Integrand size = 27, antiderivative size = 82 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {2 (1+a x)}{3 \left (c-a^2 c x^2\right )^{3/2}}+\frac {3+4 a x}{3 c \sqrt {c-a^2 c x^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}} \] Output:
2/3*(a*x+1)/(-a^2*c*x^2+c)^(3/2)+1/3*(4*a*x+3)/c/(-a^2*c*x^2+c)^(1/2)-arct anh((-a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(3/2)
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.91 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {(5-4 a x) \sqrt {c-a^2 c x^2}}{3 c^2 (-1+a x)^2}+\frac {\log (x)}{c^{3/2}}-\frac {\log \left (c+\sqrt {c} \sqrt {c-a^2 c x^2}\right )}{c^{3/2}} \] Input:
Integrate[E^(2*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^(3/2)),x]
Output:
((5 - 4*a*x)*Sqrt[c - a^2*c*x^2])/(3*c^2*(-1 + a*x)^2) + Log[x]/c^(3/2) - Log[c + Sqrt[c]*Sqrt[c - a^2*c*x^2]]/c^(3/2)
Time = 0.41 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {6701, 532, 25, 532, 27, 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.
\(\displaystyle \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 6701 |
\(\displaystyle c \int \frac {(a x+1)^2}{x \left (c-a^2 c x^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 532 |
\(\displaystyle c \left (\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}-\frac {\int -\frac {4 a x+3}{x \left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle c \left (\frac {\int \frac {4 a x+3}{x \left (c-a^2 c x^2\right )^{3/2}}dx}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 532 |
\(\displaystyle c \left (\frac {\frac {4 a x+3}{c \sqrt {c-a^2 c x^2}}-\frac {\int -\frac {3}{x \sqrt {c-a^2 c x^2}}dx}{c}}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle c \left (\frac {\frac {3 \int \frac {1}{x \sqrt {c-a^2 c x^2}}dx}{c}+\frac {4 a x+3}{c \sqrt {c-a^2 c x^2}}}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle c \left (\frac {\frac {3 \int \frac {1}{x^2 \sqrt {c-a^2 c x^2}}dx^2}{2 c}+\frac {4 a x+3}{c \sqrt {c-a^2 c x^2}}}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle c \left (\frac {\frac {4 a x+3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \int \frac {1}{\frac {1}{a^2}-\frac {x^4}{a^2 c}}d\sqrt {c-a^2 c x^2}}{a^2 c^2}}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle c \left (\frac {\frac {4 a x+3}{c \sqrt {c-a^2 c x^2}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {c-a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}}{3 c}+\frac {2 (a x+1)}{3 c \left (c-a^2 c x^2\right )^{3/2}}\right )\) |
Input:
Int[E^(2*ArcTanh[a*x])/(x*(c - a^2*c*x^2)^(3/2)),x]
Output:
c*((2*(1 + a*x))/(3*c*(c - a^2*c*x^2)^(3/2)) + ((3 + 4*a*x)/(c*Sqrt[c - a^ 2*c*x^2]) - (3*ArcTanh[Sqrt[c - a^2*c*x^2]/Sqrt[c]])/c^(3/2))/(3*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)^(p_), x_Symbo l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) *((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1)) Int[x^m *(a + b*x^2)^(p + 1)*ExpandToSum[2*a*(p + 1)*(Qx/x^m) + e*((2*p + 3)/x^m), x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && ILtQ[m, 0] && LtQ[p, -1] && IntegerQ[2*p]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(151\) vs. \(2(68)=136\).
