Optimal. Leaf size=553 \[ -\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{a^2}+\frac {\left (a^4+\sqrt {a^4+1}+1\right ) \tan ^{-1}\left (\frac {a \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {-\sqrt {a^4+1}-1}}\right )}{a^2 \sqrt {a^4+1} \sqrt {-\sqrt {a^4+1}-1}}+\frac {\left (-a^4+\sqrt {a^4+1}-1\right ) \tan ^{-1}\left (\frac {a \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {a^4+1}-1}}\right )}{a^2 \sqrt {a^4+1} \sqrt {\sqrt {a^4+1}-1}}+\frac {\left (-\sqrt {2} \sqrt {-\sqrt {a^4+1}-a^2} a^2+\sqrt {2} \sqrt {a^4+1} \sqrt {-\sqrt {a^4+1}-a^2}+\sqrt {2} \sqrt {-\sqrt {a^4+1}-a^2}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {-\sqrt {a^4+1}-a^2} \left (\sqrt {x^4+1}+x^2-1\right )}\right )}{2 a^2}+\frac {\left (\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2} a^2+\sqrt {2} \sqrt {a^4+1} \sqrt {\sqrt {a^4+1}-a^2}-\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a^4+1}-a^2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{2 a^2}+\frac {\sqrt {\sqrt {x^4+1}+x^2}}{a} \]
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Rubi [F] time = 0.10, 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 {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx\\ \end {align*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{1+a x} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 5.75, size = 553, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^2+\sqrt {1+x^4}}}{a}+\frac {\left (1+a^4+\sqrt {1+a^4}\right ) \tan ^{-1}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1-\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1-\sqrt {1+a^4}}}+\frac {\left (-1-a^4+\sqrt {1+a^4}\right ) \tan ^{-1}\left (\frac {a \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-1+\sqrt {1+a^4}}}\right )}{a^2 \sqrt {1+a^4} \sqrt {-1+\sqrt {1+a^4}}}+\frac {\left (\sqrt {2} \sqrt {-a^2-\sqrt {1+a^4}}-\sqrt {2} a^2 \sqrt {-a^2-\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2-\sqrt {1+a^4}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {-a^2-\sqrt {1+a^4}} \left (-1+x^2+\sqrt {1+x^4}\right )}\right )}{2 a^2}+\frac {\left (-\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} a^2 \sqrt {-a^2+\sqrt {1+a^4}}+\sqrt {2} \sqrt {1+a^4} \sqrt {-a^2+\sqrt {1+a^4}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-a^2+\sqrt {1+a^4}} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{2 a^2}-\frac {\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}}{1+x^2+\sqrt {1+x^4}}\right )}{a^2} \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*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{a x +1}\, dx\]
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*} \int \frac {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 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 = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{a\,x+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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{a x + 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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