Optimal. Leaf size=169 \[ \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \tan ^{-1}\left (\frac {\sqrt {3} d+2 i c x}{\sqrt {2 i c^2-\sqrt {3} d^2} \sqrt {\sqrt {3}-2 i x^2}}\right )}{\sqrt {2 i c^2-\sqrt {3} d^2}}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \tanh ^{-1}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {2 i c^2+\sqrt {3} d^2} \sqrt {\sqrt {3}+2 i x^2}}\right )}{\sqrt {2 i c^2+\sqrt {3} d^2}} \]
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Rubi [A]
time = 0.18, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2158, 739, 210,
212} \begin {gather*} \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \text {ArcTan}\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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 212
Rule 739
Rule 2158
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 ) \text {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 ) \text {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*}
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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 4.00, size = 531, normalized size = 3.14 \begin {gather*} \frac {-\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}} \tan ^{-1}\left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2-\sqrt {4 c^4+3 d^4}}}\right )+\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}} \tan ^{-1}\left (\frac {d \sqrt {2 x^2+\sqrt {3+4 x^4}}}{\sqrt {-2 c^2+\sqrt {4 c^4+3 d^4}}}\right )+2 i c \sqrt {12 c^4+9 d^4} \text {RootSum}\left [12 d^2+16 i \sqrt {3} c^2 \text {$\#$1}+24 d^2 \text {$\#$1}+24 i \sqrt {3} c^2 \text {$\#$1}^2+24 d^2 \text {$\#$1}^2+8 i \sqrt {3} c^2 \text {$\#$1}^3+12 d^2 \text {$\#$1}^3+3 d^2 \text {$\#$1}^4\&,\frac {\log \left (2 x^2+\sqrt {3+4 x^4}\right )-\log \left (i \sqrt {3}-2 x^2+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-2 x^2 \text {$\#$1}-\sqrt {3+4 x^4} (1+\text {$\#$1})\right )+\log \left (2 x^2+\sqrt {3+4 x^4}\right ) \text {$\#$1}-\log \left (i \sqrt {3}-2 x^2+2 x \sqrt {2 x^2+\sqrt {3+4 x^4}}-2 x^2 \text {$\#$1}-\sqrt {3+4 x^4} (1+\text {$\#$1})\right ) \text {$\#$1}}{4 i \sqrt {3} c^2+6 d^2+12 i \sqrt {3} c^2 \text {$\#$1}+12 d^2 \text {$\#$1}+6 i \sqrt {3} c^2 \text {$\#$1}^2+9 d^2 \text {$\#$1}^2+3 d^2 \text {$\#$1}^3}\&\right ]}{\sqrt {4 c^4+3 d^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {2 x^{2}+\sqrt {4 x^{4}+3}}}{\left (d x +c \right ) \sqrt {4 x^{4}+3}}\, dx\]
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 [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {2 x^{2} + \sqrt {4 x^{4} + 3}}}{\left (c + d x\right ) \sqrt {4 x^{4} + 3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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 = -0.01 \begin {gather*} \int \frac {\sqrt {2\,x^2+\sqrt {4\,x^4+3}}}{\sqrt {4\,x^4+3}\,\left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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