3.8.39 \(\int \frac {\sqrt {-1+x^2-2 x^3+x^4} (1-x^3+x^4)}{(-1-2 x^3+x^4) (-2-x^2-4 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=57 \[ \tanh ^{-1}\left (\frac {x}{\sqrt {x^4-2 x^3+x^2-1}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^4-2 x^3+x^2-1}}\right ) \]

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Rubi [F]  time = 1.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]

[Out]

Defer[Int][Sqrt[-1 + x^2 - 2*x^3 + x^4]/(2 + x^2 + 4*x^3 - 2*x^4), x] + 3*Defer[Int][(x*Sqrt[-1 + x^2 - 2*x^3
+ x^4])/(-1 - 2*x^3 + x^4), x] - 2*Defer[Int][(x^2*Sqrt[-1 + x^2 - 2*x^3 + x^4])/(-1 - 2*x^3 + x^4), x] - 6*De
fer[Int][(x*Sqrt[-1 + x^2 - 2*x^3 + x^4])/(-2 - x^2 - 4*x^3 + 2*x^4), x] + 4*Defer[Int][(x^2*Sqrt[-1 + x^2 - 2
*x^3 + x^4])/(-2 - x^2 - 4*x^3 + 2*x^4), x]

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^2-2 x^3+x^4} \left (1-x^3+x^4\right )}{\left (-1-2 x^3+x^4\right ) \left (-2-x^2-4 x^3+2 x^4\right )} \, dx &=\int \left (-\frac {x (-3+2 x) \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}+\frac {\left (-1-6 x+4 x^2\right ) \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}\right ) \, dx\\ &=-\int \frac {x (-3+2 x) \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx+\int \frac {\left (-1-6 x+4 x^2\right ) \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx\\ &=-\int \left (-\frac {3 x \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}+\frac {2 x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4}\right ) \, dx+\int \left (\frac {\sqrt {-1+x^2-2 x^3+x^4}}{2+x^2+4 x^3-2 x^4}-\frac {6 x \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}+\frac {4 x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4}\right ) \, dx\\ &=-\left (2 \int \frac {x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx\right )+3 \int \frac {x \sqrt {-1+x^2-2 x^3+x^4}}{-1-2 x^3+x^4} \, dx+4 \int \frac {x^2 \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx-6 \int \frac {x \sqrt {-1+x^2-2 x^3+x^4}}{-2-x^2-4 x^3+2 x^4} \, dx+\int \frac {\sqrt {-1+x^2-2 x^3+x^4}}{2+x^2+4 x^3-2 x^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 6.68, size = 59573, normalized size = 1045.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^4)),x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.44, size = 57, normalized size = 1.00 \begin {gather*} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2-2 x^3+x^4}}\right )-\sqrt {\frac {3}{2}} \tanh ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^2-2 x^3+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[-1 + x^2 - 2*x^3 + x^4]*(1 - x^3 + x^4))/((-1 - 2*x^3 + x^4)*(-2 - x^2 - 4*x^3 + 2*x^
4)),x]

[Out]

ArcTanh[x/Sqrt[-1 + x^2 - 2*x^3 + x^4]] - Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-1 + x^2 - 2*x^3 + x^4]]

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fricas [B]  time = 0.61, size = 180, normalized size = 3.16 \begin {gather*} \frac {1}{8} \, \sqrt {3} \sqrt {2} \log \left (-\frac {4 \, x^{8} - 16 \, x^{7} + 60 \, x^{6} - 88 \, x^{5} + 41 \, x^{4} + 16 \, x^{3} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{5} - 4 \, x^{4} + 5 \, x^{3} - 2 \, x\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} - 44 \, x^{2} + 4}{4 \, x^{8} - 16 \, x^{7} + 12 \, x^{6} + 8 \, x^{5} - 7 \, x^{4} + 16 \, x^{3} + 4 \, x^{2} + 4}\right ) + \frac {1}{2} \, \log \left (-\frac {x^{4} - 2 \, x^{3} + 2 \, x^{2} + 2 \, \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1} x - 1}{x^{4} - 2 \, x^{3} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="fricas")

[Out]

