Optimal. Leaf size=214 \[ 2 \sqrt {x+1}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}} \]
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Rubi [A] time = 0.25, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {825, 827, 1169, 634, 618, 204, 628} \[ 2 \sqrt {x+1}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {x+1}+\sqrt {2}+1\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {x+1}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}}-\frac {\tan ^{-1}\left (\frac {2 \sqrt {x+1}+\sqrt {2 \left (1+\sqrt {2}\right )}}{\sqrt {2 \left (\sqrt {2}-1\right )}}\right )}{\sqrt {2 \left (1+\sqrt {2}\right )}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 628
Rule 634
Rule 825
Rule 827
Rule 1169
Rubi steps
\begin {align*} \int \frac {x \sqrt {1+x}}{1+x^2} \, dx &=2 \sqrt {1+x}+\int \frac {-1+x}{\sqrt {1+x} \left (1+x^2\right )} \, dx\\ &=2 \sqrt {1+x}+2 \operatorname {Subst}\left (\int \frac {-2+x^2}{2-2 x^2+x^4} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x}+\frac {\operatorname {Subst}\left (\int \frac {-2 \sqrt {2 \left (1+\sqrt {2}\right )}-\left (-2-\sqrt {2}\right ) x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {\operatorname {Subst}\left (\int \frac {-2 \sqrt {2 \left (1+\sqrt {2}\right )}+\left (-2-\sqrt {2}\right ) x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )}{2 \sqrt {1+\sqrt {2}}}\\ &=2 \sqrt {1+x}-\frac {1}{2} \sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} \sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {-\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}-\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \operatorname {Subst}\left (\int \frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 x}{\sqrt {2}+\sqrt {2 \left (1+\sqrt {2}\right )} x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=2 \sqrt {1+x}+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )+\sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,-\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}\right )+\sqrt {3-2 \sqrt {2}} \operatorname {Subst}\left (\int \frac {1}{2 \left (1-\sqrt {2}\right )-x^2} \, dx,x,\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}\right )\\ &=2 \sqrt {1+x}+\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}-2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {2}\right )} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )}+2 \sqrt {1+x}}{\sqrt {2 \left (-1+\sqrt {2}\right )}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (1+\sqrt {2}+x-\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {2}\right )} \log \left (1+\sqrt {2}+x+\sqrt {2 \left (1+\sqrt {2}\right )} \sqrt {1+x}\right )\\ \end {align*}
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Mathematica [C] time = 0.04, size = 60, normalized size = 0.28 \[ 2 \sqrt {x+1}-\sqrt {1-i} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1-i}}\right )-\sqrt {1+i} \tanh ^{-1}\left (\frac {\sqrt {x+1}}{\sqrt {1+i}}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 307, normalized size = 1.43 \[ -\frac {1}{8} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (\frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} + x + \sqrt {2} + 1\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} \log \left (-\frac {1}{2} \cdot 2^{\frac {1}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} + x + \sqrt {2} + 1\right ) + \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {2^{\frac {1}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} + 2 \, x + 2 \, \sqrt {2} + 2} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} - \sqrt {2} - 1\right ) + \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {-2 \, \sqrt {2} + 4} \arctan \left (\frac {1}{4} \cdot 2^{\frac {3}{4}} \sqrt {-2^{\frac {1}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} + 2 \, x + 2 \, \sqrt {2} + 2} {\left (\sqrt {2} + 2\right )} \sqrt {-2 \, \sqrt {2} + 4} - \frac {1}{2} \cdot 2^{\frac {3}{4}} \sqrt {x + 1} {\left (\sqrt {2} + 1\right )} \sqrt {-2 \, \sqrt {2} + 4} + \sqrt {2} + 1\right ) + 2 \, \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.90, size = 167, normalized size = 0.78 \[ -\frac {1}{2} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{2} \, \sqrt {2 \, \sqrt {2} - 2} \arctan \left (-\frac {2^{\frac {3}{4}} {\left (2^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - 2 \, \sqrt {x + 1}\right )}}{2 \, \sqrt {-\sqrt {2} + 2}}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) + \frac {1}{4} \, \sqrt {2 \, \sqrt {2} + 2} \log \left (-2^{\frac {1}{4}} \sqrt {x + 1} \sqrt {\sqrt {2} + 2} + x + \sqrt {2} + 1\right ) + 2 \, \sqrt {x + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 240, normalized size = 1.12 \[ -\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {x +1}-\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}-\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x +1}+\sqrt {2+2 \sqrt {2}}}{\sqrt {-2+2 \sqrt {2}}}\right )}{\sqrt {-2+2 \sqrt {2}}}+\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}-\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}-\frac {\sqrt {2+2 \sqrt {2}}\, \ln \left (x +1+\sqrt {2}+\sqrt {x +1}\, \sqrt {2+2 \sqrt {2}}\right )}{4}+2 \sqrt {x +1} \]
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 \[ \int \frac {\sqrt {x + 1} x}{x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 201, normalized size = 0.94 \[ 2\,\sqrt {x+1}+\mathrm {atanh}\left (\frac {\sqrt {x+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {x+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {x+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {x+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}-2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\right )-\mathrm {atanh}\left (\frac {\sqrt {x+1}}{4\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {x+1}}{4\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}-\frac {\sqrt {2}\,\sqrt {x+1}}{8\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}}+\frac {\sqrt {2}\,\sqrt {x+1}}{8\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}}\right )\,\left (2\,\sqrt {\frac {1}{8}-\frac {\sqrt {2}}{8}}+2\,\sqrt {\frac {\sqrt {2}}{8}+\frac {1}{8}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.16, size = 68, normalized size = 0.32 \[ 2 \sqrt {x + 1} - 4 \operatorname {RootSum} {\left (512 t^{4} + 32 t^{2} + 1, \left (t \mapsto t \log {\left (- 128 t^{3} + \sqrt {x + 1} \right )} \right )\right )} + 2 \operatorname {RootSum} {\left (128 t^{4} + 16 t^{2} + 1, \left (t \mapsto t \log {\left (64 t^{3} + 4 t + \sqrt {x + 1} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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