Optimal. Leaf size=53 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+9}}{\sqrt {5}}\right )}{\sqrt {5}} \]
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Rubi [A] time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1010, 377, 203, 444, 63, 207} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {x^2+9}}\right )}{2 \sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {x^2+9}}{\sqrt {5}}\right )}{\sqrt {5}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 207
Rule 377
Rule 444
Rule 1010
Rubi steps
\begin {align*} \int \frac {1+x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx &=\int \frac {1}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx+\int \frac {x}{\left (4+x^2\right ) \sqrt {9+x^2}} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{(4+x) \sqrt {9+x}} \, dx,x,x^2\right )+\operatorname {Subst}\left (\int \frac {1}{4+5 x^2} \, dx,x,\frac {x}{\sqrt {9+x^2}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {9+x^2}}\right )}{2 \sqrt {5}}+\operatorname {Subst}\left (\int \frac {1}{-5+x^2} \, dx,x,\sqrt {9+x^2}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {5} x}{2 \sqrt {9+x^2}}\right )}{2 \sqrt {5}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {9+x^2}}{\sqrt {5}}\right )}{\sqrt {5}}\\ \end {align*}
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Mathematica [C] time = 0.04, size = 64, normalized size = 1.21 \[ -\frac {(2+i) \tanh ^{-1}\left (\frac {9-2 i x}{\sqrt {5} \sqrt {x^2+9}}\right )+(2-i) \tanh ^{-1}\left (\frac {9+2 i x}{\sqrt {5} \sqrt {x^2+9}}\right )}{4 \sqrt {5}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.44, size = 182, normalized size = 3.43 \[ \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - \sqrt {x^{2} + 9} {\left (x + \sqrt {5}\right )} + \sqrt {5} x + 9} + \frac {1}{2} \, x + \frac {1}{2} \, \sqrt {5} - \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) - \frac {1}{5} \, \sqrt {5} \arctan \left (\frac {1}{2} \, \sqrt {2} \sqrt {x^{2} - \sqrt {x^{2} + 9} {\left (x - \sqrt {5}\right )} - \sqrt {5} x + 9} + \frac {1}{2} \, x - \frac {1}{2} \, \sqrt {5} - \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) + \frac {1}{10} \, \sqrt {5} \log \left (50 \, x^{2} - 50 \, \sqrt {x^{2} + 9} {\left (x + \sqrt {5}\right )} + 50 \, \sqrt {5} x + 450\right ) - \frac {1}{10} \, \sqrt {5} \log \left (50 \, x^{2} - 50 \, \sqrt {x^{2} + 9} {\left (x - \sqrt {5}\right )} - 50 \, \sqrt {5} x + 450\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.44, size = 123, normalized size = 2.32 \[ -\frac {1}{10} \, \sqrt {5} \arctan \left (\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {5} - \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) - \frac {1}{10} \, \sqrt {5} \arctan \left (-\frac {1}{2} \, x - \frac {1}{2} \, \sqrt {5} + \frac {1}{2} \, \sqrt {x^{2} + 9}\right ) + \frac {1}{10} \, \sqrt {5} \log \left ({\left (x - \sqrt {x^{2} + 9}\right )}^{2} + 2 \, \sqrt {5} {\left (x - \sqrt {x^{2} + 9}\right )} + 9\right ) - \frac {1}{10} \, \sqrt {5} \log \left ({\left (x - \sqrt {x^{2} + 9}\right )}^{2} - 2 \, \sqrt {5} {\left (x - \sqrt {x^{2} + 9}\right )} + 9\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 39, normalized size = 0.74 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\sqrt {x^{2}+9}\, \sqrt {5}}{5}\right )}{5}+\frac {\sqrt {5}\, \arctan \left (\frac {\sqrt {5}\, x}{2 \sqrt {x^{2}+9}}\right )}{10} \]
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 \[ \int \frac {x + 1}{\sqrt {x^{2} + 9} {\left (x^{2} + 4\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.59, size = 67, normalized size = 1.26 \[ \sqrt {5}\,\left (\ln \left (x-2{}\mathrm {i}\right )-\ln \left (\sqrt {5}\,\sqrt {x^2+9}+9+x\,2{}\mathrm {i}\right )\right )\,\left (\frac {1}{10}-\frac {1}{20}{}\mathrm {i}\right )+\sqrt {5}\,\left (\ln \left (x+2{}\mathrm {i}\right )-\ln \left (\sqrt {5}\,\sqrt {x^2+9}+9-x\,2{}\mathrm {i}\right )\right )\,\left (\frac {1}{10}+\frac {1}{20}{}\mathrm {i}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x + 1}{\left (x^{2} + 4\right ) \sqrt {x^{2} + 9}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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