Optimal. Leaf size=224 \[ \frac {32}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {32}{105} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\sqrt {1+x} \left (-\frac {48}{35} \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{7} \sqrt {1+\sqrt {1+x}} \sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\& ,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{-\text {$\#$1}+4 \text {$\#$1}^5-4 \text {$\#$1}^7+\text {$\#$1}^9}\& \right ] \]
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Rubi [F]
time = 1.93, 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 {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \text {Subst}\left (\int \frac {x^3 \left (-2+x^2\right )}{\left (2-2 x^2+x^4\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \text {Subst}\left (\int \frac {x \left (-1+x^2\right )^3 \left (-1-2 x^2+x^4\right )}{\sqrt {1+x} \left (1+4 x^4-4 x^6+x^8\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \text {Subst}\left (\int \frac {(-1+x)^3 x (1+x)^{5/2} \left (-1-2 x^2+x^4\right )}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=8 \text {Subst}\left (\int \frac {x^6 \left (-2+x^2\right )^3 \left (-1+x^2\right ) \left (-2+4 x^4-4 x^6+x^8\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=8 \text {Subst}\left (\int \left (2 x^2-3 x^4+x^6-\frac {2 x^2 \left (2-3 x^2+x^4\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-16 \text {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\right )}{1+4 \left (-1+x^2\right )^4-4 \left (-1+x^2\right )^6+\left (-1+x^2\right )^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-16 \text {Subst}\left (\int \left (\frac {2 x^2}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}-\frac {3 x^4}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^6}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}\right ) \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=\frac {16}{3} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{3/2}-\frac {24}{5} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{5/2}+\frac {8}{7} \left (1+\sqrt {1+\sqrt {1+x}}\right )^{7/2}-16 \text {Subst}\left (\int \frac {x^6}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-32 \text {Subst}\left (\int \frac {x^2}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+48 \text {Subst}\left (\int \frac {x^4}{2-8 x^4+8 x^6+14 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 168, normalized size = 0.75 \begin {gather*} \frac {8}{105} \sqrt {1+\sqrt {1+\sqrt {1+x}}} \left (4-18 \sqrt {1+x}+4 \sqrt {1+\sqrt {1+x}}+15 \sqrt {1+x} \sqrt {1+\sqrt {1+x}}\right )-\text {RootSum}\left [2-8 \text {$\#$1}^4+8 \text {$\#$1}^6+14 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{-\text {$\#$1}+4 \text {$\#$1}^5-4 \text {$\#$1}^7+\text {$\#$1}^9}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
1.
time = 0.18, size = 157, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\) | \(157\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}-\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+14 \textit {\_Z}^{8}+8 \textit {\_Z}^{6}-8 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+7 \textit {\_R}^{7}+3 \textit {\_R}^{5}-2 \textit {\_R}^{3}}\right )\) | \(157\) |
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 [F(-1)] Timed out
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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt {\sqrt {\sqrt {x + 1} + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x^2-1}{\left (x^2+1\right )\,\sqrt {\sqrt {\sqrt {x+1}+1}+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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