Optimal. Leaf size=31 \[ -\frac {\sqrt {-b x^2} \log \left (\cos ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]
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Rubi [A] time = 0.06, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {4835, 4640} \[ -\frac {\sqrt {-b x^2} \log \left (\cos ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{b x} \]
Antiderivative was successfully verified.
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Rule 4640
Rule 4835
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {1+b x^2} \cos ^{-1}\left (\sqrt {1+b x^2}\right )} \, dx &=\frac {\sqrt {-b x^2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \cos ^{-1}(x)} \, dx,x,\sqrt {1+b x^2}\right )}{b x}\\ &=-\frac {\sqrt {-b x^2} \log \left (\cos ^{-1}\left (\sqrt {1+b x^2}\right )\right )}{b x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 25, normalized size = 0.81 \[ \frac {x \log \left (\cos ^{-1}\left (\sqrt {b x^2+1}\right )\right )}{\sqrt {-b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 27, normalized size = 0.87 \[ -\frac {\sqrt {-b x^{2}} \log \left (\arccos \left (\sqrt {b x^{2} + 1}\right )\right )}{b x} \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \arccos \left (\sqrt {b x^{2} + 1}\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {1}{\arccos \left (\sqrt {b \,x^{2}+1}\right ) \sqrt {b \,x^{2}+1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 25, normalized size = 0.81 \[ \frac {\ln \left (\mathrm {acos}\left (\sqrt {b\,x^2+1}\right )\right )\,\sqrt {x^2}}{\sqrt {-b}\,x} \]
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 \[ \int \frac {1}{\sqrt {b x^{2} + 1} \operatorname {acos}{\left (\sqrt {b x^{2} + 1} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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