Integrand size = 22, antiderivative size = 233 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {7}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {7}{24} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2-\frac {7}{60} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{20} a^3 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2} x^4+\frac {1}{5} a^4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{9/2} x^5-\frac {7 c^2 \text {arctanh}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a} \]
-7/8*c^2*arctanh((1-1/a/x)^(1/2)*(1+1/a/x)^(1/2))/a-7/24*a*c^2*(1+1/a/x)^( 3/2)*x^2*(1-1/a/x)^(1/2)-7/60*a^2*c^2*(1+1/a/x)^(5/2)*x^3*(1-1/a/x)^(1/2)- 1/20*a^3*c^2*(1+1/a/x)^(7/2)*x^4*(1-1/a/x)^(1/2)+1/5*a^4*c^2*(1+1/a/x)^(9/ 2)*x^5*(1-1/a/x)^(1/2)-7/8*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 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-136+15 a x+112 a^2 x^2+90 a^3 x^3+24 a^4 x^4\right )-105 \log \left (\left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{120 a} \]
(c^2*(a*Sqrt[1 - 1/(a^2*x^2)]*x*(-136 + 15*a*x + 112*a^2*x^2 + 90*a^3*x^3 + 24*a^4*x^4) - 105*Log[(1 + Sqrt[1 - 1/(a^2*x^2)])*x]))/(120*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.
| \(\displaystyle \int \left (c-a^2 c x^2\right )^2 e^{3 \coth ^{-1}(a x)} \, dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
| \(\Big \downarrow \) 6745 |
| \(\displaystyle a^4 c^2 \int \frac {e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4}{a^4}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2-\frac {1}{x^2}\right )^2 x^4dx\) |
| \(\Big \downarrow \) 2005 |
| \(\displaystyle c^2 \int e^{3 \coth ^{-1}(a x)} \left (a^2 x^2-1\right )^2dx\) |
3.6.75.3.1 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 | \(\frac {\left (24 a^{4} x^{4}+90 a^{3} x^{3}+112 a^{2} x^{2}+15 a x -136\right ) \left (a x -1\right ) c^{2}}{120 a \sqrt {\frac {a x -1}{a x +1}}}-\frac {7 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{8 \sqrt {a^{2}}\, \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(128\) |
| default | \(\frac {\left (a x -1\right )^{2} c^{2} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{2} x^{2}+90 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a x +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+105 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +120 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-105 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(192\) |
1/120*(24*a^4*x^4+90*a^3*x^3+112*a^2*x^2+15*a*x-136)*(a*x-1)/a*c^2/((a*x-1 )/(a*x+1))^(1/2)-7/8*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)/(a*x+1))^(1/2)*((a*x-1)*(a*x+1))^(1/2)
Time = 0.25 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.54 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (24 \, a^{5} c^{2} x^{5} + 114 \, a^{4} c^{2} x^{4} + 202 \, a^{3} c^{2} x^{3} + 127 \, a^{2} c^{2} x^{2} - 121 \, a c^{2} x - 136 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a} \]
-1/120*(105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1) - 105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) - 1) - (24*a^5*c^2*x^5 + 114*a^4*c^2*x^4 + 202*a^3*c^2*x^ 3 + 127*a^2*c^2*x^2 - 121*a*c^2*x - 136*c^2)*sqrt((a*x - 1)/(a*x + 1)))/a
\[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{4} x^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right ) \]
c**2*(Integral(-2*a**2*x**2/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x) + Integral(a**4*x**4 /(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1 /(a*x + 1))/(a*x + 1)), x) + Integral(1/(a*x*sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1) - sqrt(a*x/(a*x + 1) - 1/(a*x + 1))/(a*x + 1)), x))
Time = 0.19 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.11 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=-\frac {1}{120} \, a {\left (\frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (105 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 490 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 896 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 790 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \]
-1/120*a*(105*c^2*log(sqrt((a*x - 1)/(a*x + 1)) + 1)/a^2 - 105*c^2*log(sqr t((a*x - 1)/(a*x + 1)) - 1)/a^2 - 2*(105*c^2*((a*x - 1)/(a*x + 1))^(9/2) - 490*c^2*((a*x - 1)/(a*x + 1))^(7/2) + 896*c^2*((a*x - 1)/(a*x + 1))^(5/2) - 790*c^2*((a*x - 1)/(a*x + 1))^(3/2) - 105*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 = 138, normalized size of antiderivative = 0.59 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {1}{120} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (2 \, {\left (3 \, {\left (\frac {4 \, a^{3} c^{2} x}{\mathrm {sgn}\left (a x + 1\right )} + \frac {15 \, a^{2} c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x + \frac {56 \, a c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x + \frac {15 \, c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {136 \, c^{2}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} + \frac {7 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{8 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
1/120*sqrt(a^2*x^2 - 1)*((2*(3*(4*a^3*c^2*x/sgn(a*x + 1) + 15*a^2*c^2/sgn( a*x + 1))*x + 56*a*c^2/sgn(a*x + 1))*x + 15*c^2/sgn(a*x + 1))*x - 136*c^2/ (a*sgn(a*x + 1))) + 7/8*c^2*log(abs(-x*abs(a) + sqrt(a^2*x^2 - 1)))/(abs(a )*sgn(a*x + 1))
Time = 3.99 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.92 \[ \int e^{3 \coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx=\frac {\frac {7\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {79\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6}-\frac {224\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{15}+\frac {49\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{6}-\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}-\frac {7\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
((7*c^2*((a*x - 1)/(a*x + 1))^(1/2))/4 + (79*c^2*((a*x - 1)/(a*x + 1))^(3/ 2))/6 - (224*c^2*((a*x - 1)/(a*x + 1))^(5/2))/15 + (49*c^2*((a*x - 1)/(a*x + 1))^(7/2))/6 - (7*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) - (7* c^2*atanh(((a*x - 1)/(a*x + 1))^(1/2)))/(4*a)