Optimal. Leaf size=395 \[ -\frac{\log \left (\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{\sqrt{2}}-\frac{2 \sqrt [4]{a+i} \tan ^{-1}\left (\frac{\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{-a+i}}-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\frac{2 \sqrt [4]{a+i} \tanh ^{-1}\left (\frac{\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{-a+i}} \]
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Rubi [A] time = 0.205775, antiderivative size = 395, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5094, 481, 211, 1165, 628, 1162, 617, 204, 212, 208, 205} \[ -\frac{\log \left (\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{\sqrt{2}}+\frac{\log \left (\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}+1\right )}{\sqrt{2}}-\frac{2 \sqrt [4]{a+i} \tan ^{-1}\left (\frac{\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{-a+i}}-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\frac{2 \sqrt [4]{a+i} \tanh ^{-1}\left (\frac{\sqrt [4]{-a+i} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{a+i} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{-a+i}} \]
Antiderivative was successfully verified.
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Rule 5094
Rule 481
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rule 212
Rule 208
Rule 205
Rubi steps
\begin{align*} \int \frac{e^{-\frac{1}{2} i \tan ^{-1}(a+b x)}}{x} \, dx &=-\left (8 \operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^4\right ) \left (1-i a-(1+i a) x^4\right )} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\right )\\ &=4 \operatorname{Subst}\left (\int \frac{1}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-(4 (1-i a)) \operatorname{Subst}\left (\int \frac{1}{1-i a+(-1-i a) x^4} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+2 \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\left (2 \sqrt{i+a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i+a}-\sqrt{i-a} x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\left (2 \sqrt{i+a}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{i+a}+\sqrt{i-a} x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\\ &=-\frac{2 \sqrt [4]{i+a} \tan ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\frac{2 \sqrt [4]{i+a} \tanh ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}+\operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+\operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\frac{\sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\\ &=-\frac{2 \sqrt [4]{i+a} \tan ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\frac{2 \sqrt [4]{i+a} \tanh ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\frac{\log \left (1+\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}+\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\sqrt{2} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )\\ &=-\frac{2 \sqrt [4]{i+a} \tan ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )+\sqrt{2} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )-\frac{2 \sqrt [4]{i+a} \tanh ^{-1}\left (\frac{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}\right )}{\sqrt [4]{i-a}}-\frac{\log \left (1+\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}-\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}+\frac{\log \left (1+\frac{\sqrt{1-i (a+b x)}}{\sqrt{1+i (a+b x)}}+\frac{\sqrt{2} \sqrt [4]{1-i (a+b x)}}{\sqrt [4]{1+i (a+b x)}}\right )}{\sqrt{2}}\\ \end{align*}
Mathematica [C] time = 0.0325684, size = 126, normalized size = 0.32 \[ \frac{2 \sqrt [4]{-i (a+b x+i)} \left (2^{3/4} \sqrt [4]{i a+i b x+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{1}{2} i (a+b x+i)\right )-2 \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};\frac{a^2+b x a-i b x+1}{a^2+b x a+i b x+1}\right )\right )}{\sqrt [4]{i a+i b x+1}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{x}{\frac{1}{\sqrt{{(1+i \left ( bx+a \right ) ){\frac{1}{\sqrt{1+ \left ( bx+a \right ) ^{2}}}}}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.82535, size = 1296, normalized size = 3.28 \begin{align*} -\frac{1}{2} \, \sqrt{4 i} \log \left (\frac{1}{2} i \, \sqrt{4 i} + \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \frac{1}{2} \, \sqrt{4 i} \log \left (-\frac{1}{2} i \, \sqrt{4 i} + \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \frac{1}{2} \, \sqrt{-4 i} \log \left (\frac{1}{2} i \, \sqrt{-4 i} + \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) - \frac{1}{2} \, \sqrt{-4 i} \log \left (-\frac{1}{2} i \, \sqrt{-4 i} + \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}\right ) + \left (-\frac{a + i}{a - i}\right )^{\frac{1}{4}} \log \left (\frac{{\left (a - i\right )} \left (-\frac{a + i}{a - i}\right )^{\frac{3}{4}} +{\left (a + i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) - \left (-\frac{a + i}{a - i}\right )^{\frac{1}{4}} \log \left (-\frac{{\left (a - i\right )} \left (-\frac{a + i}{a - i}\right )^{\frac{3}{4}} -{\left (a + i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) - i \, \left (-\frac{a + i}{a - i}\right )^{\frac{1}{4}} \log \left (\frac{{\left (i \, a + 1\right )} \left (-\frac{a + i}{a - i}\right )^{\frac{3}{4}} +{\left (a + i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) + i \, \left (-\frac{a + i}{a - i}\right )^{\frac{1}{4}} \log \left (\frac{{\left (-i \, a - 1\right )} \left (-\frac{a + i}{a - i}\right )^{\frac{3}{4}} +{\left (a + i\right )} \sqrt{\frac{i \, \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}}}{a + i}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{\frac{i \, b x + i \, a + 1}{\sqrt{{\left (b x + a\right )}^{2} + 1}}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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