Optimal. Leaf size=140 \[ -\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\& ,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^7}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\& \right ] \]
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Rubi [C] Result contains complex when optimal does not.
time = 0.89, antiderivative size = 513, normalized size of antiderivative = 3.66, number of steps
used = 39, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {6857, 2142,
14, 2144, 1642, 840, 1180, 210, 212, 213, 209} \begin {gather*} -(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \text {ArcTan}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )+(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \text {ArcTan}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )-\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \text {ArcTan}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )+\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \text {ArcTan}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )+\frac {1}{3} \left (\sqrt {x^2+1}+x\right )^{3/2}-\frac {1}{\sqrt {\sqrt {x^2+1}+x}}+(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )-(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )+\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )-\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^2+1}+x}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 209
Rule 210
Rule 212
Rule 213
Rule 840
Rule 1180
Rule 1642
Rule 2142
Rule 2144
Rule 6857
Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right ) \sqrt {x+\sqrt {1+x^2}}}{1+x^4} \, dx &=\int \left (\sqrt {x+\sqrt {1+x^2}}-\frac {2 \sqrt {x+\sqrt {1+x^2}}}{1+x^4}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {x+\sqrt {1+x^2}}}{1+x^4} \, dx\right )+\int \sqrt {x+\sqrt {1+x^2}} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1+x^2}{x^{3/2}} \, dx,x,x+\sqrt {1+x^2}\right )-2 \int \left (\frac {i \sqrt {x+\sqrt {1+x^2}}}{2 \left (i-x^2\right )}+\frac {i \sqrt {x+\sqrt {1+x^2}}}{2 \left (i+x^2\right )}\right ) \, dx\\ &=-\left (i \int \frac {\sqrt {x+\sqrt {1+x^2}}}{i-x^2} \, dx\right )-i \int \frac {\sqrt {x+\sqrt {1+x^2}}}{i+x^2} \, dx+\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-i \int \left (-\frac {(-1)^{3/4} \sqrt {x+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}-x\right )}-\frac {(-1)^{3/4} \sqrt {x+\sqrt {1+x^2}}}{2 \left (\sqrt [4]{-1}+x\right )}\right ) \, dx-i \int \left (-\frac {\sqrt [4]{-1} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}-x\right )}-\frac {\sqrt [4]{-1} \sqrt {x+\sqrt {1+x^2}}}{2 \left (-(-1)^{3/4}+x\right )}\right ) \, dx\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt [4]{-1}-x} \, dx-\frac {1}{2} \sqrt [4]{-1} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt [4]{-1}+x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{-(-1)^{3/4}-x} \, dx+\frac {1}{2} (-1)^{3/4} \int \frac {\sqrt {x+\sqrt {1+x^2}}}{-(-1)^{3/4}+x} \, dx\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \frac {1+x^2}{\sqrt {x} \left (-1-2 (-1)^{3/4} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1+\sqrt [4]{-1} x\right )}{\sqrt {x} \left (1+2 \sqrt [4]{-1} x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )-\frac {1}{2} \sqrt [4]{-1} \text {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1-\sqrt [4]{-1} x\right )}{\sqrt {x} \left (-1+2 \sqrt [4]{-1} x+x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \left (-\frac {1}{\sqrt {x}}+\frac {2 \left (1-(-1)^{3/4} x\right )}{\sqrt {x} \left (1-2 (-1)^{3/4} x-x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )+\frac {1}{2} (-1)^{3/4} \text {Subst}\left (\int \left (\frac {1}{\sqrt {x}}+\frac {2 \left (1+(-1)^{3/4} x\right )}{\sqrt {x} \left (-1-2 (-1)^{3/4} x+x^2\right )}\right ) \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1+\sqrt [4]{-1} x}{\sqrt {x} \left (1+2 \sqrt [4]{-1} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )-\sqrt [4]{-1} \text {Subst}\left (\int \frac {1-\sqrt [4]{-1} x}{\sqrt {x} \left (-1+2 \sqrt [4]{-1} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1-(-1)^{3/4} x}{\sqrt {x} \left (1-2 (-1)^{3/4} x-x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )+(-1)^{3/4} \text {Subst}\left (\int \frac {1+(-1)^{3/4} x}{\sqrt {x} \left (-1-2 (-1)^{3/4} x+x^2\right )} \, dx,x,x+\sqrt {1+x^2}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-\left (2 \sqrt [4]{-1}\right ) \text {Subst}\left (\int \frac {1+\sqrt [4]{-1} x^2}{1+2 \sqrt [4]{-1} x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (2 \sqrt [4]{-1}\right ) \text {Subst}\left (\int \frac {1-\sqrt [4]{-1} x^2}{-1+2 \sqrt [4]{-1} x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (2 (-1)^{3/4}\right ) \text {Subst}\left (\int \frac {1-(-1)^{3/4} x^2}{1-2 (-1)^{3/4} x^2-x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (2 (-1)^{3/4}\right ) \text {Subst}\left (\int \frac {1+(-1)^{3/4} x^2}{-1-2 (-1)^{3/4} x^2+x^4} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+\left ((-1)^{3/4} \left (-(-1)^{3/4}-\sqrt {1-i}\right )\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left ((-1)^{3/4} \left ((-1)^{3/4}+\sqrt {1-i}\right )\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}-\sqrt {1-i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (i+(-1)^{3/4} \sqrt {1-i}\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (-\sqrt [4]{-1}-\sqrt {1+i}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (\sqrt [4]{-1}-\sqrt {1+i}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (-\sqrt [4]{-1}+\sqrt {1+i}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}-\sqrt {1+i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )-\left (\sqrt [4]{-1} \left (\sqrt [4]{-1}+\sqrt {1+i}\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{-1}+\sqrt {1+i}-x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )+\left (\left (\frac {1}{2}-\frac {i}{2}\right ) \left ((1-i)+\sqrt {2-2 i}\right )\right ) \text {Subst}\left (\int \frac {1}{-(-1)^{3/4}+\sqrt {1-i}+x^2} \, dx,x,\sqrt {x+\sqrt {1+x^2}}\right )\\ &=-\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}+(1-i) \sqrt {\frac {1}{2} \left (-(-1)^{3/4}+\sqrt {1-i}\right )} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )+(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )-(1+i) \sqrt {\frac {1}{2} \left (-\sqrt [4]{-1}+\sqrt {1+i}\right )} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )+\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )+(-1)^{3/4} \sqrt {-(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-(-1)^{3/4}+\sqrt {1-i}}}\right )-(-1)^{3/4} \sqrt {(-1)^{3/4}+\sqrt {1-i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {(-1)^{3/4}+\sqrt {1-i}}}\right )+\sqrt [4]{-1} \sqrt {-\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {-\sqrt [4]{-1}+\sqrt {1+i}}}\right )-\sqrt [4]{-1} \sqrt {\sqrt [4]{-1}+\sqrt {1+i}} \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {1+x^2}}}{\sqrt {\sqrt [4]{-1}+\sqrt {1+i}}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 140, normalized size = 1.00 \begin {gather*} -\frac {1}{\sqrt {x+\sqrt {1+x^2}}}+\frac {1}{3} \left (x+\sqrt {1+x^2}\right )^{3/2}-2 \text {RootSum}\left [1-4 \text {$\#$1}^4+22 \text {$\#$1}^8-4 \text {$\#$1}^{12}+\text {$\#$1}^{16}\&,\frac {\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^3+\log \left (\sqrt {x+\sqrt {1+x^2}}-\text {$\#$1}\right ) \text {$\#$1}^7}{-1+11 \text {$\#$1}^4-3 \text {$\#$1}^8+\text {$\#$1}^{12}}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right ) \sqrt {x +\sqrt {x^{2}+1}}}{x^{4}+1}\, dx\]
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 [C] Result contains higher order function than in optimal. Order 3 vs. order
1.
time = 1.93, size = 3304, normalized size = 23.60 \begin {gather*} \text {Too large to display} \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 ) \sqrt {x + \sqrt {x^{2} + 1}} \left (x^{2} + 1\right )}{x^{4} + 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 = -0.01 \begin {gather*} \int \frac {\left (x^4-1\right )\,\sqrt {x+\sqrt {x^2+1}}}{x^4+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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