Integrand size = 27, antiderivative size = 109 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\frac {2 a^2 (1+a x)}{\sqrt {c-a^2 c x^2}}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}-\frac {2 a \sqrt {c-a^2 c x^2}}{c x}-\frac {5 a^2 \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{2 \sqrt {c}} \]
-5/2*a^2*arctanh((-a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(1/2)+2*a^2*(a*x+1)/(-a^2 *c*x^2+c)^(1/2)-1/2*(-a^2*c*x^2+c)^(1/2)/c/x^2-2*a*(-a^2*c*x^2+c)^(1/2)/c/ x
Time = 0.15 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.86 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\frac {\frac {\left (1+3 a x-8 a^2 x^2\right ) \sqrt {c-a^2 c x^2}}{x^2 (-1+a x)}+5 a^2 \sqrt {c} \log (x)-5 a^2 \sqrt {c} \log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )}{2 c} \]
(((1 + 3*a*x - 8*a^2*x^2)*Sqrt[c - a^2*c*x^2])/(x^2*(-1 + a*x)) + 5*a^2*Sq rt[c]*Log[x] - 5*a^2*Sqrt[c]*Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]])/(2*c)
Time = 0.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6701, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {(a x+1)^2}{x^3 \left (c-a^2 c x^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 528 |
\(\displaystyle c \left (\frac {\int \frac {2 a^2 x^2+2 a x+1}{x^3 \sqrt {c-a^2 c x^2}}dx}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 2338 |
\(\displaystyle c \left (\frac {-\frac {\int -\frac {a c (5 a x+4)}{x^2 \sqrt {c-a^2 c x^2}}dx}{2 c}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\frac {\int \frac {a c (5 a x+4)}{x^2 \sqrt {c-a^2 c x^2}}dx}{2 c}-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {\frac {1}{2} a \int \frac {5 a x+4}{x^2 \sqrt {c-a^2 c x^2}}dx-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 534 |
\(\displaystyle c \left (\frac {\frac {1}{2} a \left (5 a \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx-\frac {4 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c \left (\frac {\frac {1}{2} a \left (\frac {5}{2} a \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2-\frac {4 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {\frac {1}{2} a \left (-\frac {5 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a c}-\frac {4 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {\frac {1}{2} a \left (-\frac {5 a \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {4 \sqrt {c-a^2 c x^2}}{c x}\right )-\frac {\sqrt {c-a^2 c x^2}}{2 c x^2}}{c}+\frac {2 a^2 (a x+1)}{c \sqrt {c-a^2 c x^2}}\right )\) |
c*((2*a^2*(1 + a*x))/(c*Sqrt[c - a^2*c*x^2]) + (-1/2*Sqrt[c - a^2*c*x^2]/( c*x^2) + (a*((-4*Sqrt[c - a^2*c*x^2])/(c*x) - (5*a*ArcTanh[Sqrt[c - a^2*c* x^2]/Sqrt[c]])/Sqrt[c]))/2)/c)
3.12.14.3.1 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^(n/2) Int[x^m*(c + d*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/2, 0]
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11
method | result | size |
risch | \(\frac {4 a^{3} x^{3}+a^{2} x^{2}-4 a x -1}{2 x^{2} \sqrt {-c \left (a^{2} x^{2}-1\right )}}-\frac {5 a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2 \sqrt {c}}-\frac {2 a \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}\) | \(121\) |
default | \(-\frac {\sqrt {-a^{2} c \,x^{2}+c}}{2 c \,x^{2}}-\frac {5 a^{2} \ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{2 \sqrt {c}}-\frac {2 a \sqrt {-a^{2} c \,x^{2}+c}}{c x}-\frac {2 a \sqrt {-a^{2} c \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a c}}{c \left (x -\frac {1}{a}\right )}\) | \(124\) |
1/2*(4*a^3*x^3+a^2*x^2-4*a*x-1)/x^2/(-c*(a^2*x^2-1))^(1/2)-5/2*a^2/c^(1/2) *ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2))/x)-2*a/c/(x-1/a)*(-a^2*c*(x-1/a)^ 2-2*(x-1/a)*a*c)^(1/2)
Time = 0.27 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.91 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\left [\frac {5 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (8 \, a^{2} x^{2} - 3 \, a x - 1\right )}}{4 \, {\left (a c x^{3} - c x^{2}\right )}}, -\frac {5 \, {\left (a^{3} x^{3} - a^{2} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{a^{2} c x^{2} - c}\right ) + \sqrt {-a^{2} c x^{2} + c} {\left (8 \, a^{2} x^{2} - 3 \, a x - 1\right )}}{2 \, {\left (a c x^{3} - c x^{2}\right )}}\right ] \]
[1/4*(5*(a^3*x^3 - a^2*x^2)*sqrt(c)*log(-(a^2*c*x^2 + 2*sqrt(-a^2*c*x^2 + c)*sqrt(c) - 2*c)/x^2) - 2*sqrt(-a^2*c*x^2 + c)*(8*a^2*x^2 - 3*a*x - 1))/( a*c*x^3 - c*x^2), -1/2*(5*(a^3*x^3 - a^2*x^2)*sqrt(-c)*arctan(sqrt(-a^2*c* x^2 + c)*sqrt(-c)/(a^2*c*x^2 - c)) + sqrt(-a^2*c*x^2 + c)*(8*a^2*x^2 - 3*a *x - 1))/(a*c*x^3 - c*x^2)]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=- \int \frac {a x}{a x^{4} \sqrt {- a^{2} c x^{2} + c} - x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{a x^{4} \sqrt {- a^{2} c x^{2} + c} - x^{3} \sqrt {- a^{2} c x^{2} + c}}\, dx \]
-Integral(a*x/(a*x**4*sqrt(-a**2*c*x**2 + c) - x**3*sqrt(-a**2*c*x**2 + c) ), x) - Integral(1/(a*x**4*sqrt(-a**2*c*x**2 + c) - x**3*sqrt(-a**2*c*x**2 + c)), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{3}} \,d x } \]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{\sqrt {-a^{2} c x^{2} + c} {\left (a^{2} x^{2} - 1\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x^3 \sqrt {c-a^2 c x^2}} \, dx=-\int \frac {{\left (a\,x+1\right )}^2}{x^3\,\sqrt {c-a^2\,c\,x^2}\,\left (a^2\,x^2-1\right )} \,d x \]