Optimal. Leaf size=29 \[ \tan ^{-1}\left (\sqrt [4]{x^2+2 x+2}\right )-\tanh ^{-1}\left (\sqrt [4]{x^2+2 x+2}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 0.93, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {694, 266, 63, 298, 203, 206} \begin {gather*} \tan ^{-1}\left (\sqrt [4]{(x+1)^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{(x+1)^2+1}\right ) \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+x) \sqrt [4]{2+2 x+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x^2}} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,(1+x)^2\right )\\ &=2 \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+(1+x)^2}\right )\\ &=-\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+(1+x)^2}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+(1+x)^2}\right )\\ &=\tan ^{-1}\left (\sqrt [4]{1+(1+x)^2}\right )-\tanh ^{-1}\left (\sqrt [4]{1+(1+x)^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.93 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{(x+1)^2+1}\right )-\tanh ^{-1}\left (\sqrt [4]{(x+1)^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 29, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\sqrt [4]{2+2 x+x^2}\right )-\tanh ^{-1}\left (\sqrt [4]{2+2 x+x^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 42, normalized size = 1.45 \begin {gather*} \arctan \left ({\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \, \log \left ({\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{2} \, \log \left ({\left (x^{2} + 2 \, x + 2\right )}^{\frac {1}{4}} - 1\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 (x^{2} + 2 \, x + 2\right )}^{\frac {1}{4}} {\left (x + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.15, size = 143, normalized size = 4.93
method | result | size |
trager | \(-\frac {\ln \left (-\frac {2 \left (x^{2}+2 x +2\right )^{\frac {3}{4}}+2 \sqrt {x^{2}+2 x +2}+x^{2}+2 \left (x^{2}+2 x +2\right )^{\frac {1}{4}}+2 x +3}{\left (1+x \right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{2}+2 x +2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{2}+2 x +2\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -3 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{2}+2 x +2\right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{2}\) | \(143\) |
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 (x^{2} + 2 \, x + 2\right )}^{\frac {1}{4}} {\left (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 = -0.03 \begin {gather*} \int \frac {1}{\left (x+1\right )\,{\left (x^2+2\,x+2\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 (x + 1\right ) \sqrt [4]{x^{2} + 2 x + 2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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