Integrand size = 20, antiderivative size = 168 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {10 \sqrt {1-a^2 x^2}}{a c^4 (1-a x)}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}-\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {184 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a c^4 (1-a x)^2}+\frac {5 \arcsin (a x)}{a c^4} \]
1/7*(-a^2*x^2+1)^(3/2)/a/c^4/(-a*x+1)^5-26/35*(-a^2*x^2+1)^(3/2)/a/c^4/(-a *x+1)^4+184/105*(-a^2*x^2+1)^(3/2)/a/c^4/(-a*x+1)^3+(-a^2*x^2+1)^(3/2)/a/c ^4/(-a*x+1)^2+5*arcsin(a*x)/a/c^4-10*(-a^2*x^2+1)^(1/2)/a/c^4/(-a*x+1)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.57 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {\sqrt {1+a x} \left (124+29 a x-244 a^2 x^2-44 a^3 x^3+105 a^4 x^4\right )-700 \sqrt {2} (-1+a x)^2 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},-\frac {3}{2},-\frac {1}{2},\frac {1}{2} (1-a x)\right )}{105 a c^4 (1-a x)^{7/2}} \]
-1/105*(Sqrt[1 + a*x]*(124 + 29*a*x - 244*a^2*x^2 - 44*a^3*x^3 + 105*a^4*x ^4) - 700*Sqrt[2]*(-1 + a*x)^2*Hypergeometric2F1[-3/2, -3/2, -1/2, (1 - a* x)/2])/(a*c^4*(1 - a*x)^(7/2))
Time = 0.59 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx\) |
| \(\Big \downarrow \) 6681 |
| \(\displaystyle \frac {a^4 \int \frac {e^{\text {arctanh}(a x)} x^4}{(1-a x)^4}dx}{c^4}\) |
| \(\Big \downarrow \) 6678 |
| \(\displaystyle \frac {a^4 \int \frac {x^4 \sqrt {1-a^2 x^2}}{(1-a x)^5}dx}{c^4}\) |
| \(\Big \downarrow \) 581 |
| \(\displaystyle \frac {a^4 \left (\frac {\int -\frac {\sqrt {1-a^2 x^2} \left (-5 a^3 x^3+9 a^2 x^2-7 a x+2\right )}{(1-a x)^5}dx}{a^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{a^5 (1-a x)^2}\right )}{c^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {a^4 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^5 (1-a x)^2}-\frac {\int \frac {\sqrt {1-a^2 x^2} \left (-5 a^3 x^3+9 a^2 x^2-7 a x+2\right )}{(1-a x)^5}dx}{a^4}\right )}{c^4}\) |
| \(\Big \downarrow \) 2168 |
| \(\displaystyle \frac {a^4 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^5 (1-a x)^2}-\frac {\int \left (\frac {5 \sqrt {1-a^2 x^2}}{(a x-1)^2}+\frac {6 \sqrt {1-a^2 x^2}}{(a x-1)^3}+\frac {4 \sqrt {1-a^2 x^2}}{(a x-1)^4}+\frac {\sqrt {1-a^2 x^2}}{(a x-1)^5}\right )dx}{a^4}\right )}{c^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^4 \left (\frac {\left (1-a^2 x^2\right )^{3/2}}{a^5 (1-a x)^2}-\frac {-\frac {184 \left (1-a^2 x^2\right )^{3/2}}{105 a (1-a x)^3}+\frac {26 \left (1-a^2 x^2\right )^{3/2}}{35 a (1-a x)^4}-\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a (1-a x)^5}+\frac {10 \sqrt {1-a^2 x^2}}{a (1-a x)}-\frac {5 \arcsin (a x)}{a}}{a^4}\right )}{c^4}\) |
(a^4*((1 - a^2*x^2)^(3/2)/(a^5*(1 - a*x)^2) - ((10*Sqrt[1 - a^2*x^2])/(a*( 1 - a*x)) - (1 - a^2*x^2)^(3/2)/(7*a*(1 - a*x)^5) + (26*(1 - a^2*x^2)^(3/2 ))/(35*a*(1 - a*x)^4) - (184*(1 - a^2*x^2)^(3/2))/(105*a*(1 - a*x)^3) - (5 *ArcSin[a*x])/a)/a^4))/c^4
3.5.55.3.1 Defintions of rubi rules used
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 = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.38
| method | result | size |
| risch | \(\frac {a^{2} x^{2}-1}{a \sqrt {-a^{2} x^{2}+1}\, c^{4}}+\frac {\left (\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {57 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{35 a^{8} \left (x -\frac {1}{a}\right )^{3}}+\frac {446 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{105 a^{7} \left (x -\frac {1}{a}\right )^{2}}+\frac {1024 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{105 a^{6} \left (x -\frac {1}{a}\right )}+\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a^{9} \left (x -\frac {1}{a}\right )^{4}}\right ) a^{4}}{c^{4}}\) | \(232\) |
| default | \(\frac {a^{4} \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{a^{5}}+\frac {5 \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2}}}+\frac {\frac {9 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {18 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{7}}+\frac {\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {6 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{8}}+\frac {\frac {16 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {16 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{3 \left (x -\frac {1}{a}\right )}}{a^{6}}+\frac {14 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a^{6} \left (x -\frac {1}{a}\right )}\right )}{c^{4}}\) | \(487\) |
