Optimal. Leaf size=395 \[ -\frac {2 \sqrt [4]{i-a} \text {ArcTan}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}\right )}{\sqrt [4]{i+a}}-\sqrt {2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}\right )+\sqrt {2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}\right )-\frac {2 \sqrt [4]{i-a} \tanh ^{-1}\left (\frac {\sqrt [4]{i+a} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{i-a} \sqrt [4]{1-i (a+b x)}}\right )}{\sqrt [4]{i+a}}-\frac {\log \left (1-\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}} \]
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Rubi [A]
time = 0.21, antiderivative size = 395, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 12, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5202, 456,
492, 217, 1179, 642, 1176, 631, 210, 218, 214, 211} \begin {gather*} -\frac {2 \sqrt [4]{-a+i} \text {ArcTan}\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} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}\right )+\sqrt {2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}\right )-\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} \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}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 211
Rule 214
Rule 217
Rule 218
Rule 456
Rule 492
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 5202
Rubi steps
\begin {align*} \int \frac {e^{\frac {1}{2} i \tan ^{-1}(a+b x)}}{x} \, dx &=8 \text {Subst}\left (\int \frac {1}{\left (1+\frac {1}{x^4}\right ) \left (1-i a-\frac {1+i a}{x^4}\right ) x^4} \, dx,x,\frac {\sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}\right )\\ &=8 \text {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 )\\ &=4 \text {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)) \text {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 \text {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 \text {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 ) \text {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 ) \text {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 {\text {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 {\text {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}}+\text {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 )+\text {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 {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}}+\sqrt {2} \text {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} \text {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 {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {2} \sqrt [4]{1+i (a+b x)}}{\sqrt [4]{1-i (a+b x)}}+\frac {\sqrt {1+i (a+b x)}}{\sqrt {1-i (a+b x)}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.06, size = 124, normalized size = 0.31 \begin {gather*} \frac {2}{3} (-i (i+a+b x))^{3/4} \left (-\sqrt [4]{2} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-\frac {1}{2} i (i+a+b x)\right )+\frac {2 (-i+a) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1+a^2-i b x+a b x}{1+a^2+i b x+a b x}\right )}{(i+a) (1+i a+i b x)^{3/4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}}{x}\, dx\]
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.81, size = 414, normalized size = 1.05 \begin {gather*} \frac {1}{2} \, \sqrt {4 i} \log \left (\frac {1}{2} \, \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} \, \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} \, \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} \, \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 (\sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}}\right ) - i \, \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}} \log \left (\sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} + i \, \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}}\right ) + i \, \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}} \log \left (\sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - i \, \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}}\right ) + \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}} \log \left (\sqrt {\frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b x + a + i}} - \left (-\frac {a - i}{a + i}\right )^{\frac {1}{4}}\right ) \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 {\sqrt {\frac {i \left (a + b x - i\right )}{\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {\sqrt {\frac {1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}}{\sqrt {{\left (a+b\,x\right )}^2+1}}}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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