Integrand size = 23, antiderivative size = 128 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=\frac {1+a x}{c x^3 \sqrt {1-a^2 x^2}}-\frac {4 \sqrt {1-a^2 x^2}}{3 c x^3}-\frac {3 a \sqrt {1-a^2 x^2}}{2 c x^2}-\frac {8 a^2 \sqrt {1-a^2 x^2}}{3 c x}-\frac {3 a^3 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \]
-3/2*a^3*arctanh((-a^2*x^2+1)^(1/2))/c+(a*x+1)/c/x^3/(-a^2*x^2+1)^(1/2)-4/ 3*(-a^2*x^2+1)^(1/2)/c/x^3-3/2*a*(-a^2*x^2+1)^(1/2)/c/x^2-8/3*a^2*(-a^2*x^ 2+1)^(1/2)/c/x
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.71 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=-\frac {2+3 a x+8 a^2 x^2-9 a^3 x^3-16 a^4 x^4+9 a^3 x^3 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{6 c x^3 \sqrt {1-a^2 x^2}} \]
-1/6*(2 + 3*a*x + 8*a^2*x^2 - 9*a^3*x^3 - 16*a^4*x^4 + 9*a^3*x^3*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/(c*x^3*Sqrt[1 - a^2*x^2])
Time = 0.53 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {6698, 528, 2338, 25, 2338, 25, 27, 534, 243, 73, 221}
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.
| \(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx\) |
| \(\Big \downarrow \) 6698 |
| \(\displaystyle \frac {\int \frac {a x+1}{x^4 \left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
| \(\Big \downarrow \) 528 |
| \(\displaystyle \frac {\int \frac {a^3 x^3+a^2 x^2+a x+1}{x^4 \sqrt {1-a^2 x^2}}dx+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {-\frac {1}{3} \int -\frac {3 x^2 a^3+5 x a^2+3 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \int \frac {3 x^2 a^3+5 x a^2+3 a}{x^3 \sqrt {1-a^2 x^2}}dx-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 2338 |
| \(\displaystyle \frac {\frac {1}{3} \left (-\frac {1}{2} \int -\frac {a^2 (9 a x+10)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {a^2 (9 a x+10)}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} a^2 \int \frac {9 a x+10}{x^2 \sqrt {1-a^2 x^2}}dx-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 534 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} a^2 \left (9 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} a^2 \left (\frac {9}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} a^2 \left (-\frac {9 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} a^2 \left (-9 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {10 \sqrt {1-a^2 x^2}}{x}\right )-\frac {3 a \sqrt {1-a^2 x^2}}{2 x^2}\right )-\frac {\sqrt {1-a^2 x^2}}{3 x^3}+\frac {a^3 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
((a^3*(1 + a*x))/Sqrt[1 - a^2*x^2] - Sqrt[1 - a^2*x^2]/(3*x^3) + ((-3*a*Sq rt[1 - a^2*x^2])/(2*x^2) + (a^2*((-10*Sqrt[1 - a^2*x^2])/x - 9*a*ArcTanh[S qrt[1 - a^2*x^2]]))/2)/3)/c
3.9.95.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((x_)^(m_)*((c_) + (d_.)*(x_))^(n_.))/((a_) + (b_.)*(x_)^2)^(3/2), x_Sy mbol] :> Simp[(-2^(n - 1))*c^(m + n - 2)*((c + d*x)/(b*d^(m - 1)*Sqrt[a + b *x^2])), x] + Simp[c^2/a Int[(x^m/Sqrt[a + b*x^2])*ExpandToSum[((c + d*x) ^(n - 1) - (2^(n - 1)*c^(m + n - 1))/(d^m*x^m))/(c - d*x), x], x], x] /; Fr eeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && EqQ[b*c^2 + a*d^2, 0]
Int[(x_)^(m_)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-c)*x^(m + 1)*((a + b*x^2)^(p + 1)/(2*a*(p + 1))), x] + Simp[d Int[ x^(m + 1)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[m, 0] && GtQ[p, -1] && EqQ[m + 2*p + 3, 0]
Int[(Pq_)*((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{ Q = PolynomialQuotient[Pq, c*x, x], R = PolynomialRemainder[Pq, c*x, x]}, S imp[R*(c*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] + Simp[1/(a*c*( m + 1)) Int[(c*x)^(m + 1)*(a + b*x^2)^p*ExpandToSum[a*c*(m + 1)*Q - b*R*( m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x] && Lt Q[m, -1] && (IntegerQ[2*p] || NeQ[Expon[Pq, x], 1])
Int[E^(ArcTanh[(a_.)*(x_)]*(n_.))*(x_)^(m_.)*((c_) + (d_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[c^p Int[x^m*(1 - a^2*x^2)^(p - n/2)*(1 + a*x)^n, x], x] /; FreeQ[{a, c, d, m, p}, x] && EqQ[a^2*c + d, 0] && (IntegerQ[p] || GtQ[c, 0]) && IGtQ[(n + 1)/2, 0] && !IntegerQ[p - n/2]
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.91
| method | result | size |
