Optimal. Leaf size=43 \[ \frac {4 \sqrt {x+1}}{\sqrt {1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {98, 21, 105, 41, 216, 92, 206} \[ \frac {4 \sqrt {x+1}}{\sqrt {1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {x+1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 41
Rule 92
Rule 98
Rule 105
Rule 206
Rule 216
Rubi steps
\begin {align*} \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx &=\frac {4 \sqrt {1+x}}{\sqrt {1-x}}-2 \int \frac {-\frac {1}{2}+\frac {x}{2}}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {4 \sqrt {1+x}}{\sqrt {1-x}}+\int \frac {\sqrt {1-x}}{x \sqrt {1+x}} \, dx\\ &=\frac {4 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x} \sqrt {1+x}} \, dx+\int \frac {1}{\sqrt {1-x} x \sqrt {1+x}} \, dx\\ &=\frac {4 \sqrt {1+x}}{\sqrt {1-x}}-\int \frac {1}{\sqrt {1-x^2}} \, dx-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt {1-x} \sqrt {1+x}\right )\\ &=\frac {4 \sqrt {1+x}}{\sqrt {1-x}}-\sin ^{-1}(x)-\tanh ^{-1}\left (\sqrt {1-x} \sqrt {1+x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 61, normalized size = 1.42 \[ \frac {2 \left (\sqrt {1-x^2} \sin ^{-1}\left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )+2 x+2\right )}{\sqrt {1-x^2}}-\tanh ^{-1}\left (\sqrt {1-x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.41, size = 74, normalized size = 1.72 \[ \frac {2 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + {\left (x - 1\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 4 \, x - 4 \, \sqrt {x + 1} \sqrt {-x + 1} - 4}{x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 70, normalized size = 1.63 \[ \frac {\left (-x \arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )-x \arcsin \relax (x )+\arctanh \left (\frac {1}{\sqrt {-x^{2}+1}}\right )+\arcsin \relax (x )-4 \sqrt {-x^{2}+1}\right ) \sqrt {-x +1}\, \sqrt {x +1}}{\left (x -1\right ) \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.99, size = 53, normalized size = 1.23 \[ \frac {4 \, x}{\sqrt {-x^{2} + 1}} + \frac {4}{\sqrt {-x^{2} + 1}} - \arcsin \relax (x) - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (x+1\right )}^{3/2}}{x\,{\left (1-x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x \left (1 - x\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________