Optimal. Leaf size=11 \[ \frac {2}{3} (x+1)^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6127, 26, 32} \[ \frac {2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 26
Rule 32
Rule 6127
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} \sqrt {1-x} \, dx &=\int \frac {\sqrt {1-x^2}}{\sqrt {1-x}} \, dx\\ &=\int \sqrt {1+x} \, dx\\ &=\frac {2}{3} (1+x)^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.00, size = 11, normalized size = 1.00 \[ \frac {2}{3} (x+1)^{3/2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.56, size = 26, normalized size = 2.36 \[ -\frac {2 \, \sqrt {-x^{2} + 1} {\left (x + 1\right )} \sqrt {-x + 1}}{3 \, {\left (x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.40, size = 13, normalized size = 1.18 \[ \frac {2}{3} \, {\left (x + 1\right )}^{\frac {3}{2}} - \frac {4}{3} \, \sqrt {2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 24, normalized size = 2.18 \[ \frac {2 \left (1+x \right )^{2} \sqrt {1-x}}{3 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.38, size = 23, normalized size = 2.09 \[ \frac {2 \, {\left (x^{2} - x - 2\right )}}{3 \, \sqrt {x + 1}} + 2 \, \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.86, size = 33, normalized size = 3.00 \[ \frac {2\,x\,\sqrt {1-x^2}+2\,\sqrt {1-x^2}}{3\,\sqrt {1-x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 - x} \left (x + 1\right )}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________