Integrand size = 13, antiderivative size = 122 \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {1+x^3}}{2 x^2}-\frac {\sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \]
[Out]
Time = 0.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {331, 224} \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {x^3+1}}{2 x^2} \]
[In]
[Out]
Rule 224
Rule 331
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {1+x^3}}{2 x^2}-\frac {1}{4} \int \frac {1}{\sqrt {1+x^3}} \, dx \\ & = -\frac {\sqrt {1+x^3}}{2 x^2}-\frac {\sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {1}{3},-x^3\right )}{2 x^2} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 5 vs. order 4.
Time = 3.94 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.14
method | result | size |
meijerg | \(-\frac {{}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (-\frac {2}{3},\frac {1}{2};\frac {1}{3};-x^{3}\right )}{2 x^{2}}\) | \(17\) |
default | \(-\frac {\sqrt {x^{3}+1}}{2 x^{2}}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\) | \(129\) |
risch | \(-\frac {\sqrt {x^{3}+1}}{2 x^{2}}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\) | \(129\) |
elliptic | \(-\frac {\sqrt {x^{3}+1}}{2 x^{2}}-\frac {\left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}\) | \(129\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.17 \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=-\frac {x^{2} {\rm weierstrassPInverse}\left (0, -4, x\right ) + \sqrt {x^{3} + 1}}{2 \, x^{2}} \]
[In]
[Out]
Time = 0.43 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=\frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} \]
[In]
[Out]
\[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=\int { \frac {1}{\sqrt {x^{3} + 1} x^{3}} \,d x } \]
[In]
[Out]
Time = 5.61 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.39 \[ \int \frac {1}{x^3 \sqrt {1+x^3}} \, dx=-\frac {\sqrt {x^3+1}}{2\,x^2}-\frac {\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{2\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
[In]
[Out]