Integrand size = 23, antiderivative size = 99 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=\frac {a^2 (1+a x)}{c \sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 c x^2}-\frac {a \sqrt {1-a^2 x^2}}{c x}-\frac {3 a^2 \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c} \] Output:
a^2*(a*x+1)/c/(-a^2*x^2+1)^(1/2)-1/2*(-a^2*x^2+1)^(1/2)/c/x^2-a*(-a^2*x^2+ 1)^(1/2)/c/x-3/2*a^2*arctanh((-a^2*x^2+1)^(1/2))/c
Time = 0.05 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.84 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=-\frac {1+2 a x-3 a^2 x^2-4 a^3 x^3+3 a^2 x^2 \sqrt {1-a^2 x^2} \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )}{2 c x^2 \sqrt {1-a^2 x^2}} \] Input:
Integrate[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)),x]
Output:
-1/2*(1 + 2*a*x - 3*a^2*x^2 - 4*a^3*x^3 + 3*a^2*x^2*Sqrt[1 - a^2*x^2]*ArcT anh[Sqrt[1 - a^2*x^2]])/(c*x^2*Sqrt[1 - a^2*x^2])
Time = 0.69 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6698, 528, 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.
\(\displaystyle \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx\) |
\(\Big \downarrow \) 6698 |
\(\displaystyle \frac {\int \frac {a x+1}{x^3 \left (1-a^2 x^2\right )^{3/2}}dx}{c}\) |
\(\Big \downarrow \) 528 |
\(\displaystyle \frac {\int \frac {a^2 x^2+a x+1}{x^3 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}}{c}\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle \frac {-\frac {1}{2} \int -\frac {a (3 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {1}{2} \int \frac {a (3 a x+2)}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {1}{2} a \int \frac {3 a x+2}{x^2 \sqrt {1-a^2 x^2}}dx+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 534 |
\(\displaystyle \frac {\frac {1}{2} a \left (3 a \int \frac {1}{x \sqrt {1-a^2 x^2}}dx-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {\frac {1}{2} a \left (\frac {3}{2} a \int \frac {1}{x^2 \sqrt {1-a^2 x^2}}dx^2-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{2} a \left (-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2}}d\sqrt {1-a^2 x^2}}{a}-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{2} a \left (-3 a \text {arctanh}\left (\sqrt {1-a^2 x^2}\right )-\frac {2 \sqrt {1-a^2 x^2}}{x}\right )+\frac {a^2 (a x+1)}{\sqrt {1-a^2 x^2}}-\frac {\sqrt {1-a^2 x^2}}{2 x^2}}{c}\) |
Input:
Int[E^ArcTanh[a*x]/(x^3*(c - a^2*c*x^2)),x]
Output:
((a^2*(1 + a*x))/Sqrt[1 - a^2*x^2] - Sqrt[1 - a^2*x^2]/(2*x^2) + (a*((-2*S qrt[1 - a^2*x^2])/x - 3*a*ArcTanh[Sqrt[1 - a^2*x^2]]))/2)/c
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 = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {\frac {a \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{x -\frac {1}{a}}+\frac {\sqrt {-a^{2} x^{2}+1}}{2 x^{2}}+\frac {3 a^{2} \operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )}{2}+\frac {a \sqrt {-a^{2} x^{2}+1}}{x}}{c}\) | \(97\) |
risch | \(\frac {2 a^{3} x^{3}+a^{2} x^{2}-2 a x -1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}\, c}+\frac {a^{2} \left (-3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )-\frac {2 \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2}-2 a \left (x -\frac {1}{a}\right )}}{a \left (x -\frac {1}{a}\right )}\right )}{2 c}\) | \(108\) |
Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x,method=_RETURNVERBOSE)
Output:
-1/c*(a/(x-1/a)*(-(x-1/a)^2*a^2-2*a*(x-1/a))^(1/2)+1/2*(-a^2*x^2+1)^(1/2)/ x^2+3/2*a^2*arctanh(1/(-a^2*x^2+1)^(1/2))+a*(-a^2*x^2+1)^(1/2)/x)
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=\frac {2 \, a^{3} x^{3} - 2 \, a^{2} x^{2} + 3 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) - {\left (4 \, a^{2} x^{2} - a x - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a c x^{3} - c x^{2}\right )}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x, algorithm="fric as")
Output:
1/2*(2*a^3*x^3 - 2*a^2*x^2 + 3*(a^3*x^3 - a^2*x^2)*log((sqrt(-a^2*x^2 + 1) - 1)/x) - (4*a^2*x^2 - a*x - 1)*sqrt(-a^2*x^2 + 1))/(a*c*x^3 - c*x^2)
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=\frac {\int \frac {a}{- a^{2} x^{4} \sqrt {- a^{2} x^{2} + 1} + x^{2} \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{2} x^{5} \sqrt {- a^{2} x^{2} + 1} + x^{3} \sqrt {- a^{2} x^{2} + 1}}\, dx}{c} \] Input:
integrate((a*x+1)/(-a**2*x**2+1)**(1/2)/x**3/(-a**2*c*x**2+c),x)
Output:
(Integral(a/(-a**2*x**4*sqrt(-a**2*x**2 + 1) + x**2*sqrt(-a**2*x**2 + 1)), x) + Integral(1/(-a**2*x**5*sqrt(-a**2*x**2 + 1) + x**3*sqrt(-a**2*x**2 + 1)), x))/c
\[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=\int { -\frac {a x + 1}{{\left (a^{2} c x^{2} - c\right )} \sqrt {-a^{2} x^{2} + 1} x^{3}} \,d x } \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x, algorithm="maxi ma")
Output:
-integrate((a*x + 1)/((a^2*c*x^2 - c)*sqrt(-a^2*x^2 + 1)*x^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (87) = 174\).
