Optimal. Leaf size=39 \[ \frac {1}{2} \log \left (2 x^2+2 \sqrt {x^4+4 x^3+13 x^2+18 x+16}+4 x+9\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1680, 1107, 621, 206} \begin {gather*} \frac {1}{2} \tanh ^{-1}\left (\frac {2 (x+1)^2+7}{2 \sqrt {(x+1)^4+7 (x+1)^2+8}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 621
Rule 1107
Rule 1680
Rubi steps
\begin {align*} \int \frac {1+x}{\sqrt {16+18 x+13 x^2+4 x^3+x^4}} \, dx &=\operatorname {Subst}\left (\int \frac {x}{\sqrt {8+7 x^2+x^4}} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {8+7 x+x^2}} \, dx,x,(1+x)^2\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{4-x^2} \, dx,x,\frac {7+2 (1+x)^2}{\sqrt {8+7 (1+x)^2+(1+x)^4}}\right )\\ &=\frac {1}{2} \tanh ^{-1}\left (\frac {7+2 (1+x)^2}{2 \sqrt {8+7 (1+x)^2+(1+x)^4}}\right )\\ \end {align*}
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Mathematica [C] time = 6.08, size = 2092, normalized size = 53.64 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.17, size = 39, normalized size = 1.00 \begin {gather*} \frac {1}{2} \log \left (9+4 x+2 x^2+2 \sqrt {16+18 x+13 x^2+4 x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 35, normalized size = 0.90 \begin {gather*} \frac {1}{2} \, \log \left (2 \, x^{2} + 4 \, x + 2 \, \sqrt {x^{4} + 4 \, x^{3} + 13 \, x^{2} + 18 \, x + 16} + 9\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 46, normalized size = 1.18 \begin {gather*} -\frac {1}{2} \, \log \left (\sqrt {2} {\left (\sqrt {2} {\left (x^{2} + 2 \, x\right )} - 2 \, \sqrt {\frac {1}{2} \, {\left (x^{2} + 2 \, x\right )}^{2} + \frac {9}{2} \, x^{2} + 9 \, x + 8}\right )} + 9\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.72, size = 36, normalized size = 0.92
method | result | size |
trager | \(-\frac {\ln \left (-2 x^{2}+2 \sqrt {x^{4}+4 x^{3}+13 x^{2}+18 x +16}-4 x -9\right )}{2}\) | \(36\) |
default | \(-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (\left (-1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )-i \sqrt {14+2 \sqrt {17}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \frac {\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}}{\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}\) | \(1422\) |
elliptic | \(-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}-\frac {2 i \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )^{2} \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \sqrt {\frac {i \sqrt {14+2 \sqrt {17}}\, \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\, \left (\left (-1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \EllipticF \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )-i \sqrt {14+2 \sqrt {17}}\, \EllipticPi \left (\sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}, \frac {\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}}{\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}}, \sqrt {\frac {\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14+2 \sqrt {17}}}{2}-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (-\frac {i \sqrt {14-2 \sqrt {17}}}{2}+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right )}}\right )\right )}{\left (\frac {i \sqrt {14-2 \sqrt {17}}}{2}-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \sqrt {14+2 \sqrt {17}}\, \sqrt {\left (x +1+\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14+2 \sqrt {17}}}{2}\right ) \left (x +1+\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right ) \left (x +1-\frac {i \sqrt {14-2 \sqrt {17}}}{2}\right )}}\) | \(1422\) |
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*} \int \frac {x + 1}{\sqrt {x^{4} + 4 \, x^{3} + 13 \, x^{2} + 18 \, x + 16}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {x+1}{\sqrt {x^4+4\,x^3+13\,x^2+18\,x+16}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\sqrt {x^{4} + 4 x^{3} + 13 x^{2} + 18 x + 16}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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