Optimal. Leaf size=130 \[ -\text {RootSum}\left [\text {$\#$1}^8-4 \text {$\#$1}^6+4 \text {$\#$1}^4+1\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^5-2 \text {$\#$1}^3}\& \right ]+\frac {4}{15} \sqrt {\sqrt {x+1}+1} (3 x+11)-\frac {16}{15} \sqrt {x+1} \sqrt {\sqrt {x+1}+1} \]
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Rubi [F] time = 0.98, 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} \left (-1+x^2\right )}{\left (1+x^2\right ) \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 {\sqrt {1+x} \left (-1+x^2\right )}{\left (1+x^2\right ) \sqrt {1+\sqrt {1+x}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^4 \left (-2+x^2\right )}{\sqrt {1+x} \left (2-2 x^2+x^4\right )} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^4 \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 )\\ &=4 \operatorname {Subst}\left (\int \left (1-2 x^2+x^4-\frac {2 \left (1-2 x^2+x^4\right )}{1+4 x^4-4 x^6+x^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-\frac {8}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \frac {1-2 x^2+x^4}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-\frac {8}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \frac {\left (-1+x^2\right )^2}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-\frac {8}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \left (\frac {1}{1+4 x^4-4 x^6+x^8}-\frac {2 x^2}{1+4 x^4-4 x^6+x^8}+\frac {x^4}{1+4 x^4-4 x^6+x^8}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \sqrt {1+\sqrt {1+x}}-\frac {8}{3} \left (1+\sqrt {1+x}\right )^{3/2}+\frac {4}{5} \left (1+\sqrt {1+x}\right )^{5/2}-8 \operatorname {Subst}\left (\int \frac {1}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-8 \operatorname {Subst}\left (\int \frac {x^4}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+16 \operatorname {Subst}\left (\int \frac {x^2}{1+4 x^4-4 x^6+x^8} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.71, size = 241, normalized size = 1.85 \begin {gather*} -\frac {4}{15} \sqrt {\sqrt {x+1}+1} \left (-3 x+4 \sqrt {x+1}-11\right )-\frac {4 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1-\sqrt {1-i}}}\right )}{(1-i)^{3/2} \sqrt {1-\sqrt {1-i}}}+\frac {4 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1+\sqrt {1-i}}}\right )}{(1-i)^{3/2} \sqrt {1+\sqrt {1-i}}}+2 i \sqrt {\frac {1+i}{1-\sqrt {1+i}}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1-\sqrt {1+i}}}\right )+\frac {4 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x+1}+1}}{\sqrt {1+\sqrt {1+i}}}\right )}{(1+i)^{3/2} \sqrt {1+\sqrt {1+i}}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.12, size = 117, normalized size = 0.90 \begin {gather*} \frac {4}{15} \sqrt {1+\sqrt {1+x}} \left (8-4 \sqrt {1+x}+3 (1+x)\right )-\text {RootSum}\left [1+4 \text {$\#$1}^4-4 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.69, size = 1268, normalized size = 9.75
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*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.09, size = 97, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(97\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}+4 \sqrt {1+\sqrt {1+x}}+\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{8}-4 \textit {\_Z}^{6}+4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (-\textit {\_R}^{4}+2 \textit {\_R}^{2}-1\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{7}-3 \textit {\_R}^{5}+2 \textit {\_R}^{3}}\right )\) | \(97\) |
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 {{\left (x^{2} - 1\right )} \sqrt {x + 1}}{{\left (x^{2} + 1\right )} \sqrt {\sqrt {x + 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 {\left (x^2-1\right )\,\sqrt {x+1}}{\left (x^2+1\right )\,\sqrt {\sqrt {x+1}+1}} \,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 {\left (x - 1\right ) \left (x + 1\right )^{\frac {3}{2}}}{\left (x^{2} + 1\right ) \sqrt {\sqrt {x + 1} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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