Integrand size = 8, antiderivative size = 84 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=-(1-x)^{2/3} \sqrt [3]{1+x}+\frac {2 \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{1+x}}\right )}{\sqrt {3}}+\frac {1}{3} \log (1+x)+\log \left (1+\frac {\sqrt [3]{1-x}}{\sqrt [3]{1+x}}\right ) \] Output:
-(1-x)^(2/3)*(1+x)^(1/3)-2/3*arctan(-1/3*3^(1/2)+2/3*(1-x)^(1/3)*3^(1/2)/( 1+x)^(1/3))*3^(1/2)+1/3*ln(1+x)+ln(1+(1-x)^(1/3)/(1+x)^(1/3))
Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=-\frac {2 e^{\frac {2 \text {arctanh}(x)}{3}}}{1+e^{2 \text {arctanh}(x)}}+\frac {2 \arctan \left (\frac {-1+2 e^{\frac {2 \text {arctanh}(x)}{3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {2}{3} \log \left (1+e^{\frac {2 \text {arctanh}(x)}{3}}\right )-\frac {1}{3} \log \left (1-e^{\frac {2 \text {arctanh}(x)}{3}}+e^{\frac {4 \text {arctanh}(x)}{3}}\right ) \] Input:
Integrate[E^((2*ArcTanh[x])/3),x]
Output:
(-2*E^((2*ArcTanh[x])/3))/(1 + E^(2*ArcTanh[x])) + (2*ArcTan[(-1 + 2*E^((2 *ArcTanh[x])/3))/Sqrt[3]])/Sqrt[3] + (2*Log[1 + E^((2*ArcTanh[x])/3)])/3 - Log[1 - E^((2*ArcTanh[x])/3) + E^((4*ArcTanh[x])/3)]/3
Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6675, 60, 72}
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.
\(\displaystyle \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx\) |
\(\Big \downarrow \) 6675 |
\(\displaystyle \int \frac {\sqrt [3]{x+1}}{\sqrt [3]{1-x}}dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {2}{3} \int \frac {1}{\sqrt [3]{1-x} (x+1)^{2/3}}dx-(1-x)^{2/3} \sqrt [3]{x+1}\) |
\(\Big \downarrow \) 72 |
\(\displaystyle \frac {2}{3} \left (\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{1-x}}{\sqrt {3} \sqrt [3]{x+1}}\right )+\frac {1}{2} \log (x+1)+\frac {3}{2} \log \left (\frac {\sqrt [3]{1-x}}{\sqrt [3]{x+1}}+1\right )\right )-(1-x)^{2/3} \sqrt [3]{x+1}\) |
Input:
Int[E^((2*ArcTanh[x])/3),x]
Output:
-((1 - x)^(2/3)*(1 + x)^(1/3)) + (2*(Sqrt[3]*ArcTan[1/Sqrt[3] - (2*(1 - x) ^(1/3))/(Sqrt[3]*(1 + x)^(1/3))] + Log[1 + x]/2 + (3*Log[1 + (1 - x)^(1/3) /(1 + x)^(1/3)])/2))/3
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[1/(((a_.) + (b_.)*(x_))^(1/3)*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[{q = Rt[-d/b, 3]}, Simp[Sqrt[3]*(q/d)*ArcTan[1/Sqrt[3] - 2*q*((a + b* x)^(1/3)/(Sqrt[3]*(c + d*x)^(1/3)))], x] + (Simp[3*(q/(2*d))*Log[q*((a + b* x)^(1/3)/(c + d*x)^(1/3)) + 1], x] + Simp[(q/(2*d))*Log[c + d*x], x])] /; F reeQ[{a, b, c, d}, x] && NegQ[d/b]
Int[E^(ArcTanh[(a_.)*(x_)]*(n_)), x_Symbol] :> Int[(1 + a*x)^(n/2)/(1 - a*x )^(n/2), x] /; FreeQ[{a, n}, x] && !IntegerQ[(n - 1)/2]
\[\int {\left (\frac {1+x}{\sqrt {-x^{2}+1}}\right )}^{\frac {2}{3}}d x\]
Input:
int(((1+x)/(-x^2+1)^(1/2))^(2/3),x)
Output:
int(((1+x)/(-x^2+1)^(1/2))^(2/3),x)
Leaf count of result is larger than twice the leaf count of optimal. 146 vs. \(2 (66) = 132\).
Time = 0.09 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.74 \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx={\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + \frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {2}{3} \, \log \left (\left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} + 1\right ) - \frac {1}{3} \, \log \left (-\frac {{\left (x - 1\right )} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {2}{3}} - x + \sqrt {-x^{2} + 1} \left (-\frac {\sqrt {-x^{2} + 1}}{x - 1}\right )^{\frac {1}{3}} + 1}{x - 1}\right ) \] Input:
integrate(((1+x)/(-x^2+1)^(1/2))^(2/3),x, algorithm="fricas")
Output:
(x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) + 2/3*sqrt(3)*arctan(2/3*sqrt(3)*( -sqrt(-x^2 + 1)/(x - 1))^(2/3) - 1/3*sqrt(3)) + 2/3*log((-sqrt(-x^2 + 1)/( x - 1))^(2/3) + 1) - 1/3*log(-((x - 1)*(-sqrt(-x^2 + 1)/(x - 1))^(2/3) - x + sqrt(-x^2 + 1)*(-sqrt(-x^2 + 1)/(x - 1))^(1/3) + 1)/(x - 1))
\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=\int \left (\frac {x + 1}{\sqrt {1 - x^{2}}}\right )^{\frac {2}{3}}\, dx \] Input:
integrate(((1+x)/(-x**2+1)**(1/2))**(2/3),x)
Output:
Integral(((x + 1)/sqrt(1 - x**2))**(2/3), x)
\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=\int { \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}} \,d x } \] Input:
integrate(((1+x)/(-x^2+1)^(1/2))^(2/3),x, algorithm="maxima")
Output:
integrate(((x + 1)/sqrt(-x^2 + 1))^(2/3), x)
\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=\int { \left (\frac {x + 1}{\sqrt {-x^{2} + 1}}\right )^{\frac {2}{3}} \,d x } \] Input:
integrate(((1+x)/(-x^2+1)^(1/2))^(2/3),x, algorithm="giac")
Output:
integrate(((x + 1)/sqrt(-x^2 + 1))^(2/3), x)
Timed out. \[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=\int {\left (\frac {x+1}{\sqrt {1-x^2}}\right )}^{2/3} \,d x \] Input:
int(((x + 1)/(1 - x^2)^(1/2))^(2/3),x)
Output:
int(((x + 1)/(1 - x^2)^(1/2))^(2/3), x)
\[ \int e^{\frac {2 \text {arctanh}(x)}{3}} \, dx=\int \frac {\left (x +1\right )^{\frac {2}{3}}}{\left (-x^{2}+1\right )^{\frac {1}{3}}}d x \] Input:
int(((1+x)/(-x^2+1)^(1/2))^(2/3),x)
Output:
int((x + 1)**(2/3)/( - x**2 + 1)**(1/3),x)