Optimal. Leaf size=161 \[ -\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+16 \text {$\#$1}^8+1\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^{11}-6 \text {$\#$1}^9+12 \text {$\#$1}^7-8 \text {$\#$1}^5}\& \right ]+\frac {4}{105} \sqrt {\sqrt {x+1}+1} (3 x+11)+\frac {4}{105} \sqrt {x+1} (15 x+11) \sqrt {\sqrt {x+1}+1} \]
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Rubi [F] time = 1.03, 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^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, 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^4\right ) \sqrt {1+\sqrt {1+x}}}{1+x^4} \, dx &=2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {1+x} \left (-1+\left (-1+x^2\right )^4\right )}{1+\left (-1+x^2\right )^4} \, dx,x,\sqrt {1+x}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2 \left (-1+x^8 \left (-2+x^2\right )^4\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=4 \operatorname {Subst}\left (\int \left (x^2-2 x^4+x^6-\frac {2 x^2 \left (1-2 x^2+x^4\right )}{1+x^8 \left (-2+x^2\right )^4}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {1+x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {1+x}\right )^{7/2}-8 \operatorname {Subst}\left (\int \frac {x^2 \left (1-2 x^2+x^4\right )}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {1+x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {1+x}\right )^{7/2}-8 \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{1+x^8 \left (-2+x^2\right )^4} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {1+x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {1+x}\right )^{7/2}-8 \operatorname {Subst}\left (\int \left (\frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}-\frac {2 x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}+\frac {x^6}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}}\right ) \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ &=\frac {4}{3} \left (1+\sqrt {1+x}\right )^{3/2}-\frac {8}{5} \left (1+\sqrt {1+x}\right )^{5/2}+\frac {4}{7} \left (1+\sqrt {1+x}\right )^{7/2}-8 \operatorname {Subst}\left (\int \frac {x^2}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )-8 \operatorname {Subst}\left (\int \frac {x^6}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )+16 \operatorname {Subst}\left (\int \frac {x^4}{1+16 x^8-32 x^{10}+24 x^{12}-8 x^{14}+x^{16}} \, dx,x,\sqrt {1+\sqrt {1+x}}\right )\\ \end {align*}
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Mathematica [A] time = 0.26, size = 153, normalized size = 0.95 \begin {gather*} \frac {4}{105} \sqrt {\sqrt {x+1}+1} \left (11 \left (\sqrt {x+1}+1\right )+3 x \left (5 \sqrt {x+1}+1\right )\right )-\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^{16}-8 \text {$\#$1}^{14}+24 \text {$\#$1}^{12}-32 \text {$\#$1}^{10}+16 \text {$\#$1}^8+1\&,\frac {\text {$\#$1}^2 \log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )-\log \left (\sqrt {\sqrt {x+1}+1}-\text {$\#$1}\right )}{\text {$\#$1}^{11}-6 \text {$\#$1}^9+12 \text {$\#$1}^7-8 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 152, normalized size = 0.94 \begin {gather*} \frac {4}{105} \sqrt {1+\sqrt {1+x}} \left (8-4 \sqrt {1+x}+3 (1+x)+15 (1+x)^{3/2}\right )-\frac {1}{2} \text {RootSum}\left [1+16 \text {$\#$1}^8-32 \text {$\#$1}^{10}+24 \text {$\#$1}^{12}-8 \text {$\#$1}^{14}+\text {$\#$1}^{16}\&,\frac {-\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right )+\log \left (\sqrt {1+\sqrt {1+x}}-\text {$\#$1}\right ) \text {$\#$1}^2}{-8 \text {$\#$1}^5+12 \text {$\#$1}^7-6 \text {$\#$1}^9+\text {$\#$1}^{11}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [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.
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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 = 0.10, size = 119, normalized size = 0.74
method | result | size |
derivativedivides | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {7}{2}}}{7}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}+\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4}+\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(119\) |
default | \(\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {7}{2}}}{7}-\frac {8 \left (1+\sqrt {1+x}\right )^{\frac {5}{2}}}{5}+\frac {4 \left (1+\sqrt {1+x}\right )^{\frac {3}{2}}}{3}-\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{16}-8 \textit {\_Z}^{14}+24 \textit {\_Z}^{12}-32 \textit {\_Z}^{10}+16 \textit {\_Z}^{8}+1\right )}{\sum }\frac {\left (\textit {\_R}^{6}-2 \textit {\_R}^{4}+\textit {\_R}^{2}\right ) \ln \left (\sqrt {1+\sqrt {1+x}}-\textit {\_R} \right )}{\textit {\_R}^{15}-7 \textit {\_R}^{13}+18 \textit {\_R}^{11}-20 \textit {\_R}^{9}+8 \textit {\_R}^{7}}\right )}{2}\) | \(119\) |
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^{4} - 1\right )} \sqrt {x + 1} \sqrt {\sqrt {x + 1} + 1}}{x^{4} + 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^4-1\right )\,\sqrt {\sqrt {x+1}+1}\,\sqrt {x+1}}{x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SympifyError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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