1/8*sqrt(3)*sqrt(2)*log(-(4*x^8 - 16*x^7 + 60*x^6 - 88*x^5 + 41*x^4 + 16*x^3 - 4*sqrt(3)*sqrt(2)*(2*x^5 - 4*x^
4 + 5*x^3 - 2*x)*sqrt(x^4 - 2*x^3 + x^2 - 1) - 44*x^2 + 4)/(4*x^8 - 16*x^7 + 12*x^6 + 8*x^5 - 7*x^4 + 16*x^3 +
4*x^2 + 4)) + 1/2*log(-(x^4 - 2*x^3 + 2*x^2 + 2*sqrt(x^4 - 2*x^3 + x^2 - 1)*x - 1)/(x^4 - 2*x^3 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^2 - 2)*(x^4 - 2*x^3 - 1)), x)

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maple [C]  time = 1.44, size = 142, normalized size = 2.49

method	result	size
trager	\(\frac {\ln \left (-\frac {x^{4}-2 x^{3}+2 \sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x +2 x^{2}-1}{x^{4}-2 x^{3}-1}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{4}+4 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{3}-5 \RootOf \left (\textit {\_Z}^{2}-6\right ) x^{2}+12 \sqrt {x^{4}-2 x^{3}+x^{2}-1}\, x +2 \RootOf \left (\textit {\_Z}^{2}-6\right )}{2 x^{4}-4 x^{3}-x^{2}-2}\right )}{4}\)	\(142\)
default	\(-\frac {i \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right )}{\left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {i \sqrt {3}\, \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {3}\, \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )}{3 \left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {\sqrt {5}}{2}\right ) \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right )}}+\frac {i \sqrt {16}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}-2 \textit {\_Z}^{3}-1\right )}{\sum }\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (-i \sqrt {3}-\sqrt {5}\right ) \sqrt {\frac {\left (\sqrt {5}-i \sqrt {3}\right ) \left (i \sqrt {3}+2 x -1\right )}{\left (\sqrt {5}+i \sqrt {3}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \left (-1+2 x -i \sqrt {3}\right )^{2} \sqrt {\frac {i \left (2 x -1+\sqrt {5}\right )}{\left (i \sqrt {3}-\sqrt {5}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \sqrt {\frac {i \left (-1+2 x -\sqrt {5}\right )}{\left (\sqrt {5}+i \sqrt {3}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3+i \sqrt {3}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}-2 \underline {\hspace {1.25 ex}}\alpha -1\right )\right ) \left (2 \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )+i \sqrt {3}\, \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-3 \underline {\hspace {1.25 ex}}\alpha ^{2}+3+i \sqrt {3}\, \left (-\underline {\hspace {1.25 ex}}\alpha ^{3}+\underline {\hspace {1.25 ex}}\alpha ^{2}+2 \underline {\hspace {1.25 ex}}\alpha +1\right )\right ) \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \frac {3 i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}\, \sqrt {3}}{8}-\frac {i \sqrt {5}\, \sqrt {3}}{8}+\frac {3 i \sqrt {3}}{8}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{8}-\frac {9 \sqrt {5}}{8}-\frac {3 i \underline {\hspace {1.25 ex}}\alpha \sqrt {5}\, \sqrt {3}}{4}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{8}+\frac {9 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}}{8}-\frac {3 i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}\, \sqrt {3}}{8}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{3}}{8}-\frac {1}{8}-\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2}}{8}-\frac {3 \underline {\hspace {1.25 ex}}\alpha }{4}-\frac {3 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{8}, \sqrt {\frac {\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\sqrt {5}-i \sqrt {3}\right ) \sqrt {\left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right ) \left (2 x -1+\sqrt {5}\right ) \left (-1+2 x -\sqrt {5}\right )}}\right )}{16}-\frac {i \sqrt {16}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (2 \textit {\_Z}^{4}-4 \textit {\_Z}^{3}-\textit {\_Z}^{2}-2\right )}{\sum }\frac {2 \underline {\hspace {1.25 ex}}\alpha \left (-i \sqrt {3}-\sqrt {5}\right ) \sqrt {\frac {\left (\sqrt {5}-i \sqrt {3}\right ) \left (i \sqrt {3}+2 x -1\right )}{\left (\sqrt {5}+i \sqrt {3}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \left (-1+2 x -i \sqrt {3}\right )^{2} \sqrt {\frac {i \left (2 x -1+\sqrt {5}\right )}{\left (i \sqrt {3}-\sqrt {5}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \sqrt {\frac {i \left (-1+2 x -\sqrt {5}\right )}{\left (\sqrt {5}+i \sqrt {3}\right ) \left (-1+2 x -i \sqrt {3}\right )}}\, \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-6 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +7+i \sqrt {3}\, \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-2 \underline {\hspace {1.25 ex}}\alpha ^{2}-5 \underline {\hspace {1.25 ex}}\alpha -3\right )\right ) \left (6 \EllipticF \left (\sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, \sqrt {\frac {\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )+i \sqrt {3}\, \left (2 \underline {\hspace {1.25 ex}}\alpha ^{3}-6 \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +7+i \sqrt {3}\, \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{3}+2 \underline {\hspace {1.25 ex}}\alpha ^{2}+5 \underline {\hspace {1.25 ex}}\alpha +3\right )\right ) \EllipticPi \left (\sqrt {\frac {\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (x -\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}}, -\frac {1}{8}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{24}+\frac {\underline {\hspace {1.25 ex}}\alpha \sqrt {5}}{8}+\frac {3 \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}}{4}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}}{4}-\frac {7 \sqrt {5}}{8}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {5}\, \sqrt {3}}{4}-\frac {5 \underline {\hspace {1.25 ex}}\alpha }{8}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {5}\, \sqrt {3}}{4}-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{4}-\frac {i \sqrt {5}\, \sqrt {3}}{8}+\frac {\underline {\hspace {1.25 ex}}\alpha ^{3}}{4}-\frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{4}-\frac {5 i \underline {\hspace {1.25 ex}}\alpha \sqrt {5}\, \sqrt {3}}{8}+\frac {7 i \sqrt {3}}{24}+\frac {i \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}}{12}, \sqrt {\frac {\left (\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right ) \left (-\frac {i \sqrt {3}}{2}-\frac {\sqrt {5}}{2}\right )}{\left (\frac {\sqrt {5}}{2}-\frac {i \sqrt {3}}{2}\right ) \left (-\frac {\sqrt {5}}{2}+\frac {i \sqrt {3}}{2}\right )}}\right )\right )}{\left (\sqrt {5}-i \sqrt {3}\right ) \sqrt {\left (i \sqrt {3}+2 x -1\right ) \left (-1+2 x -i \sqrt {3}\right ) \left (2 x -1+\sqrt {5}\right ) \left (-1+2 x -\sqrt {5}\right )}}\right )}{96}\)	\(1574\)
elliptic	\(\text {Expression too large to display}\)	\(15410\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x,method=_RETURNVERBOSE)

