Optimal. Leaf size=268 \[ \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}+\frac {(1+i) c \tan ^{-1}\left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\left (-\sqrt {3} d^2+2 i c^2\right )^{3/2}}+\frac {(1-i) c \tanh ^{-1}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {\sqrt {3}+2 i x^2} \sqrt {\sqrt {3} d^2+2 i c^2}}\right )}{\left (\sqrt {3} d^2+2 i c^2\right )^{3/2}} \]
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Rubi [A] time = 0.31, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2133, 731, 725, 204, 206} \[ \frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (-\sqrt {3} d^2+2 i c^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (\sqrt {3} d^2+2 i c^2\right ) (c+d x)}+\frac {(1+i) c \tan ^{-1}\left (\frac {\sqrt {3} d+2 i c x}{\sqrt {\sqrt {3}-2 i x^2} \sqrt {-\sqrt {3} d^2+2 i c^2}}\right )}{\left (-\sqrt {3} d^2+2 i c^2\right )^{3/2}}+\frac {(1-i) c \tanh ^{-1}\left (\frac {\sqrt {3} d-2 i c x}{\sqrt {\sqrt {3}+2 i x^2} \sqrt {\sqrt {3} d^2+2 i c^2}}\right )}{\left (\sqrt {3} d^2+2 i c^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 206
Rule 725
Rule 731
Rule 2133
Rubi steps
\begin {align*} \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx &=\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {\sqrt {3}-2 i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(c+d x)^2 \sqrt {\sqrt {3}+2 i x^2}} \, dx\\ &=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (2 i c^2-\sqrt {3} d^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (2 i c^2+\sqrt {3} d^2\right ) (c+d x)}+\frac {((1+i) c) \int \frac {1}{(c+d x) \sqrt {\sqrt {3}+2 i x^2}} \, dx}{2 c^2-i \sqrt {3} d^2}+\frac {((1-i) c) \int \frac {1}{(c+d x) \sqrt {\sqrt {3}-2 i x^2}} \, dx}{2 c^2+i \sqrt {3} d^2}\\ &=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (2 i c^2-\sqrt {3} d^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (2 i c^2+\sqrt {3} d^2\right ) (c+d x)}+-\frac {((1+i) c) \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 )}{2 c^2-i \sqrt {3} d^2}+-\frac {((1-i) c) \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 )}{2 c^2+i \sqrt {3} d^2}\\ &=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) d \sqrt {\sqrt {3}-2 i x^2}}{\left (2 i c^2-\sqrt {3} d^2\right ) (c+d x)}-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) d \sqrt {\sqrt {3}+2 i x^2}}{\left (2 i c^2+\sqrt {3} d^2\right ) (c+d x)}+\frac {(1+i) c \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 )}{\left (2 i c^2-\sqrt {3} d^2\right )^{3/2}}+\frac {(1-i) c \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 )}{\left (2 i c^2+\sqrt {3} d^2\right )^{3/2}}\\ \end {align*}
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Mathematica [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {2 x^2+\sqrt {3+4 x^4}}}{(c+d x)^2 \sqrt {3+4 x^4}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.07, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {2 x^{2}+\sqrt {4 x^{4}+3}}}{\left (d x +c \right )^{2} \sqrt {4 x^{4}+3}}\, dx \]
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 \[ \int \frac {\sqrt {2 \, x^{2} + \sqrt {4 \, x^{4} + 3}}}{\sqrt {4 \, x^{4} + 3} {\left (d x + c\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {2\,x^2+\sqrt {4\,x^4+3}}}{\sqrt {4\,x^4+3}\,{\left (c+d\,x\right )}^2} \,d x \]
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 \[ \int \frac {\sqrt {2 x^{2} + \sqrt {4 x^{4} + 3}}}{\left (c + d x\right )^{2} \sqrt {4 x^{4} + 3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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