Optimal. Leaf size=181 \[ 2 \text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4-2\& ,\frac {\log \left (\sqrt {\sqrt {\sqrt {x+1}+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}}\& \right ]+\sqrt {x+1} \left (\frac {8}{7} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}-\frac {48}{35} \sqrt {\sqrt {\sqrt {x+1}+1}+1}\right )+\frac {32}{105} \sqrt {\sqrt {x+1}+1} \sqrt {\sqrt {\sqrt {x+1}+1}+1}+\frac {32}{105} \sqrt {\sqrt {\sqrt {x+1}+1}+1} \]
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Rubi [F] time = 1.58, 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}{(-1+x) \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}{(-1+x) \sqrt {1+\sqrt {1+\sqrt {1+x}}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^3}{\left (-2+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x \left (-1+x^2\right )^3}{\sqrt {1+x} \left (-2+\left (-1+x^2\right )^2\right )} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {(-1+x)^3 x (1+x)^{5/2}}{-2+\left (-1+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=8 \operatorname {Subst}\left (\int \frac {x^6 \left (-2+x^2\right )^3 \left (-1+x^2\right )}{-2+x^4 \left (-2+x^2\right )^2} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ &=8 \operatorname {Subst}\left (\int \left (2 x^2-3 x^4+x^6+\frac {2 x^2 \left (2-3 x^2+x^4\right )}{-2+x^4 \left (-2+x^2\right )^2}\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 \operatorname {Subst}\left (\int \frac {x^2 \left (2-3 x^2+x^4\right )}{-2+x^4 \left (-2+x^2\right )^2} \, 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 \operatorname {Subst}\left (\int \left (\frac {2 x^2}{-2+4 x^4-4 x^6+x^8}-\frac {3 x^4}{-2+4 x^4-4 x^6+x^8}+\frac {x^6}{-2+4 x^4-4 x^6+x^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 \operatorname {Subst}\left (\int \frac {x^6}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )+32 \operatorname {Subst}\left (\int \frac {x^2}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )-48 \operatorname {Subst}\left (\int \frac {x^4}{-2+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+\sqrt {1+x}}}\right )\\ \end {align*}
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Mathematica [A] time = 0.59, size = 205, normalized size = 1.13 \begin {gather*} 8 \left (\frac {1}{4} \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4+4 \text {$\#$1}^2-1\&,\frac {2 \text {$\#$1}^4 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )-3 \text {$\#$1}^2 \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )+\log \left (\frac {1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}}-\text {$\#$1}\right )}{2 \text {$\#$1}^7-2 \text {$\#$1}^3+\text {$\#$1}}\&\right ]+\frac {1}{7} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{7/2}-\frac {3}{5} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{5/2}+\frac {2}{3} \left (\sqrt {\sqrt {x+1}+1}+1\right )^{3/2}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 135, normalized size = 0.75 \begin {gather*} -\frac {16}{105} \left (-2+9 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}+\frac {8}{105} \sqrt {1+\sqrt {1+x}} \left (4+15 \sqrt {1+x}\right ) \sqrt {1+\sqrt {1+\sqrt {1+x}}}+2 \text {RootSum}\left [-2+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.89, size = 2096, normalized size = 11.58
result too large to display
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*} \int \frac {x + 1}{{\left (x - 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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maple [A] time = 0.18, size = 117, normalized size = 0.65
method | result | size |
derivativedivides | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(117\) |
default | \(\frac {8 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {7}{2}}}{7}-\frac {24 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {5}{2}}}{5}+\frac {16 \left (1+\sqrt {1+\sqrt {1+x}}\right )^{\frac {3}{2}}}{3}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}-2\right )}{\sum }\frac {\left (\textit {\_R}^{6}-3 \textit {\_R}^{4}+2 \textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+\sqrt {1+x}}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(117\) |
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}{{\left (x - 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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x+1}{\sqrt {\sqrt {\sqrt {x+1}+1}+1}\,\left (x-1\right )} \,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}{\left (x - 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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