Optimal. Leaf size=153 \[ \frac {2 x}{\sqrt {\sqrt {a x^2+b^2}+b}}-\frac {4 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {\sqrt {a x^2+b^2}+b}}-\frac {\sqrt {\sqrt {a x^2+b^2}+b}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}} \]
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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 {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx &=\int \left (\frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}}-\frac {2 b^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\\ &=-\left (\left (2 b^2\right ) \int \frac {1}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=-\left (\left (2 b^2\right ) \int \left (\frac {1}{2 b \left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}+\frac {1}{2 b \left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}}\right ) \, dx\right )+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=-\left (b \int \frac {1}{\left (b-\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx\right )-b \int \frac {1}{\left (b+\sqrt {-a} x\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.63, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b^2+a x^2}{\left (b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.26, size = 120, normalized size = 0.78 \begin {gather*} \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}-\frac {4 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}+\frac {\sqrt {2} \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}} \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 {a x^{2} - b^{2}}{{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {a \,x^{2}-b^{2}}{\left (a \,x^{2}+b^{2}\right ) \sqrt {b +\sqrt {a \,x^{2}+b^{2}}}}\, dx\]
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 {a x^{2} - b^{2}}{{\left (a x^{2} + b^{2}\right )} \sqrt {b + \sqrt {a x^{2} + b^{2}}}}\,{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 {a\,x^2-b^2}{\left (b^2+a\,x^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}} \,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 {a x^{2} - b^{2}}{\sqrt {b + \sqrt {a x^{2} + b^{2}}} \left (a x^{2} + b^{2}\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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