Time = 0.13 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.26 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=-\frac {{\left (a^{3} + \frac {3 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a}{x} - \frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a x^{2}}\right )} a^{4} x^{2}}{8 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )} {\left | a \right |}} - \frac {3 \, a^{3} \log \left (\frac {{\left | -2 \, \sqrt {-a^{2} x^{2} + 1} {\left | a \right |} - 2 \, a \right |}}{2 \, a^{2} {\left | x \right |}}\right )}{2 \, c {\left | a \right |}} - \frac {\frac {4 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} a c {\left | a \right |}}{x} + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2} c {\left | a \right |}}{a x^{2}}}{8 \, a^{2} c^{2}} \] Input:
integrate((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x, algorithm="giac ")
Output:
-1/8*(a^3 + 3*(sqrt(-a^2*x^2 + 1)*abs(a) + a)*a/x - 20*(sqrt(-a^2*x^2 + 1) *abs(a) + a)^2/(a*x^2))*a^4*x^2/((sqrt(-a^2*x^2 + 1)*abs(a) + a)^2*c*((sqr t(-a^2*x^2 + 1)*abs(a) + a)/(a^2*x) - 1)*abs(a)) - 3/2*a^3*log(1/2*abs(-2* sqrt(-a^2*x^2 + 1)*abs(a) - 2*a)/(a^2*abs(x)))/(c*abs(a)) - 1/8*(4*(sqrt(- a^2*x^2 + 1)*abs(a) + a)*a*c*abs(a)/x + (sqrt(-a^2*x^2 + 1)*abs(a) + a)^2* c*abs(a)/(a*x^2))/(a^2*c^2)
Time = 0.07 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.18 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=-\frac {\sqrt {1-a^2\,x^2}}{2\,c\,x^2}-\frac {a\,\sqrt {1-a^2\,x^2}}{c\,x}-\frac {a^3\,\sqrt {1-a^2\,x^2}}{\left (\frac {c\,\sqrt {-a^2}}{a}-c\,x\,\sqrt {-a^2}\right )\,\sqrt {-a^2}}+\frac {a^2\,\mathrm {atan}\left (\sqrt {1-a^2\,x^2}\,1{}\mathrm {i}\right )\,3{}\mathrm {i}}{2\,c} \] Input:
int((a*x + 1)/(x^3*(c - a^2*c*x^2)*(1 - a^2*x^2)^(1/2)),x)
Output:
(a^2*atan((1 - a^2*x^2)^(1/2)*1i)*3i)/(2*c) - (1 - a^2*x^2)^(1/2)/(2*c*x^2 ) - (a*(1 - a^2*x^2)^(1/2))/(c*x) - (a^3*(1 - a^2*x^2)^(1/2))/(((c*(-a^2)^ (1/2))/a - c*x*(-a^2)^(1/2))*(-a^2)^(1/2))
Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.53 \[ \int \frac {e^{\text {arctanh}(a x)}}{x^3 \left (c-a^2 c x^2\right )} \, dx=\frac {-8 \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 \sqrt {-a^{2} x^{2}+1}\, a x +2 \sqrt {-a^{2} x^{2}+1}+3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{3} x^{3}-3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}-1\right ) a^{2} x^{2}-3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{3} x^{3}+3 \,\mathrm {log}\left (\sqrt {-a^{2} x^{2}+1}+1\right ) a^{2} x^{2}}{4 c \,x^{2} \left (a x -1\right )} \] Input:
int((a*x+1)/(-a^2*x^2+1)^(1/2)/x^3/(-a^2*c*x^2+c),x)
Output:
( - 8*sqrt( - a**2*x**2 + 1)*a**2*x**2 + 2*sqrt( - a**2*x**2 + 1)*a*x + 2* sqrt( - a**2*x**2 + 1) + 3*log(sqrt( - a**2*x**2 + 1) - 1)*a**3*x**3 - 3*l og(sqrt( - a**2*x**2 + 1) - 1)*a**2*x**2 - 3*log(sqrt( - a**2*x**2 + 1) + 1)*a**3*x**3 + 3*log(sqrt( - a**2*x**2 + 1) + 1)*a**2*x**2)/(4*c*x**2*(a*x - 1))