Optimal. Leaf size=70 \[ \frac{\sqrt{x^2+x-1}}{2 \left (1-x^2\right )}-\frac{1}{8} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-\frac{5}{8} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0514388, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {976, 1033, 724, 206, 204} \[ \frac{\sqrt{x^2+x-1}}{2 \left (1-x^2\right )}-\frac{1}{8} \tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-\frac{5}{8} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 976
Rule 1033
Rule 724
Rule 206
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{\left (-1+x^2\right )^2 \sqrt{-1+x+x^2}} \, dx &=\frac{\sqrt{-1+x+x^2}}{2 \left (1-x^2\right )}-\frac{1}{4} \int \frac{3+2 x}{\left (-1+x^2\right ) \sqrt{-1+x+x^2}} \, dx\\ &=\frac{\sqrt{-1+x+x^2}}{2 \left (1-x^2\right )}+\frac{1}{8} \int \frac{1}{(1+x) \sqrt{-1+x+x^2}} \, dx-\frac{5}{8} \int \frac{1}{(-1+x) \sqrt{-1+x+x^2}} \, dx\\ &=\frac{\sqrt{-1+x+x^2}}{2 \left (1-x^2\right )}-\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{-4-x^2} \, dx,x,\frac{-3-x}{\sqrt{-1+x+x^2}}\right )+\frac{5}{4} \operatorname{Subst}\left (\int \frac{1}{4-x^2} \, dx,x,\frac{-1+3 x}{\sqrt{-1+x+x^2}}\right )\\ &=\frac{\sqrt{-1+x+x^2}}{2 \left (1-x^2\right )}-\frac{1}{8} \tan ^{-1}\left (\frac{3+x}{2 \sqrt{-1+x+x^2}}\right )-\frac{5}{8} \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{-1+x+x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.0834212, size = 66, normalized size = 0.94 \[ \frac{1}{8} \left (-\frac{4 \sqrt{x^2+x-1}}{x^2-1}-\tan ^{-1}\left (\frac{x+3}{2 \sqrt{x^2+x-1}}\right )-5 \tanh ^{-1}\left (\frac{1-3 x}{2 \sqrt{x^2+x-1}}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 84, normalized size = 1.2 \begin{align*}{\frac{1}{8}\arctan \left ({\frac{-3-x}{2}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}-2-x}}}} \right ) }-{\frac{1}{-4+4\,x}\sqrt{ \left ( -1+x \right ) ^{2}-2+3\,x}}+{\frac{5}{8}{\it Artanh} \left ({\frac{3\,x-1}{2}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}-2+3\,x}}}} \right ) }+{\frac{1}{4+4\,x}\sqrt{ \left ( 1+x \right ) ^{2}-2-x}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{x^{2} + x - 1}{\left (x^{2} - 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.808353, size = 235, normalized size = 3.36 \begin{align*} \frac{2 \,{\left (x^{2} - 1\right )} \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) + 5 \,{\left (x^{2} - 1\right )} \log \left (-x + \sqrt{x^{2} + x - 1} + 2\right ) - 5 \,{\left (x^{2} - 1\right )} \log \left (-x + \sqrt{x^{2} + x - 1}\right ) - 4 \, \sqrt{x^{2} + x - 1}}{8 \,{\left (x^{2} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (x - 1\right )^{2} \left (x + 1\right )^{2} \sqrt{x^{2} + x - 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.16836, size = 193, normalized size = 2.76 \begin{align*} \frac{2 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{3} + 3 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{2} - x + \sqrt{x^{2} + x - 1} - 1}{2 \,{\left ({\left (x - \sqrt{x^{2} + x - 1}\right )}^{4} - 2 \,{\left (x - \sqrt{x^{2} + x - 1}\right )}^{2} - 4 \, x + 4 \, \sqrt{x^{2} + x - 1}\right )}} + \frac{1}{4} \, \arctan \left (-x + \sqrt{x^{2} + x - 1} - 1\right ) + \frac{5}{8} \, \log \left ({\left | -x + \sqrt{x^{2} + x - 1} + 2 \right |}\right ) - \frac{5}{8} \, \log \left ({\left | -x + \sqrt{x^{2} + x - 1} \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]