Optimal. Leaf size=63 \[ -\frac {2 \tan ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )}{a^{3/2}} \]
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Rubi [A]
time = 0.13, 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 = {2145, 335, 218,
212, 209} \begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 212
Rule 218
Rule 335
Rule 2145
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt {a^2+x^2} \sqrt {x+\sqrt {a^2+x^2}}} \, dx &=2 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (-a^2+x^2\right )} \, dx,x,x+\sqrt {a^2+x^2}\right )\\ &=4 \text {Subst}\left (\int \frac {1}{-a^2+x^4} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {x+\sqrt {a^2+x^2}}\right )}{a}-\frac {2 \text {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*}
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Mathematica [A]
time = 0.06, size = 56, normalized size = 0.89 \begin {gather*} -\frac {2 \left (\tan ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )+\tanh ^{-1}\left (\frac {\sqrt {x+\sqrt {a^2+x^2}}}{\sqrt {a}}\right )\right )}{a^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 2.80, size = 27, normalized size = 0.43 \begin {gather*} -\frac {\sqrt {2} \text {hyper}\left [\left \{\frac {3}{4},\frac {3}{4},\frac {5}{4}\right \},\left \{\frac {3}{2},\frac {7}{4}\right \},\frac {a^2 \text {exp\_polar}\left [I \text {Pi}\right ]}{x^2}\right ]}{3 x^{\frac {3}{2}}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {1}{x \sqrt {a^{2}+x^{2}}\, \sqrt {x +\sqrt {a^{2}+x^{2}}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] Leaf count of result is larger than twice the leaf count of optimal. 105 vs.
\(2 (47) = 94\).
time = 0.32, size = 198, normalized size = 3.14 \begin {gather*} \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 ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.08, size = 46, normalized size = 0.73 \begin {gather*} - \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,\sqrt {x+\sqrt {a^2+x^2}}\,\sqrt {a^2+x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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