Optimal. Leaf size=56 \[ \frac {\tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{2 x^2+2 x+1}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{2 x^2+2 x+1}\right )}{2^{3/4}} \]
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Rubi [A] time = 0.04, antiderivative size = 42, normalized size of antiderivative = 0.75, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {694, 266, 63, 298, 203, 206} \begin {gather*} \frac {\tan ^{-1}\left (\sqrt [4]{(2 x+1)^2+1}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\sqrt [4]{(2 x+1)^2+1}\right )}{2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(1+2 x) \sqrt [4]{1+2 x+2 x^2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{\frac {1}{2}+\frac {x^2}{2}}} \, dx,x,1+2 x\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{\frac {1}{2}+\frac {x}{2}} x} \, dx,x,(1+2 x)^2\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\sqrt [4]{1+2 x+2 x^2}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+2 x+2 x^2}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\sqrt [4]{1+2 x+2 x^2}\right )}{\sqrt {2}}\\ &=\frac {\tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{1+2 x+2 x^2}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{1+2 x+2 x^2}\right )}{2^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 39, normalized size = 0.70 \begin {gather*} \frac {\tan ^{-1}\left (\sqrt [4]{4 x^2+4 x+2}\right )-\tanh ^{-1}\left (\sqrt [4]{4 x^2+4 x+2}\right )}{2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.11, size = 56, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\sqrt [4]{2} \sqrt [4]{1+2 x+2 x^2}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\sqrt [4]{2} \sqrt [4]{1+2 x+2 x^2}\right )}{2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 102, normalized size = 1.82 \begin {gather*} -\frac {1}{4} \cdot 8^{\frac {3}{4}} \arctan \left (\frac {1}{8} \cdot 8^{\frac {3}{4}} \sqrt {2 \, \sqrt {2} + 4 \, \sqrt {2 \, x^{2} + 2 \, x + 1}} - \frac {1}{4} \cdot 8^{\frac {3}{4}} {\left (2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (8^{\frac {1}{4}} + 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{16} \cdot 8^{\frac {3}{4}} \log \left (-8^{\frac {1}{4}} + 2 \, {\left (2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{4}}\right ) \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*} \int \frac {1}{{\left (2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{4}} {\left (2 \, x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (1+2 x \right ) \left (2 x^{2}+2 x +1\right )^{\frac {1}{4}}}\, dx\]
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*} \int \frac {1}{{\left (2 \, x^{2} + 2 \, x + 1\right )}^{\frac {1}{4}} {\left (2 \, x + 1\right )}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {1}{\left (2\,x+1\right )\,{\left (2\,x^2+2\,x+1\right )}^{1/4}} \,d x \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 {1}{\left (2 x + 1\right ) \sqrt [4]{2 x^{2} + 2 x + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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