Optimal. Leaf size=176 \[ -\sqrt {2 \left (\sqrt {2}-1\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
________________________________________________________________________________________
Rubi [C] time = 1.01, antiderivative size = 180, normalized size of antiderivative = 1.02, number of steps used = 16, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6725, 2132, 206, 2133, 725} \begin {gather*} \frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 206
Rule 725
Rule 2132
Rule 2133
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx &=\int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}-\frac {2 \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=-\left (2 \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx\right )+\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx\\ &=-\left (2 \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x) \sqrt {1+x^4}}\right ) \, dx\right )+\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i-x) \sqrt {1+x^4}} \, dx-i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i+x) \sqrt {1+x^4}} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(i-x) \sqrt {1+i x^2}} \, dx-\left (-\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(i+x) \sqrt {1+i x^2}} \, dx-\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(i-x) \sqrt {1-i x^2}} \, dx-\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(i+x) \sqrt {1-i x^2}} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\left (-\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-x}{\sqrt {1-i x^2}}\right )-\left (-\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-x}{\sqrt {1-i x^2}}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+x}{\sqrt {1+i x^2}}\right )-\left (\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+x}{\sqrt {1+i x^2}}\right )\\ &=\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{2} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {1}{2} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.92, size = 247, normalized size = 1.40 \begin {gather*} -\sqrt {2 \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )-\sqrt {2 \left (1+\sqrt {2}\right )} \tanh ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 3.91, size = 368, normalized size = 2.09 \begin {gather*} -\sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {{\left (4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{2} + 1\right )} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 2\right )} \sqrt {-8 \, \sqrt {2} + 12} - 2 \, \sqrt {2} - 4\right )} - {\left (2 \, x^{2} + \sqrt {2} {\left (x^{2} + 3\right )} + 4\right )} \sqrt {-8 \, \sqrt {2} + 12}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} - 2}}{8 \, x}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} + {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} + 2} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (\frac {2 \, \sqrt {2} x^{2} + 4 \, x^{2} - {\left (\sqrt {2} \sqrt {x^{4} + 1} x - \sqrt {2} {\left (x^{3} + x\right )} - 2 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {2 \, \sqrt {2} + 2} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}-1\right ) \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{2}+1\right ) \sqrt {x^{4}+1}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - 1\right )}}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-1\right )\,\sqrt {\sqrt {x^4+1}+x^2}}{\left (x^2+1\right )\,\sqrt {x^4+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2} + \sqrt {x^{4} + 1}}}{\left (x^{2} + 1\right ) \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________