^3 \sqrt {1-a^2 x^2}}{(1-a x)^4}dx}{c^3}\) |
| \(\Big \downarrow \) 581 |
| \(\displaystyle -\frac {a^3 \left (-\frac {\int -\frac {\sqrt {1-a^2 x^2} \left (4 a^2 x^2-5 a x+2\right )}{(1-a x)^4}dx}{a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 (1-a x)^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {a^3 \left (\frac {\int \frac {\sqrt {1-a^2 x^2} \left (4 a^2 x^2-5 a x+2\right )}{(1-a x)^4}dx}{a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 (1-a x)^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 2168 |
| \(\displaystyle -\frac {a^3 \left (\frac {\int \left (\frac {4 \sqrt {1-a^2 x^2}}{(a x-1)^2}+\frac {3 \sqrt {1-a^2 x^2}}{(a x-1)^3}+\frac {\sqrt {1-a^2 x^2}}{(a x-1)^4}\right )dx}{a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 (1-a x)^2}\right )}{c^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \left (\frac {-\frac {14 \left (1-a^2 x^2\right )^{3/2}}{15 a (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a (1-a x)^4}+\frac {8 \sqrt {1-a^2 x^2}}{a (1-a x)}-\frac {4 \arcsin (a x)}{a}}{a^3}-\frac {\left (1-a^2 x^2\right )^{3/2}}{a^4 (1-a x)^2}\right )}{c^3}\) |
Input:
Int[E^ArcTanh[a*x]/(c - c/(a*x))^3,x]
Output:
-((a^3*(-((1 - a^2*x^2)^(3/2)/(a^4*(1 - a*x)^2)) + ((8*Sqrt[1 - a^2*x^2])/ (a*(1 - a*x)) + (1 - a^2*x^2)^(3/2)/(5*a*(1 - a*x)^4) - (14*(1 - a^2*x^2)^ (3/2))/(15*a*(1 - a*x)^3) - (4*ArcSin[a*x])/a)/a^3))/c^3)
Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[(c + d*x)^(m + n - 1)*((a + b*x^2)^(p + 1)/(b*d^(m - 1)*(m + n + 2*p + 1))), x] + Simp[1/(d^m*(m + n + 2*p + 1)) Int[(c + d*x)^n*(a + b*x^ 2)^p*ExpandToSum[d^m*(m + n + 2*p + 1)*x^m - (m + n + 2*p + 1)*(c + d*x)^m + c*(c + d*x)^(m - 2)*(c*(m + n - 1) + c*(m + n + 2*p + 1) + 2*d*(m + n + p )*x), x], x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] & & IGtQ[m, 1] && NeQ[m + n + 2*p + 1, 0] && (IntegerQ[2*p] || ILtQ[m + n, 0] )
Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2)^p, (d + e*x)^m*Pq, x], x] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x] && EqQ[b*d^2 + a*e^2, 0] && EqQ[m + Expon[Pq, x] + 2*p + 1, 0] && ILtQ[m, 0]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*((c_) + (d_.)*(x_))^(p_.)*((e_.) + (f_.)* (x_))^(m_.), x_Symbol] :> Simp[c^n Int[(e + f*x)^m*(c + d*x)^(p - n)*(1 - a^2*x^2)^(n/2), x], x] /; FreeQ[{a, c, d, e, f, m, p}, x] && EqQ[a*c + d, 0] && IntegerQ[(n - 1)/2] && (IntegerQ[p] || EqQ[p, n/2] || EqQ[p - n/2 - 1 , 0]) && IntegerQ[2*p]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)/(x_))^(p_.), x_Symbol ] :> Simp[d^p Int[u*(1 + c*(x/d))^p*(E^(n*ArcTanh[a*x])/x^p), x], x] /; F reeQ[{a, c, d, n}, x] && EqQ[c^2 - a^2*d^2, 0] && IntegerQ[p]
Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.40
| method | result | size |
| risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{3}}+\frac {\left (\frac {4 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a^{7} \left (x -\frac {1}{a}\right )^{3}}+\frac {31 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{6} \left (x -\frac {1}{a}\right )^{2}}+\frac {104 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{15 a^{5} \left (x -\frac {1}{a}\right )}\right ) a^{3}}{c^{3}}\) | \(191\) |
| default | \(\frac {a^{3} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{4}}+\frac {4 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{3} \sqrt {a^{2}}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {4 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{6}}+\frac {\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {7 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{5}}+\frac {9 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a^{5} \left (x -\frac {1}{a}\right )}\right )}{c^{3}}\) | \(312\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x,method=_RETURNVERBOSE)
Output:
1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^3+(4/a^3/(a^2)^(1/2)*arctan((a^2)^(1/ 2)*x/(-a^2*x^2+1)^(1/2))+2/5/a^7/(x-1/a)^3*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1 /2)+31/15/a^6/(x-1/a)^2*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+104/15/a^5/(x-1 /a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2))*a^3/c^3
Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=-\frac {94 \, a^{3} x^{3} - 282 \, a^{2} x^{2} + 282 \, a x + 120 \, {\left (a^{3} x^{3} - 3 \, a^{2} x^{2} + 3 \, a x - 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (15 \, a^{3} x^{3} - 149 \, a^{2} x^{2} + 222 \, a x - 94\right )} \sqrt {-a^{2} x^{2} + 1} - 94}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x, algorithm="fricas")
