Optimal. Leaf size=63 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{a^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.22, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2120, 329, 212, 206, 203} \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 212
Rule 329
Rule 2120
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2+x^2} \sqrt {x+\sqrt {a^2+x^2}}} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {x} \left (-a^2+x^2\right )} \, dx,x,x+\sqrt {a^2+x^2}\right )\\ &=4 \operatorname {Subst}\left (\int \frac {1}{-a^2+x^4} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )\\ &=-\frac {2 \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )}{a}-\frac {2 \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )}{a}\\ &=-\frac {2 \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )}{a^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.20, size = 56, normalized size = 0.89 \[ -\frac {2 \left (\tan ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )+\tanh ^{-1}\left (\frac {\sqrt {\sqrt {a^2+x^2}+x}}{\sqrt {a}}\right )\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.09, size = 198, normalized size = 3.14 \[ \left [-\frac {2 \, \sqrt {a} \arctan \left (\frac {\sqrt {x + \sqrt {a^{2} + x^{2}}}}{\sqrt {a}}\right ) - \sqrt {a} \log \left (\frac {a^{2} + \sqrt {a^{2} + x^{2}} a - {\left ({\left (a - x\right )} \sqrt {a} + \sqrt {a^{2} + x^{2}} \sqrt {a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right )}{a^{2}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {-a} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{a}\right ) - \sqrt {-a} \log \left (-\frac {a^{2} - \sqrt {a^{2} + x^{2}} a - {\left (\sqrt {-a} {\left (a + x\right )} - \sqrt {a^{2} + x^{2}} \sqrt {-a}\right )} \sqrt {x + \sqrt {a^{2} + x^{2}}}}{x}\right )}{a^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2} + x^{2}} \sqrt {x + \sqrt {a^{2} + x^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2}+x^{2}}\, \sqrt {x +\sqrt {a^{2}+x^{2}}}\, x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a^{2} + x^{2}} \sqrt {x + \sqrt {a^{2} + x^{2}}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{x\,\sqrt {x+\sqrt {a^2+x^2}}\,\sqrt {a^2+x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 1.47, size = 46, normalized size = 0.73 \[ - \frac {\Gamma ^{2}\left (\frac {3}{4}\right ) \Gamma \left (\frac {5}{4}\right ) {{}_{3}F_{2}\left (\begin {matrix} \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\ \frac {3}{2}, \frac {7}{4} \end {matrix}\middle | {\frac {a^{2} e^{i \pi }}{x^{2}}} \right )}}{\pi x^{\frac {3}{2}} \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________