1/a*(a^2*x^2-1)/(-a^2*x^2+1)^(1/2)/c^4+(5/a^4/(a^2)^(1/2)*arctan((a^2)^(1/ 2)*x/(-a^2*x^2+1)^(1/2))+57/35/a^8/(x-1/a)^3*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^ (1/2)+446/105/a^7/(x-1/a)^2*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)+1024/105/a^ 6/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a)*a)^(1/2)+2/7/a^9/(x-1/a)^4*(-(x-1/a)^2 *a^2-2*(x-1/a)*a)^(1/2))*a^4/c^4
Time = 0.27 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.05 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=-\frac {824 \, a^{4} x^{4} - 3296 \, a^{3} x^{3} + 4944 \, a^{2} x^{2} - 3296 \, a x + 1050 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + {\left (105 \, a^{4} x^{4} - 1444 \, a^{3} x^{3} + 3256 \, a^{2} x^{2} - 2771 \, a x + 824\right )} \sqrt {-a^{2} x^{2} + 1} + 824}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
-1/105*(824*a^4*x^4 - 3296*a^3*x^3 + 4944*a^2*x^2 - 3296*a*x + 1050*(a^4*x ^4 - 4*a^3*x^3 + 6*a^2*x^2 - 4*a*x + 1)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a *x)) + (105*a^4*x^4 - 1444*a^3*x^3 + 3256*a^2*x^2 - 2771*a*x + 824)*sqrt(- a^2*x^2 + 1) + 824)/(a^5*c^4*x^4 - 4*a^4*c^4*x^3 + 6*a^3*c^4*x^2 - 4*a^2*c ^4*x + a*c^4)
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {a^{4} \left (\int \frac {x^{4}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a x^{5}}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx\right )}{c^{4}} \]
a**4*(Integral(x**4/(a**4*x**4*sqrt(-a**2*x**2 + 1) - 4*a**3*x**3*sqrt(-a* *2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x) + Integral(a*x**5/(a**4*x**4*sqrt(-a**2*x** 2 + 1) - 4*a**3*x**3*sqrt(-a**2*x**2 + 1) + 6*a**2*x**2*sqrt(-a**2*x**2 + 1) - 4*a*x*sqrt(-a**2*x**2 + 1) + sqrt(-a**2*x**2 + 1)), x))/c**4
\[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\int { \frac {a x + 1}{\sqrt {-a^{2} x^{2} + 1} {\left (c - \frac {c}{a x}\right )}^{4}} \,d x } \]
Exception generated. \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\text {Exception raised: TypeError} \]
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
Time = 0.09 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.32 \[ \int \frac {e^{\text {arctanh}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx=\frac {16\,a\,\sqrt {1-a^2\,x^2}}{3\,\left (a^4\,c^4\,x^2-2\,a^3\,c^4\,x+a^2\,c^4\right )}+\frac {5\,\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )}{c^4\,\sqrt {-a^2}}+\frac {4\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^4\,x^2-2\,a^5\,c^4\,x+a^4\,c^4\right )}-\frac {6\,a^4\,\sqrt {1-a^2\,x^2}}{5\,\left (a^7\,c^4\,x^2-2\,a^6\,c^4\,x+a^5\,c^4\right )}-\frac {\sqrt {1-a^2\,x^2}}{a\,c^4}+\frac {2\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^4\,x^4-4\,a^5\,c^4\,x^3+6\,a^4\,c^4\,x^2-4\,a^3\,c^4\,x+a^2\,c^4\right )}-\frac {1024\,\sqrt {1-a^2\,x^2}}{105\,\sqrt {-a^2}\,\left (c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}\right )}-\frac {57\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^4\,x\,\sqrt {-a^2}-\frac {c^4\,\sqrt {-a^2}}{a}+a^2\,c^4\,x^3\,\sqrt {-a^2}-3\,a\,c^4\,x^2\,\sqrt {-a^2}\right )} \]
(16*a*(1 - a^2*x^2)^(1/2))/(3*(a^2*c^4 - 2*a^3*c^4*x + a^4*c^4*x^2)) + (5* asinh(x*(-a^2)^(1/2)))/(c^4*(-a^2)^(1/2)) + (4*a^3*(1 - a^2*x^2)^(1/2))/(3 5*(a^4*c^4 - 2*a^5*c^4*x + a^6*c^4*x^2)) - (6*a^4*(1 - a^2*x^2)^(1/2))/(5* (a^5*c^4 - 2*a^6*c^4*x + a^7*c^4*x^2)) - (1 - a^2*x^2)^(1/2)/(a*c^4) + (2* a*(1 - a^2*x^2)^(1/2))/(7*(a^2*c^4 - 4*a^3*c^4*x + 6*a^4*c^4*x^2 - 4*a^5*c ^4*x^3 + a^6*c^4*x^4)) - (1024*(1 - a^2*x^2)^(1/2))/(105*(-a^2)^(1/2)*(c^4 *x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a)) - (57*(1 - a^2*x^2)^(1/2))/(35*(- a^2)^(1/2)*(3*c^4*x*(-a^2)^(1/2) - (c^4*(-a^2)^(1/2))/a + a^2*c^4*x^3*(-a^ 2)^(1/2) - 3*a*c^4*x^2*(-a^2)^(1/2)))