Optimal. Leaf size=63 \[ \frac {\sqrt {1+x^3} \left (2+5 x^3\right )}{6 x^6}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {5}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 2.43, antiderivative size = 1234, normalized size of antiderivative = 19.59, number of
steps used = 41, number of rules used = 16, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used =
{1600, 6857, 2161, 224, 2167, 2138, 551, 585, 95, 212, 272, 44, 65, 213, 6860, 211}
\begin {gather*} \frac {5 i (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \text {ArcTan}\left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(3-6 i)-(2-3 i) \sqrt {3}}{(4+6 i)-(2+4 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {10 \sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \text {ArcTan}\left (\frac {\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3^{3/4} \left (3 i-\sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {5 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}}}{\sqrt {2} \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {x^3+1}\right )-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {10 (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \Pi \left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2};\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {40 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \Pi \left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2};\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {20 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} \Pi \left (97+56 \sqrt {3};\text {ArcSin}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {5 \sqrt {x^3+1}}{6 x^3}+\frac {\sqrt {x^3+1}}{3 x^6} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 44
Rule 65
Rule 95
Rule 211
Rule 212
Rule 213
Rule 224
Rule 272
Rule 551
Rule 585
Rule 1600
Rule 2138
Rule 2161
Rule 2167
Rule 6857
Rule 6860
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^3} \left (2+2 x^3+x^6\right )}{x^7 \left (-1+x^6\right )} \, dx &=\int \frac {2+2 x^3+x^6}{x^7 \left (-1+x^3\right ) \sqrt {1+x^3}} \, dx\\ &=\int \left (\frac {5}{3 (-1+x) \sqrt {1+x^3}}-\frac {2}{x^7 \sqrt {1+x^3}}-\frac {4}{x^4 \sqrt {1+x^3}}-\frac {5}{x \sqrt {1+x^3}}+\frac {5 (1+2 x)}{3 \left (1+x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx\\ &=\frac {5}{3} \int \frac {1}{(-1+x) \sqrt {1+x^3}} \, dx+\frac {5}{3} \int \frac {1+2 x}{\left (1+x+x^2\right ) \sqrt {1+x^3}} \, dx-2 \int \frac {1}{x^7 \sqrt {1+x^3}} \, dx-4 \int \frac {1}{x^4 \sqrt {1+x^3}} \, dx-5 \int \frac {1}{x \sqrt {1+x^3}} \, dx\\ &=-\left (\frac {2}{3} \text {Subst}\left (\int \frac {1}{x^3 \sqrt {1+x}} \, dx,x,x^3\right )\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\frac {5}{3} \int \left (\frac {2}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {2}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx-\frac {5}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )-\frac {5 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )}+\frac {5 \int \frac {1+\sqrt {3}+x}{(-1+x) \sqrt {1+x^3}} \, dx}{3 \left (2+\sqrt {3}\right )}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {4 \sqrt {1+x^3}}{3 x^3}-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^3\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+\frac {10}{3} \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {10}{3} \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx-\frac {10}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )+\frac {\left (20 \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 (2-\sqrt {3}+\left (2+\sqrt {3}\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 )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+\frac {10}{3} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^3\right )+\frac {4}{3} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2-i) \sqrt {3}\right )}-\frac {10 \int \frac {1}{\sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}+\frac {20 \int \frac {1+\sqrt {3}+x}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx}{3 \left (1+(2+i) \sqrt {3}\right )}-\frac {\left (20 \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 {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (20 \left (2-\sqrt {3}\right )^{3/2} (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 (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (2+\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+2 \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97+56 \sqrt {3};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^3}\right )-\frac {\left (10 \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}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (2-\sqrt {3}\right )^2-\left (2+\sqrt {3}\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \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 (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\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 )}{3^{3/4} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \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 (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )+\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\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 )}{3^{3/4} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97+56 \sqrt {3};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (20 \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 (2-\sqrt {3}\right )^2+\left (2+\sqrt {3}\right )^2-\left (\left (2-\sqrt {3}\right )^2+\left (7-4 \sqrt {3}\right ) \left (2+\sqrt {3}\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 )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (80 \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 {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (80 \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 {x}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \left (1-(2+i) \sqrt {3}\right ) \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}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {\left (80 \left (1-(2-i) \sqrt {3}\right ) \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}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2} \left (\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x^2\right )} \, dx,x,\frac {-1+\sqrt {3}-x}{1+\sqrt {3}+x}\right )}{3^{3/4} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}-\frac {5 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97+56 \sqrt {3};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (40 \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}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (40 \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}{\sqrt {1-x} \sqrt {7-4 \sqrt {3}+x} \left (\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2-\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2 x\right )} \, dx,x,\frac {\left (-1+\sqrt {3}-x\right )^2}{\left (1+\sqrt {3}+x\right )^2}\right )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}-\frac {5 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97+56 \sqrt {3};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (80 \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 (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2+\left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2-\left (\left (-1-i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2+\left (7-4 \sqrt {3}\right ) \left (-1-i \sqrt {3}+2 \left (1+\sqrt {3}\right )\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 )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (80 \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 (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2+\left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\right )^2-\left (\left (-1+i \sqrt {3}+2 \left (1-\sqrt {3}\right )\right )^2+\left (7-4 \sqrt {3}\right ) \left (-1+i \sqrt {3}+2 \left (1+\sqrt {3}\right )\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 )}{3^{3/4} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ &=\frac {\sqrt {1+x^3}}{3 x^6}+\frac {5 \sqrt {1+x^3}}{6 x^3}+\frac {5 i (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {2-\sqrt {3}} \sqrt {\frac {3 i+\sqrt {3}}{(-4-6 i)+(2+4 i) \sqrt {3}}} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {10 \sqrt {\frac {(6-3 i)-(3-2 i) \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tan ^{-1}\left (\frac {\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {2-\sqrt {3}} \sqrt {\frac {3+i \sqrt {3}}{(-6-4 i)+(4+2 i) \sqrt {3}}} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{3^{3/4} \left (3 i-\sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {5 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}}}{\sqrt {2} \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}}}\right )}{3 \sqrt {2} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {5}{2} \tanh ^{-1}\left (\sqrt {1+x^3}\right )-\frac {10 (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 )}{3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2-i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 \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 )}{3 \sqrt [4]{3} \left (1+(2+i) \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (i+(1+2 i) \sqrt {3}\right )^2}{\left (1-(2+i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7+i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {40 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (-\frac {\left (1+(2+i) \sqrt {3}\right )^2}{\left (i-(1+2 i) \sqrt {3}\right )^2};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \left (7-i \sqrt {3}\right ) \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {20 (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} \Pi \left (97+56 \sqrt {3};\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{3^{3/4} \sqrt {2-\sqrt {3}} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 60, normalized size = 0.95 \begin {gather*} \frac {1}{6} \left (\frac {\sqrt {1+x^3} \left (2+5 x^3\right )}{x^6}+15 \tanh ^{-1}\left (\sqrt {1+x^3}\right )-10 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+x^3}}{\sqrt {2}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 1.66, size = 449, normalized size = 7.13
method | result | size |
trager | \(\frac {\sqrt {x^{3}+1}\, \left (5 x^{3}+2\right )}{6 x^{6}}-\frac {5 \ln \left (-\frac {-x^{3}+2 \sqrt {x^{3}+1}-2}{x^{3}}\right )}{4}-\frac {5 \RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-2\right )+4 \sqrt {x^{3}+1}}{\left (-1+x \right ) \left (x^{2}+x +1\right )}\right )}{6}\) | \(97\) |
default | \(\frac {10 \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}}}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}-\frac {3 i \sqrt {3}}{4}+\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {10 \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}}}\, \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}+\frac {3 i \sqrt {3}}{4}+\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}+\frac {5 \sqrt {x^{3}+1}}{6 x^{3}}+\frac {\sqrt {x^{3}+1}}{3 x^{6}}-\frac {5 \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}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}-\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}\) | \(449\) |
risch | \(\frac {5 x^{6}+7 x^{3}+2}{6 x^{6} \sqrt {x^{3}+1}}+\frac {10 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}-\frac {3 i \sqrt {3}}{4}+\frac {i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {10 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}+\frac {3 i \sqrt {3}}{4}+\frac {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{2}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}-\frac {5 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {3}{4}-\frac {i \sqrt {3}}{4}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}+1}}+\frac {5 \arctanh \left (\sqrt {x^{3}+1}\right )}{2}\) | \(449\) |
elliptic | \(\text {Expression too large to display}\) | \(920\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 86, normalized size = 1.37 \begin {gather*} \frac {10 \, \sqrt {2} x^{6} \log \left (\frac {x^{3} - 2 \, \sqrt {2} \sqrt {x^{3} + 1} + 3}{x^{3} - 1}\right ) + 15 \, x^{6} \log \left (\sqrt {x^{3} + 1} + 1\right ) - 15 \, x^{6} \log \left (\sqrt {x^{3} + 1} - 1\right ) + 2 \, {\left (5 \, x^{3} + 2\right )} \sqrt {x^{3} + 1}}{12 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 65.39, size = 144, normalized size = 2.29 \begin {gather*} \frac {10 \left (\begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2} \sqrt {x^{3} + 1}}{2} \right )}}{2} & \text {for}\: x^{3} > 1 \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2} \sqrt {x^{3} + 1}}{2} \right )}}{2} & \text {for}\: x^{3} < 1 \end {cases}\right )}{3} - \frac {5 \log {\left (\sqrt {x^{3} + 1} - 1 \right )}}{4} + \frac {5 \log {\left (\sqrt {x^{3} + 1} + 1 \right )}}{4} + \frac {5}{12 \left (\sqrt {x^{3} + 1} + 1\right )} - \frac {1}{12 \left (\sqrt {x^{3} + 1} + 1\right )^{2}} + \frac {5}{12 \left (\sqrt {x^{3} + 1} - 1\right )} + \frac {1}{12 \left (\sqrt {x^{3} + 1} - 1\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 87, normalized size = 1.38 \begin {gather*} \frac {5}{6} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {x^{3} + 1} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {x^{3} + 1}\right )}}\right ) + \frac {5 \, {\left (x^{3} + 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {x^{3} + 1}}{6 \, x^{6}} + \frac {5}{4} \, \log \left (\sqrt {x^{3} + 1} + 1\right ) - \frac {5}{4} \, \log \left ({\left | \sqrt {x^{3} + 1} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 724, normalized size = 11.49 \begin {gather*} \frac {5\,\sqrt {x^3+1}}{6\,x^3}+\frac {\sqrt {x^3+1}}{3\,x^6}+\frac {15\,\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}}}\,\Pi \left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\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 )}}-\frac {5\,\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}}}\,\Pi \left (\frac {3}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4};\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 )}{3\,\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 )}}-\frac {10\,\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}}}\,\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}};\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 )}{3\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\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 )}}+\frac {10\,\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}}}\,\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}};\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 )}{3\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\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 )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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