Optimal. Leaf size=27 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-\pi }}\right )}{\sqrt {b}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {217, 206} \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-\pi }}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-\pi +b x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {-\pi +b x^2}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {-\pi +b x^2}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {b x^2-\pi }}\right )}{\sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 74, normalized size = 2.74 \[ \left [\frac {\log \left (-\pi + 2 \, b x^{2} + 2 \, \sqrt {-\pi + b x^{2}} \sqrt {b} x\right )}{2 \, \sqrt {b}}, -\frac {\sqrt {-b} \arctan \left (-\frac {\sqrt {-\pi + b x^{2}} \sqrt {-b} x}{\pi - b x^{2}}\right )}{b}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 25, normalized size = 0.93 \[ -\frac {\log \left ({\left | -\sqrt {b} x + \sqrt {-\pi + b x^{2}} \right |}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 23, normalized size = 0.85 \[ \frac {\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}-\pi }\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 26, normalized size = 0.96 \[ \frac {\log \left (2 \, b x + 2 \, \sqrt {-\pi + b x^{2}} \sqrt {b}\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 22, normalized size = 0.81 \[ \frac {\ln \left (\sqrt {b\,x^2-\Pi }+\sqrt {b}\,x\right )}{\sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.05, size = 46, normalized size = 1.70 \[ \begin {cases} \frac {\operatorname {acosh}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} & \text {for}\: \frac {\left |{b x^{2}}\right |}{\pi } > 1 \\- \frac {i \operatorname {asin}{\left (\frac {\sqrt {b} x}{\sqrt {\pi }} \right )}}{\sqrt {b}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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