| risch | \(\frac {10 a^{4} x^{4}+3 a^{3} x^{3}-8 a^{2} x^{2}-3 a x -2}{6 x^{3} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {a^{3} \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{a \left (x -\frac {1}{a}\right )}\right )}{2 c}\) | \(117\) |
| default | \(-\frac {\frac {\sqrt {-a^{2} x^{2}+1}}{3 x^{3}}+\frac {5 a^{2} \sqrt {-a^{2} x^{2}+1}}{3 x}+a^{3} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-a \left (-\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}-\frac {a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}\right )+\frac {a^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 \left (x -\frac {1}{a}\right ) a}}{x -\frac {1}{a}}}{c}\) | \(140\) |
1/6*(10*a^4*x^4+3*a^3*x^3-8*a^2*x^2-3*a*x-2)/x^3/(-a^2*x^2+1)^(1/2)/c+1/2* a^3*(-3*arctanh(1/(-a^2*x^2+1)^(1/2))-2/a/(x-1/a)*(-(x-1/a)^2*a^2-2*(x-1/a )*a)^(1/2))/c
Time = 0.25 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=\frac {6 \, a^{4} x^{4} - 6 \, a^{3} x^{3} + 9 \, {\left (a^{4} x^{4} - a^{3} x^{3}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (16 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - a x - 2\right )} \sqrt {-a^{2} x^{2} + 1}}{6 \, {\left (a c x^{4} - c x^{3}\right )}} \]
1/6*(6*a^4*x^4 - 6*a^3*x^3 + 9*(a^4*x^4 - a^3*x^3)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (16*a^3*x^3 - 7*a^2*x^2 - a*x - 2)*sqrt(-a^2*x^2 + 1))/(a*c*x^4 - c*x^3)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=\frac {\int \frac {a}{- a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{6} \sqrt {- a^{2} x^{2} + 1} + x^{4} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \]
(Integral(a/(-a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**6*sqrt(-a**2*x**2 + 1) + x**4*sqrt(-a**2*x**2 + 1)), x))/c
Time = 0.21 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=-\frac {\frac {3 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c} - \frac {3 \, a^{4} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c} + \frac {2 \, {\left (3 \, {\left (a^{2} x^{2} - 1\right )} a^{4} + 2 \, a^{4}\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c - \sqrt {-a^{2} x^{2} + 1} c}}{4 \, a} + \frac {8 \, a^{4} x^{4} - 4 \, a^{2} x^{2} - 1}{3 \, \sqrt {a x + 1} \sqrt {-a x + 1} c x^{3}} \]
-1/4*(3*a^4*log(sqrt(-a^2*x^2 + 1) + 1)/c - 3*a^4*log(sqrt(-a^2*x^2 + 1) - 1)/c + 2*(3*(a^2*x^2 - 1)*a^4 + 2*a^4)/((-a^2*x^2 + 1)^(3/2)*c - sqrt(-a^ 2*x^2 + 1)*c))/a + 1/3*(8*a^4*x^4 - 4*a^2*x^2 - 1)/(sqrt(a*x + 1)*sqrt(-a* x + 1)*c*x^3)
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (110) = 220\).
Time = 0.28 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.21 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=-\frac {{\left (a^{4} + \frac {2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{2}}{x} + \frac {18 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{x^{2}} - \frac {69 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{2} x^{3}}\right )} a^{6} x^{3}}{24 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {3 \, a^{4} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c {\left | a \right |}} - \frac {\frac {21 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a^{4} c^{2}}{x} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} a^{2} c^{2}}{x^{2}} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3} c^{2}}{x^{3}}}{24 \, a^{2} c^{3} {\left | a \right |}} \]
-1/24*(a^4 + 2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a^2/x + 18*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2/x^2 - 69*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^3/(a^2*x^3))*a^ 6*x^3/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^3*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a )/(a^2*x) - 1)*abs(a)) - 3/2*a^4*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c*abs(a)) - 1/24*(21*(sqrt(-a^2*x^2 + 1)*abs(a) + a) *a^4*c^2/x + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*a^2*c^2/x^2 + (sqrt(-a^2* x^2 + 1)*abs(a) + a)^3*c^2/x^3)/(a^2*c^3*abs(a))
Time = 3.82 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^4 \left (c-a^2 c x^2\right )} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{3\,c\,x^3}-\frac {a\,\sqrt {1-a^2\,x^2}}{2\,c\,x^2}-\frac {5\,a^2\,\sqrt {1-a^2\,x^2}}{3\,c\,x}-\frac {a^4\,\sqrt {1-a^2\,x^2}}{\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {a^3\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,c} \]