Optimal. Leaf size=89 \[ i \sinh ^{-1}(a+b x)-\frac{2 \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right )}{\sqrt{a+i}} \]
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Rubi [A] time = 0.0751899, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {5095, 105, 53, 619, 215, 93, 208} \[ i \sinh ^{-1}(a+b x)-\frac{2 \sqrt{-a+i} \tanh ^{-1}\left (\frac{\sqrt{a+i} \sqrt{i a+i b x+1}}{\sqrt{-a+i} \sqrt{-i a-i b x+1}}\right )}{\sqrt{a+i}} \]
Antiderivative was successfully verified.
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Rule 5095
Rule 105
Rule 53
Rule 619
Rule 215
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{e^{i \tan ^{-1}(a+b x)}}{x} \, dx &=\int \frac{\sqrt{1+i a+i b x}}{x \sqrt{1-i a-i b x}} \, dx\\ &=-\left ((-1-i a) \int \frac{1}{x \sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx\right )+(i b) \int \frac{1}{\sqrt{1-i a-i b x} \sqrt{1+i a+i b x}} \, dx\\ &=(2 (1+i a)) \operatorname{Subst}\left (\int \frac{1}{-1-i a-(-1+i a) x^2} \, dx,x,\frac{\sqrt{1+i a+i b x}}{\sqrt{1-i a-i b x}}\right )+(i b) \int \frac{1}{\sqrt{(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=-\frac{2 \sqrt{i-a} \tanh ^{-1}\left (\frac{\sqrt{i+a} \sqrt{1+i a+i b x}}{\sqrt{i-a} \sqrt{1-i a-i b x}}\right )}{\sqrt{i+a}}+\frac{i \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{4 b^2}}} \, dx,x,2 a b+2 b^2 x\right )}{2 b}\\ &=i \sinh ^{-1}(a+b x)-\frac{2 \sqrt{i-a} \tanh ^{-1}\left (\frac{\sqrt{i+a} \sqrt{1+i a+i b x}}{\sqrt{i-a} \sqrt{1-i a-i b x}}\right )}{\sqrt{i+a}}\\ \end{align*}
Mathematica [A] time = 0.134819, size = 132, normalized size = 1.48 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{-i (a+b x+i)}}{\sqrt{\frac{a+i}{a-i}} \sqrt{i a+i b x+1}}\right )}{\sqrt{\frac{a+i}{a-i}}}+\frac{2 (-1)^{3/4} \sqrt{-i b} \sinh ^{-1}\left (\frac{\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt{b} \sqrt{-i (a+b x+i)}}{\sqrt{-i b}}\right )}{\sqrt{b}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.105, size = 157, normalized size = 1.8 \begin{align*}{ib\ln \left ({({b}^{2}x+ab){\frac{1}{\sqrt{{b}^{2}}}}}+\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ){\frac{1}{\sqrt{{b}^{2}}}}}-{ia\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2+2\,xab+2\,\sqrt{{a}^{2}+1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}+1}}}}-{\ln \left ({\frac{1}{x} \left ( 2\,{a}^{2}+2+2\,xab+2\,\sqrt{{a}^{2}+1}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1} \right ) } \right ){\frac{1}{\sqrt{{a}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69298, size = 402, normalized size = 4.52 \begin{align*} \frac{1}{2} \, \sqrt{-\frac{4 \, a - 4 i}{a + i}} \log \left (-b x + \frac{1}{2} \,{\left (i \, a - 1\right )} \sqrt{-\frac{4 \, a - 4 i}{a + i}} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - \frac{1}{2} \, \sqrt{-\frac{4 \, a - 4 i}{a + i}} \log \left (-b x + \frac{1}{2} \,{\left (-i \, a + 1\right )} \sqrt{-\frac{4 \, a - 4 i}{a + i}} + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - i \, \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{i a + i b x + 1}{x \sqrt{a^{2} + 2 a b x + b^{2} x^{2} + 1}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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