Optimal. Leaf size=100 \[ \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}} \]
________________________________________________________________________________________
Rubi [A] time = 0.12, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2119, 453, 329, 298, 203, 206} \begin {gather*} \frac {2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 298
Rule 329
Rule 453
Rule 2119
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx &=\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2} \left (-b^2+x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+2 \operatorname {Subst}\left (\int \frac {\sqrt {x}}{-b^2+x^2} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+4 \operatorname {Subst}\left (\int \frac {x^2}{-b^2+x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\\ &=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}-2 \operatorname {Subst}\left (\int \frac {1}{b-x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{b+x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )\\ &=\frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 2.19, size = 204, normalized size = 2.04 \begin {gather*} \frac {2}{3} \sqrt {\sqrt {a^2 x^2+b^2}+a x} \left (\frac {\sqrt {a^2 x^2+b^2}-2 a x}{b^2}-\frac {\left (a^2 x^2+b^2\right ) \left (\sqrt {a^2 x^2+b^2}+a x\right ) \left (2 \left (2 a x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^2}{b^2}\right )-a x \left (\sqrt {a^2 x^2+b^2}+a x\right )-2 b^2\right )}{\left (a b x \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^3\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.12, size = 100, normalized size = 1.00 \begin {gather*} \frac {2}{\sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {b}}\right )}{\sqrt {b}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.67, size = 321, normalized size = 3.21 \begin {gather*} \left [\frac {2 \, b^{\frac {3}{2}} \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}{\sqrt {b}}\right ) + b^{\frac {3}{2}} \log \left (\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x - b\right )} \sqrt {b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {b}\right )} + \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}, \frac {2 \, \sqrt {-b} b \arctan \left (\frac {\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \sqrt {-b}}{b}\right ) - \sqrt {-b} b \log \left (-\frac {b^{2} + \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left ({\left (a x + b\right )} \sqrt {-b} - \sqrt {a^{2} x^{2} + b^{2}} \sqrt {-b}\right )} - \sqrt {a^{2} x^{2} + b^{2}} b}{x}\right ) - 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{b^{2}}\right ] \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*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]
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*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x\,\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}} \,d x \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 {1}{x \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________