Optimal. Leaf size=47 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2157, 212}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {\sqrt {a+b^2 x^4}+b x^2}}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2157
Rubi steps
\begin {align*} \int \frac {\sqrt {b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {a+b^2 x^4}} \, dx &=\text {Subst}\left (\int \frac {1}{1-2 b x^2} \, dx,x,\frac {x}{\sqrt {b x^2+\sqrt {a+b^2 x^4}}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {b} x}{\sqrt {b x^2+\sqrt {a+b^2 x^4}}}\right )}{\sqrt {2} \sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 49, normalized size = 1.04 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {b x^2+\sqrt {a+b^2 x^4}}}{\sqrt {2} \sqrt {b} x}\right )}{\sqrt {2} \sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {b \,x^{2}+\sqrt {b^{2} x^{4}+a}}}{\sqrt {b^{2} x^{4}+a}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 1.15, size = 135, normalized size = 2.87 \begin {gather*} \left [\frac {\sqrt {2} \log \left (4 \, b^{2} x^{4} + 4 \, \sqrt {b^{2} x^{4} + a} b x^{2} + 2 \, {\left (\sqrt {2} b^{\frac {3}{2}} x^{3} + \sqrt {2} \sqrt {b^{2} x^{4} + a} \sqrt {b} x\right )} \sqrt {b x^{2} + \sqrt {b^{2} x^{4} + a}} + a\right )}{4 \, \sqrt {b}}, -\frac {1}{2} \, \sqrt {2} \sqrt {-\frac {1}{b}} \arctan \left (\frac {\sqrt {2} \sqrt {b x^{2} + \sqrt {b^{2} x^{4} + a}} \sqrt {-\frac {1}{b}}}{2 \, x}\right )\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b x^{2} + \sqrt {a + b^{2} x^{4}}}}{\sqrt {a + b^{2} x^{4}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {\sqrt {b^2\,x^4+a}+b\,x^2}}{\sqrt {b^2\,x^4+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________