Integrand size = 22, antiderivative size = 313 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {5}{16} c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {5}{48} a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {1}{24} a^2 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3+\frac {1}{8} a^3 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4-\frac {1}{6} a^4 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {1}{6} a^5 c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x^6-\frac {1}{7} a^6 c^3 \left (1-\frac {1}{a x}\right )^{7/2} \left (1+\frac {1}{a x}\right )^{7/2} x^7-\frac {5 c^3 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{16 a} \]
-1/6*a^4*c^3*(1-1/a/x)^(3/2)*(1+1/a/x)^(7/2)*x^5+1/6*a^5*c^3*(1-1/a/x)^(5/ 2)*(1+1/a/x)^(7/2)*x^6-1/7*a^6*c^3*(1-1/a/x)^(7/2)*(1+1/a/x)^(7/2)*x^7-5/1 6*c^3*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-5/48*a*c^3*(1+1/a/x)^(3/2 )*x^2*(1-1/a/x)^(1/2)-1/24*a^2*c^3*(1+1/a/x)^(5/2)*x^3*(1-1/a/x)^(1/2)+1/8 *a^3*c^3*(1+1/a/x)^(7/2)*x^4*(1-1/a/x)^(1/2)-5/16*c^3*x*(1-1/a/x)^(1/2)*(1 +1/a/x)^(1/2)
Time = 0.21 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.30 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {c^3 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (48+231 a x-144 a^2 x^2-182 a^3 x^3+144 a^4 x^4+56 a^5 x^5-48 a^6 x^6\right )-105 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{336 a} \]
(c^3*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(48 + 231*a*x - 144*a^2*x^2 - 182*a^3*x^3 + 144*a^4*x^4 + 56*a^5*x^5 - 48*a^6*x^6) - 105*Log[(1 + Sqrt[1 - 1/(a^2*x^ 2)])*x]))/(336*a)
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.
| \(\displaystyle \int \left (c-a^2 c x^2\right )^3 e^{-\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle -a^6 c^3 \int \frac {e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6}{a^6}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^3 x^6dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle -c^3 \int e^{-\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^3dx\) |
3.6.91.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Fx_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p*Fx, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && Neg Q[n]
Int[E^(ArcCoth[(a_.)*(x_)]*(n_.))*(u_.)*((c_) + (d_.)*(x_)^2)^(p_.), x_Symb ol] :> Simp[d^p Int[u*x^(2*p)*(1 - 1/(a^2*x^2))^p*E^(n*ArcCoth[a*x]), x], x] /; FreeQ[{a, c, d, n}, x] && EqQ[a^2*c + d, 0] && !IntegerQ[n/2] && In tegerQ[p]
Time = 0.49 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(-\frac {\left (48 a^{6} x^{6}-56 a^{5} x^{5}-144 a^{4} x^{4}+182 a^{3} x^{3}+144 a^{2} x^{2}-231 a x -48\right ) \left (a x +1\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}}{336 a}-\frac {5 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{3} \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{16 \sqrt {a^{2}}\, \left (a x -1\right )}\) | \(144\) |
| default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (48 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{4} x^{4}-56 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}-96 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+126 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x -64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +112 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{336 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(231\) |
-1/336*(48*a^6*x^6-56*a^5*x^5-144*a^4*x^4+182*a^3*x^3+144*a^2*x^2-231*a*x- 48)*(a*x+1)/a*c^3*((a*x-1)/(a*x+1))^(1/2)-5/16*ln(a^2*x/(a^2)^(1/2)+(a^2*x ^2-1)^(1/2))/(a^2)^(1/2)*c^3*((a*x-1)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/ 2)/(a*x-1)
Time = 0.26 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.47 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (48 \, a^{7} c^{3} x^{7} - 8 \, a^{6} c^{3} x^{6} - 200 \, a^{5} c^{3} x^{5} + 38 \, a^{4} c^{3} x^{4} + 326 \, a^{3} c^{3} x^{3} - 87 \, a^{2} c^{3} x^{2} - 279 \, a c^{3} x - 48 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{336 \, a} \]
-1/336*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) - 1) + (48*a^7*c^3*x^7 - 8*a^6*c^3*x^6 - 200*a^5*c^3*x^5 + 38*a^4*c^3*x^4 + 326*a^3*c^3*x^3 - 87*a^2*c^3*x^2 - 279*a*c^3*x - 48*c^3 )*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=- c^{3} \left (\int 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- 3 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx\right ) \]