Output:
-1/15*(94*a^3*x^3 - 282*a^2*x^2 + 282*a*x + 120*(a^3*x^3 - 3*a^2*x^2 + 3*a *x - 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + (15*a^3*x^3 - 149*a^2*x^2 + 222*a*x - 94)*sqrt(-a^2*x^2 + 1) - 94)/(a^4*c^3*x^3 - 3*a^3*c^3*x^2 + 3 *a^2*c^3*x - a*c^3)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {a^{3} \left (\int \frac {x^{3}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{4}}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{3}} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/(c-c/a/x)**3,x)
Output:
a**3*(Integral(x**3/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a* *2*x**2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x) + In tegral(a*x**4/(a**3*x**3*sqrt(-a**2*x**2 + 1) - 3*a**2*x**2*sqrt(-a**2*x** 2 + 1) + 3*a*x*sqrt(-a**2*x**2 + 1) - sqrt(-a**2*x**2 + 1)), x))/c**3
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{3}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x, algorithm="maxima")
Output:
integrate((a*x + 1)/(sqrt(-a^2*x^2 + 1)*(c - c/(a*x))^3), x)
Time = 0.13 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.33 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {4 \, \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{c^{3} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{a c^{3}} + \frac {2 \, {\left (\frac {335 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {505 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {285 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {60 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 79\right )}}{15 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x, algorithm="giac")
Output:
4*arcsin(a*x)*sgn(a)/(c^3*abs(a)) - sqrt(-a^2*x^2 + 1)/(a*c^3) + 2/15*(335 *(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 505*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/(a^4*x^2) + 285*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^6*x^3) - 60*(s qrt(-a^2*x^2 + 1)*abs(a) + a)^4/(a^8*x^4) - 79)/(c^3*((sqrt(-a^2*x^2 + 1)* abs(a) + a)/(a^2*x) - 1)^5*abs(a))
Time = 24.18 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.65 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {31\,a\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {4\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^3\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^3}-\frac {104\,\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {2\,\sqrt {1-a^2\,x^2}}{5\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \] Input:
int((a*x + 1)/((c - c/(a*x))^3*(1 - a^2*x^2)^(1/2)),x)
Output:
(31*a*(1 - a^2*x^2)^(1/2))/(15*(a^2*c^3 - 2*a^3*c^3*x + a^4*c^3*x^2)) + (4 *asinh(x*(-a^2)^(1/2)))/(c^3*(-a^2)^(1/2)) - (1 - a^2*x^2)^(1/2)/(a*c^3) - (104*(1 - a^2*x^2)^(1/2))/(15*(-a^2)^(1/2)*(c^3*x*(-a^2)^(1/2) - (c^3*(-a ^2)^(1/2))/a)) - (2*(1 - a^2*x^2)^(1/2))/(5*(-a^2)^(1/2)*(3*c^3*x*(-a^2)^( 1/2) - (c^3*(-a^2)^(1/2))/a + a^2*c^3*x^3*(-a^2)^(1/2) - 3*a*c^3*x^2*(-a^2 )^(1/2)))
Time = 0.15 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.97 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx=\frac {60 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a^{2} x^{2}-120 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right ) a x +60 \sqrt {-a^{2} x^{2}+1}\, \mathit {asin} \left (a x \right )+60 \mathit {asin} \left (a x \right ) a^{3} x^{3}-180 \mathit {asin} \left (a x \right ) a^{2} x^{2}+180 \mathit {asin} \left (a x \right ) a x -60 \mathit {asin} \left (a x \right )-15 \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+79 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-82 \sqrt {-a^{2} x^{2}+1}\, a x +24 \sqrt {-a^{2} x^{2}+1}+15 a^{4} x^{4}-204 a^{3} x^{3}+283 a^{2} x^{2}-82 a x -24}{15 a \,c^{3} \left (\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-2 \sqrt {-a^{2} x^{2}+1}\, a x +\sqrt {-a^{2} x^{2}+1}+a^{3} x^{3}-3 a^{2} x^{2}+3 a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/(c-c/a/x)^3,x)
Output:
(60*sqrt( - a**2*x**2 + 1)*asin(a*x)*a**2*x**2 - 120*sqrt( - a**2*x**2 + 1 )*asin(a*x)*a*x + 60*sqrt( - a**2*x**2 + 1)*asin(a*x) + 60*asin(a*x)*a**3* x**3 - 180*asin(a*x)*a**2*x**2 + 180*asin(a*x)*a*x - 60*asin(a*x) - 15*sqr t( - a**2*x**2 + 1)*a**3*x**3 + 79*sqrt( - a**2*x**2 + 1)*a**2*x**2 - 82*s qrt( - a**2*x**2 + 1)*a*x + 24*sqrt( - a**2*x**2 + 1) + 15*a**4*x**4 - 204 *a**3*x**3 + 283*a**2*x**2 - 82*a*x - 24)/(15*a*c**3*(sqrt( - a**2*x**2 + 1)*a**2*x**2 - 2*sqrt( - a**2*x**2 + 1)*a*x + sqrt( - a**2*x**2 + 1) + a** 3*x**3 - 3*a**2*x**2 + 3*a*x - 1))