Optimal. Leaf size=269 \[ \frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} (6 b-16 a x)+\sqrt {a^2 x^2-b} \left (8 \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}-16 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+(8 a x-9 b) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{12 a \sqrt {a^2 x^2-b}+12 a^2 x}+\frac {3 b \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{4 a} \]
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Rubi [F] time = 0.08, 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 {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx &=\int \frac {1}{\sqrt {1+\sqrt {a x+\sqrt {-b+a^2 x^2}}}} \, dx\\ \end {align*}
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Mathematica [A] time = 1.23, size = 188, normalized size = 0.70 \begin {gather*} \frac {2 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} \left (-\frac {9 b}{\sqrt {\sqrt {a^2 x^2-b}+a x}}+\frac {6 b}{\sqrt {a^2 x^2-b}+a x}+8 \sqrt {\sqrt {a^2 x^2-b}+a x}-16\right )-9 b \log \left (1-\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}\right )+9 b \log \left (\frac {1}{\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}+1\right )}{24 a} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.46, size = 269, normalized size = 1.00 \begin {gather*} \frac {\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1} (6 b-16 a x)+\sqrt {a^2 x^2-b} \left (8 \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}-16 \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )+(8 a x-9 b) \sqrt {\sqrt {a^2 x^2-b}+a x} \sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}}{12 a \sqrt {a^2 x^2-b}+12 a^2 x}+\frac {3 b \tanh ^{-1}\left (\sqrt {\sqrt {\sqrt {a^2 x^2-b}+a x}+1}\right )}{4 a} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 152, normalized size = 0.57 \begin {gather*} \frac {9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} + 1\right ) - 9 \, b \log \left (\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1} - 1\right ) + 2 \, {\left (6 \, a x - {\left (9 \, a x - 9 \, \sqrt {a^{2} x^{2} - b} - 8\right )} \sqrt {a x + \sqrt {a^{2} x^{2} - b}} - 6 \, \sqrt {a^{2} x^{2} - b} - 16\right )} \sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}{24 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [F] time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {1+\sqrt {a x +\sqrt {a^{2} x^{2}-b}}}}\, dx \end {gather*}
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 {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 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 {1}{\sqrt {\sqrt {a\,x+\sqrt {a^2\,x^2-b}}+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 {1}{\sqrt {\sqrt {a x + \sqrt {a^{2} x^{2} - b}} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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