Optimal. Leaf size=138 \[ -\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^2-i}}{\sqrt {2} (x+1)}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^2+i}}{\sqrt {2} (x+1)}+\frac {\tanh ^{-1}\left (\frac {x+i}{\sqrt {1-i} \sqrt {x^2-i}}\right )}{(1-i)^{3/2} \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {-x+i}{\sqrt {1+i} \sqrt {x^2+i}}\right )}{(1+i)^{3/2} \sqrt {2}} \]
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Rubi [A] time = 0.08, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {731, 725, 206} \[ -\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^2-i}}{\sqrt {2} (x+1)}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^2+i}}{\sqrt {2} (x+1)}+\frac {\tanh ^{-1}\left (\frac {x+i}{\sqrt {1-i} \sqrt {x^2-i}}\right )}{(1-i)^{3/2} \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {-x+i}{\sqrt {1+i} \sqrt {x^2+i}}\right )}{(1+i)^{3/2} \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 725
Rule 731
Rubi steps
\begin {align*} \int \left (\frac {1}{\sqrt {2} (1+x)^2 \sqrt {-i+x^2}}+\frac {1}{\sqrt {2} (1+x)^2 \sqrt {i+x^2}}\right ) \, dx &=\frac {\int \frac {1}{(1+x)^2 \sqrt {-i+x^2}} \, dx}{\sqrt {2}}+\frac {\int \frac {1}{(1+x)^2 \sqrt {i+x^2}} \, dx}{\sqrt {2}}\\ &=-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-i+x^2}}{\sqrt {2} (1+x)}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i+x^2}}{\sqrt {2} (1+x)}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {i+x^2}} \, dx}{\sqrt {2}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+x) \sqrt {-i+x^2}} \, dx}{\sqrt {2}}\\ &=-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-i+x^2}}{\sqrt {2} (1+x)}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i+x^2}}{\sqrt {2} (1+x)}+-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-i-x}{\sqrt {-i+x^2}}\right )}{\sqrt {2}}+-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {i-x}{\sqrt {i+x^2}}\right )}{\sqrt {2}}\\ &=-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-i+x^2}}{\sqrt {2} (1+x)}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {i+x^2}}{\sqrt {2} (1+x)}+\frac {\tanh ^{-1}\left (\frac {i+x}{\sqrt {1-i} \sqrt {-i+x^2}}\right )}{(1-i)^{3/2} \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {i-x}{\sqrt {1+i} \sqrt {i+x^2}}\right )}{(1+i)^{3/2} \sqrt {2}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 125, normalized size = 0.91 \[ \frac {i \left ((1+i) \left (i \sqrt {x^2-i}+\sqrt {x^2+i}\right )+\sqrt {1-i} (x+1) \tanh ^{-1}\left (\frac {x+i}{\sqrt {1-i} \sqrt {x^2-i}}\right )+\sqrt {1+i} (x+1) \tanh ^{-1}\left (\frac {(1+i)^{3/2} (1+i x)}{2 \sqrt {x^2+i}}\right )\right )}{2 \sqrt {2} (x+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 161, normalized size = 1.17 \[ \frac {\sqrt {-\frac {1}{2} i + \frac {1}{2}} {\left (-\left (i - 1\right ) \, x - i + 1\right )} \log \left (\sqrt {2} \sqrt {-\frac {1}{2} i + \frac {1}{2}} - x + \sqrt {x^{2} - i} - 1\right ) + \sqrt {-\frac {1}{2} i + \frac {1}{2}} {\left (\left (i - 1\right ) \, x + i - 1\right )} \log \left (-\sqrt {2} \sqrt {-\frac {1}{2} i + \frac {1}{2}} - x + \sqrt {x^{2} - i} - 1\right ) + \sqrt {-\frac {1}{2} i - \frac {1}{2}} {\left (-\left (i + 1\right ) \, x - i - 1\right )} \log \left (i \, \sqrt {2} \sqrt {-\frac {1}{2} i - \frac {1}{2}} - x + \sqrt {x^{2} + i} - 1\right ) + \sqrt {-\frac {1}{2} i - \frac {1}{2}} {\left (\left (i + 1\right ) \, x + i + 1\right )} \log \left (-i \, \sqrt {2} \sqrt {-\frac {1}{2} i - \frac {1}{2}} - x + \sqrt {x^{2} + i} - 1\right ) + \sqrt {2} {\left (-\left (i + 1\right ) \, x - i - 1\right )} - \sqrt {2} \sqrt {x^{2} + i} - i \, \sqrt {2} \sqrt {x^{2} - i}}{\left (2 i + 2\right ) \, x + 2 i + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.03, size = 278, normalized size = 2.01 \[ -\frac {\sqrt {2}\, \ln \left (\frac {-2 x -2 i+2 \sqrt {1-i}\, \sqrt {-2 x +\left (x +1\right )^{2}-1-i}}{x +1}\right )}{4 \sqrt {1-i}}-\frac {i \sqrt {2}\, \ln \left (\frac {-2 x -2 i+2 \sqrt {1-i}\, \sqrt {-2 x +\left (x +1\right )^{2}-1-i}}{x +1}\right )}{4 \sqrt {1-i}}-\frac {\sqrt {2}\, \ln \left (\frac {-2 x +2 i+2 \sqrt {1+i}\, \sqrt {-2 x +\left (x +1\right )^{2}-1+i}}{x +1}\right )}{4 \sqrt {1+i}}+\frac {i \sqrt {2}\, \ln \left (\frac {-2 x +2 i+2 \sqrt {1+i}\, \sqrt {-2 x +\left (x +1\right )^{2}-1+i}}{x +1}\right )}{4 \sqrt {1+i}}-\frac {\sqrt {2}\, \sqrt {-2 x +\left (x +1\right )^{2}-1-i}}{4 \left (x +1\right )}-\frac {i \sqrt {2}\, \sqrt {-2 x +\left (x +1\right )^{2}-1-i}}{4 \left (x +1\right )}-\frac {\sqrt {2}\, \sqrt {-2 x +\left (x +1\right )^{2}-1+i}}{4 \left (x +1\right )}+\frac {i \sqrt {2}\, \sqrt {-2 x +\left (x +1\right )^{2}-1+i}}{4 x +4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {\sqrt {2}}{2\,\sqrt {x^2-\mathrm {i}}\,{\left (x+1\right )}^2}+\frac {\sqrt {2}}{2\,\sqrt {x^2+1{}\mathrm {i}}\,{\left (x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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