Optimal. Leaf size=29 \[ \tan ^{-1}\left (\sqrt {4 x+3}+2\right )-\tan ^{-1}\left (2-\sqrt {4 x+3}\right ) \]
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Rubi [A] time = 0.03, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {827, 1161, 618, 204} \begin {gather*} \tan ^{-1}\left (\sqrt {4 x+3}+2\right )-\tan ^{-1}\left (2-\sqrt {4 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 827
Rule 1161
Rubi steps
\begin {align*} \int \frac {2+x}{\sqrt {3+4 x} \left (1+x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {5+x^2}{25-6 x^2+x^4} \, dx,x,\sqrt {3+4 x}\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{5-4 x+x^2} \, dx,x,\sqrt {3+4 x}\right )+\operatorname {Subst}\left (\int \frac {1}{5+4 x+x^2} \, dx,x,\sqrt {3+4 x}\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,-4+2 \sqrt {3+4 x}\right )\right )-2 \operatorname {Subst}\left (\int \frac {1}{-4-x^2} \, dx,x,4+2 \sqrt {3+4 x}\right )\\ &=-\tan ^{-1}\left (2-\sqrt {3+4 x}\right )+\tan ^{-1}\left (2+\sqrt {3+4 x}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 41, normalized size = 1.41 \begin {gather*} \tan ^{-1}\left (\left (\frac {1}{5}+\frac {2 i}{5}\right ) \sqrt {4 x+3}\right )-i \tanh ^{-1}\left (\left (\frac {2}{5}+\frac {i}{5}\right ) \sqrt {4 x+3}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.03, size = 24, normalized size = 0.83 \begin {gather*} \tan ^{-1}\left (\frac {\frac {1}{2} (4 x+3)-\frac {5}{2}}{\sqrt {4 x+3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 14, normalized size = 0.48 \begin {gather*} \arctan \left (\frac {2 \, x - 1}{\sqrt {4 \, x + 3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 21, normalized size = 0.72 \begin {gather*} \arctan \left (\sqrt {4 \, x + 3} + 2\right ) + \arctan \left (\sqrt {4 \, x + 3} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.76 \begin {gather*} \arctan \left (-2+\sqrt {4 x +3}\right )+\arctan \left (2+\sqrt {4 x +3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.20, size = 21, normalized size = 0.72 \begin {gather*} \arctan \left (\sqrt {4 \, x + 3} + 2\right ) + \arctan \left (\sqrt {4 \, x + 3} - 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 26, normalized size = 0.90 \begin {gather*} \mathrm {atan}\left (\frac {\sqrt {4\,x+3}}{2}\right )+\mathrm {atan}\left (\frac {\left (4\,x+2\right )\,\sqrt {4\,x+3}}{10}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 99.08, size = 26, normalized size = 0.90 \begin {gather*} \operatorname {atan}{\left (2 - \frac {5}{\sqrt {4 x + 3}} \right )} - \operatorname {atan}{\left (2 + \frac {5}{\sqrt {4 x + 3}} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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