Integrand size = 23, antiderivative size = 68 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {a (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{c x}-\frac {a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c} \] Output:
a*(a*x+1)/c/(-a^2*x^2+1)^(1/2)-(-a^2*x^2+1)^(1/2)/c/x-a*arctanh((-a^2*x^2+ 1)^(1/2))/c
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.99 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {-1+a x+2 a^2 x^2-a x \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{c x \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)),x]
Output:
(-1 + a*x + 2*a^2*x^2 - a*x*Sqrt[1 - a^2*x^2]*ArcTanh[Sqrt[1 - a^2*x^2]])/ (c*x*Sqrt[1 - a^2*x^2])
Time = 0.53 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {6698, 528, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {a x+1}{x^2 \left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 528 |
\(\displaystyle \frac {\int \frac {a x+1}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx+\frac {a (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}}{c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2+\frac {a (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {\int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}+\frac {a (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )+\frac {a (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{x}}{c}\) |
Input:
Int[E^ArcTanh[a*x]/(x^2*(c - a^2*c*x^2)),x]
Output:
((a*(1 + a*x))/Sqrt[1 - a^2*x^2] - Sqrt[1 - a^2*x^2]/x - a*ArcTanh[Sqrt[1 - a^2*x^2]])/c
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[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 = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.10
method | result | size |
default | \(-\frac {\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+\frac {\sqrt {-a^{2} x^{2}+1}}{x}+a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{c}\) | \(75\) |
risch | \(\frac {a^{2} x^{2}-1}{x \sqrt {-a^{2} x^{2}+1}\, c}-\frac {a \left (\frac {\sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}+\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{c}\) | \(90\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/c*(1/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+(-a^2*x^2+1)^(1/2)/x+a* arctanh(1/(-a^2*x^2+1)^(1/2)))
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {a^{2} x^{2} - a x + {\left (a^{2} x^{2} - a x\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x - 1\right )}}{a c x^{2} - c x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x, algorithm="fric as")
Output:
(a^2*x^2 - a*x + (a^2*x^2 - a*x)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - sqrt(-a ^2*x^2 + 1)*(2*a*x - 1))/(a*c*x^2 - c*x)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {\int \frac {a}{- a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} + x \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**2/(-a**2*c*x**2+c),x)
Output:
(Integral(a/(-a**2*x**3*sqrt(-a**2*x**2 + 1) + x*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1) ), x))/c
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=-\frac {\frac {a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right )}{c} - \frac {a^{2} \log \left (\sqrt {-a^{2} x^{2} + 1} - 1\right )}{c} - \frac {2 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1} c}}{2 \, a} + \frac {2 \, a^{2} x^{2} - 1}{\sqrt {a x + 1} \sqrt {-a x + 1} c x} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x, algorithm="maxi ma")
Output:
-1/2*(a^2*log(sqrt(-a^2*x^2 + 1) + 1)/c - a^2*log(sqrt(-a^2*x^2 + 1) - 1)/ c - 2*a^2/(sqrt(-a^2*x^2 + 1)*c))/a + (2*a^2*x^2 - 1)/(sqrt(a*x + 1)*sqrt( -a*x + 1)*c*x)
Leaf count of result is larger than twice the leaf count of optimal. 159 vs. \(2 (62) = 124\).
Time = 0.15 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.34 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=-\frac {a^{2} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{c {\left | a \right |}} - \frac {{\left (a^{2} - \frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{x}\right )} a^{2} x}{2 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{2 \, c x {\left | a \right |}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x, algorithm="giac ")
Output:
-a^2*log(1/2*abs(-2*sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c*abs( a)) - 1/2*(a^2 - 5*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/x)*a^2*x/((sqrt(-a^2*x^ 2 + 1)*abs(a) + a)*c*((sqrt(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a)) - 1/2*(sqrt(-a^2*x^2 + 1)*abs(a) + a)/(c*x*abs(a))
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {a^2\,\sqrt {1-a^2\,x^2}}{c\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}}{c\,x}-\frac {a\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{c} \] Input:
int((a*x + 1)/(x^2*(c - a^2*c*x^2)*(1 - a^2*x^2)^(1/2)),x)
Output:
(a^2*(1 - a^2*x^2)^(1/2))/(c*(x*(-a^2)^(1/2) - (-a^2)^(1/2)/a)*(-a^2)^(1/2 )) - (1 - a^2*x^2)^(1/2)/(c*x) - (a*atanh((1 - a^2*x^2)^(1/2)))/c
Time = 0.15 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^2 \left (c-a^2 c x^2\right )} \, dx=\frac {a \left (2 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}-2 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )+\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-6 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+1\right )}{2 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right ) c \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^2/(-a^2*c*x^2+c),x)
Output:
(a*(2*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**2 - 2*log(tan(asin(a*x)/2))* tan(asin(a*x)/2) + tan(asin(a*x)/2)**3 - 6*tan(asin(a*x)/2)**2 + 1))/(2*ta n(asin(a*x)/2)*c*(tan(asin(a*x)/2) - 1))