Optimal. Leaf size=541 \[ \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {2 \sqrt {2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}+\frac {2 \sqrt {-1+\sqrt {2}} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (-1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}}-\frac {2 \sqrt {-1+\sqrt {2}} \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}-\frac {\sqrt {b+\sqrt {b^2+a x^2}}}{\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {b}}\right )}{\sqrt {a}} \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.69, 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.
[In]
[Out]
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 (2 b^2\right ) \int \frac {1}{\left (-b^2+a x^2\right ) \sqrt {b+\sqrt {b^2+a x^2}}} \, dx+\int \frac {1}{\sqrt {b+\sqrt {b^2+a x^2}}} \, dx\\ &=\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+\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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.64, size = 208, normalized size = 0.38 \begin {gather*} \frac {2 x}{\sqrt {b+\sqrt {b^2+a x^2}}}+\frac {\sqrt {2} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {a} x}{\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {1+\sqrt {2}} \sqrt {b} \text {ArcTan}\left (\frac {\sqrt {-1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {a}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {2}} \sqrt {a} x}{\sqrt {b} \sqrt {b+\sqrt {b^2+a x^2}}}\right )}{\sqrt {1+\sqrt {2}} \sqrt {a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.02, 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.
[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 [F(-1)] Timed out
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.
[In]
[Out]
________________________________________________________________________________________
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.
[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 = -0.00 \begin {gather*} \int \frac {b^2+a\,x^2}{\left (a\,x^2-b^2\right )\,\sqrt {b+\sqrt {b^2+a\,x^2}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________