Optimal. Leaf size=71 \[ \frac{5 \sqrt{-x^3-1}}{24 x^3}-\frac{5 \sqrt{-x^3-1}}{36 x^6}+\frac{\sqrt{-x^3-1}}{9 x^9}-\frac{5}{24} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0279848, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {266, 51, 63, 204} \[ \frac{5 \sqrt{-x^3-1}}{24 x^3}-\frac{5 \sqrt{-x^3-1}}{36 x^6}+\frac{\sqrt{-x^3-1}}{9 x^9}-\frac{5}{24} \tan ^{-1}\left (\sqrt{-x^3-1}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 51
Rule 63
Rule 204
Rubi steps
\begin{align*} \int \frac{1}{x^{10} \sqrt{-1-x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^4} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{9 x^9}-\frac{5}{18} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^3} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{9 x^9}-\frac{5 \sqrt{-1-x^3}}{36 x^6}+\frac{5}{24} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x^2} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{9 x^9}-\frac{5 \sqrt{-1-x^3}}{36 x^6}+\frac{5 \sqrt{-1-x^3}}{24 x^3}-\frac{5}{48} \operatorname{Subst}\left (\int \frac{1}{\sqrt{-1-x} x} \, dx,x,x^3\right )\\ &=\frac{\sqrt{-1-x^3}}{9 x^9}-\frac{5 \sqrt{-1-x^3}}{36 x^6}+\frac{5 \sqrt{-1-x^3}}{24 x^3}+\frac{5}{24} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{-1-x^3}\right )\\ &=\frac{\sqrt{-1-x^3}}{9 x^9}-\frac{5 \sqrt{-1-x^3}}{36 x^6}+\frac{5 \sqrt{-1-x^3}}{24 x^3}-\frac{5}{24} \tan ^{-1}\left (\sqrt{-1-x^3}\right )\\ \end{align*}
Mathematica [C] time = 0.0048966, size = 28, normalized size = 0.39 \[ -\frac{2}{3} \sqrt{-x^3-1} \, _2F_1\left (\frac{1}{2},4;\frac{3}{2};x^3+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 56, normalized size = 0.8 \begin{align*} -{\frac{5}{24}\arctan \left ( \sqrt{-{x}^{3}-1} \right ) }+{\frac{1}{9\,{x}^{9}}\sqrt{-{x}^{3}-1}}-{\frac{5}{36\,{x}^{6}}\sqrt{-{x}^{3}-1}}+{\frac{5}{24\,{x}^{3}}\sqrt{-{x}^{3}-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51061, size = 100, normalized size = 1.41 \begin{align*} \frac{15 \,{\left (-x^{3} - 1\right )}^{\frac{5}{2}} + 40 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-x^{3} - 1}}{72 \,{\left ({\left (x^{3} + 1\right )}^{3} + 3 \, x^{3} - 3 \,{\left (x^{3} + 1\right )}^{2} + 2\right )}} - \frac{5}{24} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.72985, size = 111, normalized size = 1.56 \begin{align*} -\frac{15 \, x^{9} \arctan \left (\sqrt{-x^{3} - 1}\right ) -{\left (15 \, x^{6} - 10 \, x^{3} + 8\right )} \sqrt{-x^{3} - 1}}{72 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] time = 6.65864, size = 90, normalized size = 1.27 \begin{align*} - \frac{5 i \operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} + \frac{5 i}{24 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{5 i}{72 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{i}{36 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{i}{9 x^{\frac{21}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.62319, size = 80, normalized size = 1.13 \begin{align*} \frac{15 \,{\left (x^{3} + 1\right )}^{2} \sqrt{-x^{3} - 1} + 40 \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-x^{3} - 1}}{72 \, x^{9}} - \frac{5}{24} \, \arctan \left (\sqrt{-x^{3} - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]