Integrand size = 27, antiderivative size = 325 \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=-\frac {\left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \text {arctanh}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2},\arcsin \left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right ),-7-4 \sqrt {3}\right )}{\left (c+d-\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \]
[Out]
Time = 0.76 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2167, 2138, 551, 585, 95, 214} \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (c+\sqrt {3} d+d\right )^2}{\left (c-\sqrt {3} d+d\right )^2},\arcsin \left (\frac {-x-\sqrt {3}+1}{-x+\sqrt {3}+1}\right ),-7-4 \sqrt {3}\right )}{\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \left (c-\sqrt {3} d+d\right )}-\frac {(1-x) \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \left (c+\sqrt {3} d+d\right ) \text {arctanh}\left (\frac {\sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c^2-c d+d^2}}{\sqrt {d} \sqrt {\frac {x^2+x+1}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {c+d}}\right )}{\sqrt {d} \sqrt {\frac {1-x}{\left (-x+\sqrt {3}+1\right )^2}} \sqrt {x^3-1} \sqrt {c+d} \sqrt {c^2-c d+d^2}} \]
[In]
[Out]
Rule 95
Rule 214
Rule 551
Rule 585
Rule 2138
Rule 2167
Rubi steps \begin{align*} \text {integral}& = \frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\left (c+\left (1-\sqrt {3}\right ) d+\left (c+\left (1+\sqrt {3}\right ) d\right ) x\right ) \sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = \frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c+d-\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (c+\left (1-\sqrt {3}\right ) d\right )^2-\left (c+\left (1+\sqrt {3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (c+\left (1-\sqrt {3}\right ) d\right )^2-\left (c+\left (1+\sqrt {3}\right ) d\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}+x}{1+\sqrt {3}-x}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\left (c+d-\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (2 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (c+\left (1-\sqrt {3}\right ) d\right )^2-\left (c+\left (1+\sqrt {3}\right ) d\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}+x\right )^2}{\left (1+\sqrt {3}-x\right )^2}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\left (c+d-\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {\left (4 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}\right ) \text {Subst}\left (\int \frac {1}{-\left (c+\left (1-\sqrt {3}\right ) d\right )^2+\left (c+\left (1+\sqrt {3}\right ) d\right )^2-\left (\left (c+\left (1-\sqrt {3}\right ) d\right )^2+\left (7-4 \sqrt {3}\right ) \left (c+\left (1+\sqrt {3}\right ) d\right )^2\right ) x^2} \, dx,x,\frac {\sqrt [4]{3} \sqrt {-\frac {-1+x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {-\frac {\left (-2+\sqrt {3}\right ) \left (1+x+x^2\right )}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ & = -\frac {\left (c+d+\sqrt {3} d\right ) (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}}}{\sqrt {d} \sqrt {c+d} \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}}}\right )}{\sqrt {d} \sqrt {c+d} \sqrt {c^2-c d+d^2} \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}}-\frac {4 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {1+x+x^2}{\left (1+\sqrt {3}-x\right )^2}} \Pi \left (\frac {\left (c+d+\sqrt {3} d\right )^2}{\left (c+d-\sqrt {3} d\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}-x}{1+\sqrt {3}-x}\right )|-7-4 \sqrt {3}\right )}{\left (c+d-\sqrt {3} d\right ) \sqrt {\frac {1-x}{\left (1+\sqrt {3}-x\right )^2}} \sqrt {-1+x^3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 10.54 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.72 \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\frac {2 \sqrt {\frac {1-x}{1+\sqrt [3]{-1}}} \left (-\frac {3 \left (\sqrt [3]{-1}+x\right ) \sqrt {\frac {\sqrt [3]{-1}+(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac {\sqrt [3]{-1} \left (1+\sqrt [3]{-1}\right ) \left (\sqrt {3} c+\left (3+\sqrt {3}\right ) d\right ) \sqrt {1+x+x^2} \operatorname {EllipticPi}\left (\frac {i \sqrt {3} d}{-c+\sqrt [3]{-1} d},\arcsin \left (\sqrt {\frac {1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right ),\sqrt [3]{-1}\right )}{c-\sqrt [3]{-1} d}\right )}{3 d \sqrt {-1+x^3}} \]
[In]
[Out]
Time = 1.31 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.84
method | result | size |
default | \(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}+\frac {2 \left (c +d +d \sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\) | \(273\) |
elliptic | \(-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, F\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d \sqrt {x^{3}-1}}+\frac {2 \left (c +d +d \sqrt {3}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \Pi \left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{1+\frac {c}{d}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{d^{2} \sqrt {x^{3}-1}\, \left (1+\frac {c}{d}\right )}\) | \(273\) |
[In]
[Out]
Timed out. \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=- \int \left (- \frac {\sqrt {3}}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\right )\, dx - \int \frac {x}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\, dx - \int \left (- \frac {1}{c \sqrt {x^{3} - 1} + d x \sqrt {x^{3} - 1}}\right )\, dx \]
[In]
[Out]
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\int { -\frac {x - \sqrt {3} - 1}{\sqrt {x^{3} - 1} {\left (d x + c\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\int { -\frac {x - \sqrt {3} - 1}{\sqrt {x^{3} - 1} {\left (d x + c\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1+\sqrt {3}-x}{(c+d x) \sqrt {-1+x^3}} \, dx=\text {Hanged} \]
[In]
[Out]