-c**3*(Integral(3*a**2*x**2*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integr al(-3*a**4*x**4*sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(a**6*x**6 *sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(-sqrt(a*x/(a*x + 1) - 1/ (a*x + 1)), x))
Time = 0.21 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.08 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {1}{336} \, {\left (\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (105 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {13}{2}} - 700 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 1981 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 3072 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 1981 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 700 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {7 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {21 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {35 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {35 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {21 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {7 \, {\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + \frac {{\left (a x - 1\right )}^{7} a^{2}}{{\left (a x + 1\right )}^{7}} - a^{2}}\right )} a \]
-1/336*(105*c^3*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^3*log(sqrt( (a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(105*c^3*((a*x - 1)/(a*x + 1))^(13/2) - 700*c^3*((a*x - 1)/(a*x + 1))^(11/2) + 1981*c^3*((a*x - 1)/(a*x + 1))^(9/2 ) + 3072*c^3*((a*x - 1)/(a*x + 1))^(7/2) - 1981*c^3*((a*x - 1)/(a*x + 1))^ (5/2) + 700*c^3*((a*x - 1)/(a*x + 1))^(3/2) - 105*c^3*sqrt((a*x - 1)/(a*x + 1)))/(7*(a*x - 1)*a^2/(a*x + 1) - 21*(a*x - 1)^2*a^2/(a*x + 1)^2 + 35*(a *x - 1)^3*a^2/(a*x + 1)^3 - 35*(a*x - 1)^4*a^2/(a*x + 1)^4 + 21*(a*x - 1)^ 5*a^2/(a*x + 1)^5 - 7*(a*x - 1)^6*a^2/(a*x + 1)^6 + (a*x - 1)^7*a^2/(a*x + 1)^7 - a^2))*a
Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.51 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=\frac {5 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{16 \, {\left | a \right |}} + \frac {1}{336} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {48 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} + {\left (231 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (72 \, a c^{3} \mathrm {sgn}\left (a x + 1\right ) + {\left (91 \, a^{2} c^{3} \mathrm {sgn}\left (a x + 1\right ) - 4 \, {\left (18 \, a^{3} c^{3} \mathrm {sgn}\left (a x + 1\right ) - {\left (6 \, a^{5} c^{3} x \mathrm {sgn}\left (a x + 1\right ) - 7 \, a^{4} c^{3} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x\right )} x\right )} x\right )} \]
5/16*c^3*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))*sgn(a*x + 1)/abs(a) + 1/3 36*sqrt(a^2*x^2 - 1)*(48*c^3*sgn(a*x + 1)/a + (231*c^3*sgn(a*x + 1) - 2*(7 2*a*c^3*sgn(a*x + 1) + (91*a^2*c^3*sgn(a*x + 1) - 4*(18*a^3*c^3*sgn(a*x + 1) - (6*a^5*c^3*x*sgn(a*x + 1) - 7*a^4*c^3*sgn(a*x + 1))*x)*x)*x)*x)*x)
Time = 0.12 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.92 \[ \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx=-\frac {\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}-\frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8}-\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{24}+\frac {128\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{7}+\frac {283\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{24}-\frac {25\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{6}+\frac {5\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{13/2}}{8}}{a-\frac {7\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {21\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {35\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {35\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {21\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}+\frac {7\,a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}-\frac {a\,{\left (a\,x-1\right )}^7}{{\left (a\,x+1\right )}^7}}-\frac {5\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{8\,a} \]
- ((25*c^3*((a*x - 1)/(a*x + 1))^(3/2))/6 - (5*c^3*((a*x - 1)/(a*x + 1))^( 1/2))/8 - (283*c^3*((a*x - 1)/(a*x + 1))^(5/2))/24 + (128*c^3*((a*x - 1)/( a*x + 1))^(7/2))/7 + (283*c^3*((a*x - 1)/(a*x + 1))^(9/2))/24 - (25*c^3*(( a*x - 1)/(a*x + 1))^(11/2))/6 + (5*c^3*((a*x - 1)/(a*x + 1))^(13/2))/8)/(a - (7*a*(a*x - 1))/(a*x + 1) + (21*a*(a*x - 1)^2)/(a*x + 1)^2 - (35*a*(a*x - 1)^3)/(a*x + 1)^3 + (35*a*(a*x - 1)^4)/(a*x + 1)^4 - (21*a*(a*x - 1)^5) /(a*x + 1)^5 + (7*a*(a*x - 1)^6)/(a*x + 1)^6 - (a*(a*x - 1)^7)/(a*x + 1)^7 ) - (5*c^3*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(8*a)