Integrand size = 25, antiderivative size = 59 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\arctan \left (\frac {\sqrt {x+x^2+x^3}}{1+x+x^2}\right )-\frac {\text {arctanh}\left (\frac {\sqrt {3} \sqrt {x+x^2+x^3}}{1+x+x^2}\right )}{\sqrt {3}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.68 (sec) , antiderivative size = 322, normalized size of antiderivative = 5.46, number of steps used = 17, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2081, 6857, 730, 1117, 948, 175, 552, 551} \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}+\frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}} \]
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Rule 175
Rule 551
Rule 552
Rule 730
Rule 948
Rule 1117
Rule 2081
Rule 6857
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1+x^2}{\sqrt {x} \left (-1+x^2\right ) \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {1+x+x^2}}+\frac {2}{\sqrt {x} \left (-1+x^2\right ) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (-1+x^2\right ) \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \int \left (-\frac {1}{2 (1-x) \sqrt {x} \sqrt {1+x+x^2}}-\frac {1}{2 \sqrt {x} (1+x) \sqrt {1+x+x^2}}\right ) \, dx}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+x+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1+x+x^2}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {1+x+x^2}} \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{(1-x) \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{\sqrt {x+x^2+x^3}}-\frac {\left (\sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \int \frac {1}{\sqrt {x} (1+x) \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}} \, dx}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1-i \sqrt {3}+2 x} \sqrt {1+i \sqrt {3}+2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1-i \sqrt {3}+2 x^2} \sqrt {1+i \sqrt {3}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+i \sqrt {3}+2 x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+i \sqrt {3}+2 x^2} \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1-x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}}\right ) \text {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \sqrt {1+\frac {2 x^2}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x^2}{1+i \sqrt {3}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {x+x^2+x^3}} \\ & = \frac {\sqrt {x} (1+x) \sqrt {\frac {1+x+x^2}{(1+x)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{4}\right )}{\sqrt {x+x^2+x^3}}-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}}-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \operatorname {EllipticPi}\left (\frac {1}{2} \left (-1+i \sqrt {3}\right ),\arcsin \left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right ),\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x+x^2+x^3}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.34 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\frac {\sqrt {x} \sqrt {1+x+x^2} \left (3 \arctan \left (\frac {\sqrt {x}}{\sqrt {1+x+x^2}}\right )+\sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} \sqrt {x}}{\sqrt {1+x+x^2}}\right )\right )}{3 \sqrt {x \left (1+x+x^2\right )}} \]
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Time = 2.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.69
method | result | size |
default | \(\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{3}\) | \(41\) |
pseudoelliptic | \(\arctan \left (\frac {\sqrt {x \left (x^{2}+x +1\right )}}{x}\right )-\frac {\sqrt {3}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+x +1\right )}\, \sqrt {3}}{3 x}\right )}{3}\) | \(41\) |
trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{3}+x^{2}+x}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )-6 \sqrt {x^{3}+x^{2}+x}}{\left (x -1\right )^{2}}\right )}{6}\) | \(103\) |
elliptic | \(\text {Expression too large to display}\) | \(910\) |
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Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.51 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{4} + 20 \, x^{3} - 4 \, \sqrt {3} \sqrt {x^{3} + x^{2} + x} {\left (x^{2} + 4 \, x + 1\right )} + 30 \, x^{2} + 20 \, x + 1}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, x + 1}\right ) + \frac {1}{2} \, \arctan \left (\frac {x^{2} + 1}{2 \, \sqrt {x^{3} + x^{2} + x}}\right ) \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int \frac {x^{2} + 1}{\sqrt {x \left (x^{2} + x + 1\right )} \left (x - 1\right ) \left (x + 1\right )}\, dx \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{3} + x^{2} + x} {\left (x^{2} - 1\right )}} \,d x } \]
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\[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=\int { \frac {x^{2} + 1}{\sqrt {x^{3} + x^{2} + x} {\left (x^{2} - 1\right )}} \,d x } \]
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Time = 5.62 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.78 \[ \int \frac {1+x^2}{\left (-1+x^2\right ) \sqrt {x+x^2+x^3}} \, dx=-\frac {\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,1{}\mathrm {i}}{\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \]
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