Optimal. Leaf size=169 \[ \frac{\left (\frac{1}{2}-\frac{i}{2}\right ) \tan ^{-1}\left (\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right )}{\sqrt{-\sqrt{3} d^2+2 i c^2}}-\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right )}{\sqrt{\sqrt{3} d^2+2 i c^2}} \]
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Rubi [A] time = 0.265768, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2133, 725, 204, 206} \[ \frac{\left (\frac{1}{2}-\frac{i}{2}\right ) \tan ^{-1}\left (\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2} \sqrt{-\sqrt{3} d^2+2 i c^2}}\right )}{\sqrt{-\sqrt{3} d^2+2 i c^2}}-\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2} \sqrt{\sqrt{3} d^2+2 i c^2}}\right )}{\sqrt{\sqrt{3} d^2+2 i c^2}} \]
Antiderivative was successfully verified.
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Rule 2133
Rule 725
Rule 204
Rule 206
Rubi steps
\begin{align*} \int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x) \sqrt{3+4 x^4}} \, dx &=\left (\frac{1}{2}-\frac{i}{2}\right ) \int \frac{1}{(c+d x) \sqrt{\sqrt{3}-2 i x^2}} \, dx+\left (\frac{1}{2}+\frac{i}{2}\right ) \int \frac{1}{(c+d x) \sqrt{\sqrt{3}+2 i x^2}} \, dx\\ &=\left (-\frac{1}{2}-\frac{i}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{2 i c^2+\sqrt{3} d^2-x^2} \, dx,x,\frac{\sqrt{3} d-2 i c x}{\sqrt{\sqrt{3}+2 i x^2}}\right )+\left (-\frac{1}{2}+\frac{i}{2}\right ) \operatorname{Subst}\left (\int \frac{1}{-2 i c^2+\sqrt{3} d^2-x^2} \, dx,x,\frac{\sqrt{3} d+2 i c x}{\sqrt{\sqrt{3}-2 i x^2}}\right )\\ &=\frac{\left (\frac{1}{2}-\frac{i}{2}\right ) \tan ^{-1}\left (\frac{\sqrt{3} d+2 i c x}{\sqrt{2 i c^2-\sqrt{3} d^2} \sqrt{\sqrt{3}-2 i x^2}}\right )}{\sqrt{2 i c^2-\sqrt{3} d^2}}-\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \tanh ^{-1}\left (\frac{\sqrt{3} d-2 i c x}{\sqrt{2 i c^2+\sqrt{3} d^2} \sqrt{\sqrt{3}+2 i x^2}}\right )}{\sqrt{2 i c^2+\sqrt{3} d^2}}\\ \end{align*}
Mathematica [F] time = 0.109194, size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 x^2+\sqrt{3+4 x^4}}}{(c+d x) \sqrt{3+4 x^4}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c}\sqrt{2\,{x}^{2}+\sqrt{4\,{x}^{4}+3}}{\frac{1}{\sqrt{4\,{x}^{4}+3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x^{2} + \sqrt{4 \, x^{4} + 3}}}{\sqrt{4 \, x^{4} + 3}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 x^{2} + \sqrt{4 x^{4} + 3}}}{\left (c + d x\right ) \sqrt{4 x^{4} + 3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{2 \, x^{2} + \sqrt{4 \, x^{4} + 3}}}{\sqrt{4 \, x^{4} + 3}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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