Optimal. Leaf size=67 \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {5}{6} \sqrt {1-x} (x+1)^{3/2}-\frac {5}{2} \sqrt {1-x} \sqrt {x+1}+\frac {5}{2} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6129, 50, 41, 216} \[ -\frac {1}{3} \sqrt {1-x} (x+1)^{5/2}-\frac {5}{6} \sqrt {1-x} (x+1)^{3/2}-\frac {5}{2} \sqrt {1-x} \sqrt {x+1}+\frac {5}{2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 41
Rule 50
Rule 216
Rule 6129
Rubi steps
\begin {align*} \int e^{\tanh ^{-1}(x)} (1+x)^2 \, dx &=\int \frac {(1+x)^{5/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{3} \int \frac {(1+x)^{3/2}}{\sqrt {1-x}} \, dx\\ &=-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {\sqrt {1+x}}{\sqrt {1-x}} \, dx\\ &=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx\\ &=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=-\frac {5}{2} \sqrt {1-x} \sqrt {1+x}-\frac {5}{6} \sqrt {1-x} (1+x)^{3/2}-\frac {1}{3} \sqrt {1-x} (1+x)^{5/2}+\frac {5}{2} \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 44, normalized size = 0.66 \[ -\frac {1}{6} \sqrt {1-x^2} \left (2 x^2+9 x+22\right )-5 \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 40, normalized size = 0.60 \[ -\frac {1}{6} \, {\left (2 \, x^{2} + 9 \, x + 22\right )} \sqrt {-x^{2} + 1} - 5 \, \arctan \left (\frac {\sqrt {-x^{2} + 1} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.18, size = 25, normalized size = 0.37 \[ -\frac {1}{6} \, {\left ({\left (2 \, x + 9\right )} x + 22\right )} \sqrt {-x^{2} + 1} + \frac {5}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.03, size = 43, normalized size = 0.64 \[ -\frac {x^{2} \sqrt {-x^{2}+1}}{3}-\frac {11 \sqrt {-x^{2}+1}}{3}-\frac {3 x \sqrt {-x^{2}+1}}{2}+\frac {5 \arcsin \relax (x )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.49, size = 42, normalized size = 0.63 \[ -\frac {1}{3} \, \sqrt {-x^{2} + 1} x^{2} - \frac {3}{2} \, \sqrt {-x^{2} + 1} x - \frac {11}{3} \, \sqrt {-x^{2} + 1} + \frac {5}{2} \, \arcsin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 26, normalized size = 0.39 \[ \frac {5\,\mathrm {asin}\relax (x)}{2}-\sqrt {1-x^2}\,\left (\frac {x^2}{3}+\frac {3\,x}{2}+\frac {11}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.33, size = 44, normalized size = 0.66 \[ - \frac {x^{2} \sqrt {1 - x^{2}}}{3} - \frac {3 x \sqrt {1 - x^{2}}}{2} - \frac {11 \sqrt {1 - x^{2}}}{3} + \frac {5 \operatorname {asin}{\relax (x )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________