[Out]

1/2*ln(-(x^4-2*x^3+2*(x^4-2*x^3+x^2-1)^(1/2)*x+2*x^2-1)/(x^4-2*x^3-1))+1/4*RootOf(_Z^2-6)*ln(-(-2*RootOf(_Z^2-
6)*x^4+4*RootOf(_Z^2-6)*x^3-5*RootOf(_Z^2-6)*x^2+12*(x^4-2*x^3+x^2-1)^(1/2)*x+2*RootOf(_Z^2-6))/(2*x^4-4*x^3-x
^2-2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - x^{3} + 1\right )} \sqrt {x^{4} - 2 \, x^{3} + x^{2} - 1}}{{\left (2 \, x^{4} - 4 \, x^{3} - x^{2} - 2\right )} {\left (x^{4} - 2 \, x^{3} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-2*x^3+x^2-1)^(1/2)*(x^4-x^3+1)/(x^4-2*x^3-1)/(2*x^4-4*x^3-x^2-2),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 + 1)*sqrt(x^4 - 2*x^3 + x^2 - 1)/((2*x^4 - 4*x^3 - x^2 - 2)*(x^4 - 2*x^3 - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (x^4-x^3+1\right )\,\sqrt {x^4-2\,x^3+x^2-1}}{\left (-x^4+2\,x^3+1\right )\,\left (-2\,x^4+4\,x^3+x^2+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^2 + 4*x^3 - 2*x^4 + 2)),x)

[Out]

int(((x^4 - x^3 + 1)*(x^2 - 2*x^3 + x^4 - 1)^(1/2))/((2*x^3 - x^4 + 1)*(x^2 + 4*x^3 - 2*x^4 + 2)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-2*x**3+x**2-1)**(1/2)*(x**4-x**3+1)/(x**4-2*x**3-1)/(2*x**4-4*x**3-x**2-2),x)

[Out]

Timed out

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