Time = 0.38 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.85
method | result | size |
default | \(\frac {1}{c \sqrt {-a^{2} c \,x^{2}+c}}-\frac {\ln \left (\frac {2 c +2 \sqrt {c}\, \sqrt {-a^{2} c \,x^{2}+c}}{x}\right )}{c^{\frac {3}{2}}}-\frac {2}{3 a c \left (x -\frac {1}{a}\right ) \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}-\frac {2 \left (-2 \left (x -\frac {1}{a}\right ) a^{2} c -2 a c \right )}{3 a \,c^{2} \sqrt {-\left (x -\frac {1}{a}\right )^{2} a^{2} c -2 \left (x -\frac {1}{a}\right ) a c}}\) | \(152\) |
Input:
int((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/c/(-a^2*c*x^2+c)^(1/2)-1/c^(3/2)*ln((2*c+2*c^(1/2)*(-a^2*c*x^2+c)^(1/2)) /x)-2/3/a/c/(x-1/a)/(-(x-1/a)^2*a^2*c-2*(x-1/a)*a*c)^(1/2)-2/3/a/c^2*(-2*( x-1/a)*a^2*c-2*a*c)/(-(x-1/a)^2*a^2*c-2*(x-1/a)*a*c)^(1/2)
Time = 0.10 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.34 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\left [\frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {c} \log \left (-\frac {a^{2} c x^{2} + 2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {c} - 2 \, c}{x^{2}}\right ) - 2 \, \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 5\right )}}{6 \, {\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}, \frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-a^{2} c x^{2} + c} \sqrt {-c}}{c}\right ) - \sqrt {-a^{2} c x^{2} + c} {\left (4 \, a x - 5\right )}}{3 \, {\left (a^{2} c^{2} x^{2} - 2 \, a c^{2} x + c^{2}\right )}}\right ] \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="fric as")
Output:
[1/6*(3*(a^2*x^2 - 2*a*x + 1)*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)*(4*a*x - 5))/(a^2*c^2*x^ 2 - 2*a*c^2*x + c^2), 1/3*(3*(a^2*x^2 - 2*a*x + 1)*sqrt(-c)*arctan(sqrt(-a ^2*c*x^2 + c)*sqrt(-c)/c) - sqrt(-a^2*c*x^2 + c)*(4*a*x - 5))/(a^2*c^2*x^2 - 2*a*c^2*x + c^2)]
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=- \int \frac {a x}{- a^{3} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a c x^{2} \sqrt {- a^{2} c x^{2} + c} - c x \sqrt {- a^{2} c x^{2} + c}}\, dx - \int \frac {1}{- a^{3} c x^{4} \sqrt {- a^{2} c x^{2} + c} + a^{2} c x^{3} \sqrt {- a^{2} c x^{2} + c} + a c x^{2} \sqrt {- a^{2} c x^{2} + c} - c x \sqrt {- a^{2} c x^{2} + c}}\, dx \] Input:
integrate((a*x+1)**2/(-a**2*x**2+1)/x/(-a**2*c*x**2+c)**(3/2),x)
Output:
-Integral(a*x/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*sqrt(-a** 2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*c*x**2 + c) - c*x*sqrt(-a**2*c*x**2 + c)), x) - Integral(1/(-a**3*c*x**4*sqrt(-a**2*c*x**2 + c) + a**2*c*x**3*sq rt(-a**2*c*x**2 + c) + a*c*x**2*sqrt(-a**2*c*x**2 + c) - c*x*sqrt(-a**2*c* x**2 + c)), x)
\[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\int { -\frac {{\left (a x + 1\right )}^{2}}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} {\left (a^{2} x^{2} - 1\right )} x} \,d x } \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="maxi ma")
Output:
-integrate((a*x + 1)^2/((-a^2*c*x^2 + c)^(3/2)*(a^2*x^2 - 1)*x), x)
Exception generated. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x, algorithm="giac ")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=-\int \frac {{\left (a\,x+1\right )}^2}{x\,{\left (c-a^2\,c\,x^2\right )}^{3/2}\,\left (a^2\,x^2-1\right )} \,d x \] Input:
int(-(a*x + 1)^2/(x*(c - a^2*c*x^2)^(3/2)*(a^2*x^2 - 1)),x)
Output:
-int((a*x + 1)^2/(x*(c - a^2*c*x^2)^(3/2)*(a^2*x^2 - 1)), x)
Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.55 \[ \int \frac {e^{2 \text {arctanh}(a x)}}{x \left (c-a^2 c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c}\, \left (3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-9 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+9 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right ) \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-3 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )\right )-4 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}+6 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-6\right )}{3 c^{2} \left (\tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{3}-3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )^{2}+3 \tan \left (\frac {\mathit {asin} \left (a x \right )}{2}\right )-1\right )} \] Input:
int((a*x+1)^2/(-a^2*x^2+1)/x/(-a^2*c*x^2+c)^(3/2),x)
Output:
(sqrt(c)*(3*log(tan(asin(a*x)/2))*tan(asin(a*x)/2)**3 - 9*log(tan(asin(a*x )/2))*tan(asin(a*x)/2)**2 + 9*log(tan(asin(a*x)/2))*tan(asin(a*x)/2) - 3*l og(tan(asin(a*x)/2)) - 4*tan(asin(a*x)/2)**3 + 6*tan(asin(a*x)/2) - 6))/(3 *c**2*(tan(asin(a*x)/2)**3 - 3*tan(asin(a*x)/2)**2 + 3*tan(asin(a*x)/2) - 1))