Integrand size = 20, antiderivative size = 233 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x+\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{20} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {3}{20} a^3 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2} x^5+\frac {3 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a} \]
1/5*a^4*c^2*(1-1/a/x)^(3/2)*(1+1/a/x)^(7/2)*x^5+3/8*c^2*arctanh((1-1/a/x)^ (1/2)*(1+1/a/x)^(1/2))/a+1/8*a*c^2*(1+1/a/x)^(3/2)*x^2*(1-1/a/x)^(1/2)+1/2 0*a^2*c^2*(1+1/a/x)^(5/2)*x^3*(1-1/a/x)^(1/2)-3/20*a^3*c^2*(1+1/a/x)^(7/2) *x^4*(1-1/a/x)^(1/2)+3/8*c^2*x*(1-1/a/x)^(1/2)*(1+1/a/x)^(1/2)
Time = 0.14 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.34 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (8-25 a x-16 a^2 x^2+10 a^3 x^3+8 a^4 x^4\right )+15 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{40 a} \]
(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(8 - 25*a*x - 16*a^2*x^2 + 10*a^3*x^3 + 8* a^4*x^4) + 15*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(40*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.
| \(\displaystyle \int \left (c-a^2 c x^2\right )^2 e^{\coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{\coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
3.6.58.3.1 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 = 128, normalized size of antiderivative = 0.55
| method | result | size |
| risch | \(\frac {\left (8 a^{4} x^{4}+10 a^{3} x^{3}-16 a^{2} x^{2}-25 a x +8\right ) \left (a x -1\right ) c^{2}}{40 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {3 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(128\) |
| default | \(\frac {\left (a x -1\right ) c^{2} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x -40 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+45 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(183\) |
1/40*(8*a^4*x^4+10*a^3*x^3-16*a^2*x^2-25*a*x+8)*(a*x-1)/a*c^2/((a*x-1)/(a* x+1))^(1/2)+3/8*ln(a^2*x/(a^2)^(1/2)+(a^2*x^2-1)^(1/2))/(a^2)^(1/2)*c^2/(( 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 = 125, normalized size of antiderivative = 0.54 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (8 \, a^{5} c^{2} x^{5} + 18 \, a^{4} c^{2} x^{4} - 6 \, a^{3} c^{2} x^{3} - 41 \, a^{2} c^{2} x^{2} - 17 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a} \]
1/40*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 15*c^2*log(sqrt((a*x - 1 )/(a*x + 1)) - 1) + (8*a^5*c^2*x^5 + 18*a^4*c^2*x^4 - 6*a^3*c^2*x^3 - 41*a ^2*c^2*x^2 - 17*a*c^2*x + 8*c^2)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{4} x^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \]
c**2*(Integral(-2*a**2*x**2/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integr al(a**4*x**4/sqrt(a*x/(a*x + 1) - 1/(a*x + 1)), x) + Integral(1/sqrt(a*x/( a*x + 1) - 1/(a*x + 1)), x))
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.11 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {1}{40} \, a {\left (\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (15 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 128 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \]
1/40*a*(15*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 15*c^2*log(sqrt((a *x - 1)/(a*x + 1)) - 1)/a^2 - 2*(15*c^2*((a*x - 1)/(a*x + 1))^(9/2) - 70*c ^2*((a*x - 1)/(a*x + 1))^(7/2) + 128*c^2*((a*x - 1)/(a*x + 1))^(5/2) + 70* c^2*((a*x - 1)/(a*x + 1))^(3/2) - 15*c^2*sqrt((a*x - 1)/(a*x + 1)))/(5*(a* x - 1)*a^2/(a*x + 1) - 10*(a*x - 1)^2*a^2/(a*x + 1)^2 + 10*(a*x - 1)^3*a^2 /(a*x + 1)^3 - 5*(a*x - 1)^4*a^2/(a*x + 1)^4 + (a*x - 1)^5*a^2/(a*x + 1)^5 - a^2))
Time = 0.29 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.59 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {1}{40} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, {\left ({\left (\frac {4 \, a^{3} c^{2} x}{\mathrm {sgn}\left (a x + 1\right )} + \frac {5 \, a^{2} c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {8 \, a c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {25 \, c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x + \frac {8 \, c^{2}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {3 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
1/40*sqrt(a^2*x^2 - 1)*((2*((4*a^3*c^2*x/sgn(a*x + 1) + 5*a^2*c^2/sgn(a*x + 1))*x - 8*a*c^2/sgn(a*x + 1))*x - 25*c^2/sgn(a*x + 1))*x + 8*c^2/(a*sgn( a*x + 1))) - 3/8*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a)*sgn(a *x + 1))
Time = 3.89 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.92 \[ \int e^{\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}-\frac {3\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {32\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{5}-\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{2}+\frac {3\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}+\frac {3\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
((7*c^2*((a*x - 1)/(a*x + 1))^(3/2))/2 - (3*c^2*((a*x - 1)/(a*x + 1))^(1/2 ))/4 + (32*c^2*((a*x - 1)/(a*x + 1))^(5/2))/5 - (7*c^2*((a*x - 1)/(a*x + 1 ))^(7/2))/2 + (3*c^2*((a*x - 1)/(a*x + 1))^(9/2))/4)/(a - (5*a*(a*x - 1))/ (a*x + 1) + (10*a*(a*x - 1)^2)/(a*x + 1)^2 - (10*a*(a*x - 1)^3)/(a*x + 1)^ 3 + (5*a*(a*x - 1)^4)/(a*x + 1)^4 - (a*(a*x - 1)^5)/(a*x + 1)^5) + (3*c^2* atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)