Optimal. Leaf size=75 \[ -\frac{3}{2} \log \left (1-\frac{x-3}{\sqrt [3]{x^3-5 x^2+3 x+9}}\right )+\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (x-3)}{\sqrt [3]{x^3-5 x^2+3 x+9}}+1}{\sqrt{3}}\right )-\frac{1}{2} \log (x+1) \]
[Out]
________________________________________________________________________________________
Rubi [B] time = 0.118254, antiderivative size = 188, normalized size of antiderivative = 2.51, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2067, 2064, 60} \[ -\frac{(9-3 x)^{2/3} \sqrt [3]{x+1} \log \left (-\frac{32}{3} (x-3)\right )}{2\ 3^{2/3} \sqrt [3]{x^3-5 x^2+3 x+9}}-\frac{\sqrt [3]{3} (9-3 x)^{2/3} \sqrt [3]{x+1} \log \left (\frac{\sqrt [3]{3} \sqrt [3]{x+1}}{\sqrt [3]{9-3 x}}+1\right )}{2 \sqrt [3]{x^3-5 x^2+3 x+9}}-\frac{(9-3 x)^{2/3} \sqrt [3]{x+1} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{x+1}}{\sqrt [6]{3} \sqrt [3]{9-3 x}}\right )}{\sqrt [6]{3} \sqrt [3]{x^3-5 x^2+3 x+9}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2067
Rule 2064
Rule 60
Rubi steps
\begin{align*} \int \frac{1}{\sqrt [3]{9+3 x-5 x^2+x^3}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{\frac{128}{27}-\frac{16 x}{3}+x^3}} \, dx,x,-\frac{5}{3}+x\right )\\ &=\frac{\left (16\ 2^{2/3} (3-x)^{2/3} \sqrt [3]{1+x}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (\frac{128}{9}-\frac{32 x}{3}\right )^{2/3} \sqrt [3]{\frac{128}{9}+\frac{16 x}{3}}} \, dx,x,-\frac{5}{3}+x\right )}{3 \sqrt [3]{9+3 x-5 x^2+x^3}}\\ &=-\frac{\sqrt{3} (3-x)^{2/3} \sqrt [3]{1+x} \tan ^{-1}\left (\frac{1}{\sqrt{3}}-\frac{2 \sqrt [3]{1+x}}{\sqrt{3} \sqrt [3]{3-x}}\right )}{\sqrt [3]{9+3 x-5 x^2+x^3}}-\frac{(3-x)^{2/3} \sqrt [3]{1+x} \log (3-x)}{2 \sqrt [3]{9+3 x-5 x^2+x^3}}-\frac{3 (3-x)^{2/3} \sqrt [3]{1+x} \log \left (\frac{3 \left (\sqrt [3]{3-x}+\sqrt [3]{1+x}\right )}{\sqrt [3]{3-x}}\right )}{2 \sqrt [3]{9+3 x-5 x^2+x^3}}\\ \end{align*}
Mathematica [C] time = 0.0112024, size = 49, normalized size = 0.65 \[ \frac{3 (x-3) \sqrt [3]{x+1} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{3-x}{4}\right )}{2^{2/3} \sqrt [3]{(x-3)^2 (x+1)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.01, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt [3]{{x}^{3}-5\,{x}^{2}+3\,x+9}}}\, dx \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}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.81094, size = 352, normalized size = 4.69 \begin{align*} -\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (x - 3\right )} + 2 \, \sqrt{3}{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}}{3 \,{\left (x - 3\right )}}\right ) + \frac{1}{2} \, \log \left (\frac{x^{2} +{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}{\left (x - 3\right )} - 6 \, x +{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{2}{3}} + 9}{x^{2} - 6 \, x + 9}\right ) - \log \left (-\frac{x -{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}} - 3}{x - 3}\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}{\sqrt [3]{x^{3} - 5 x^{2} + 3 x + 9}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} - 5 \, x^{2} + 3 \, x + 9\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]