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 = 0.07, antiderivative size = 89, normalized size of antiderivative = 1.00, 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 53
Rule 93
Rule 105
Rule 208
Rule 215
Rule 619
Rule 5095
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*}
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Mathematica [A] time = 0.08, size = 142, normalized size = 1.60 \[ \frac {2 \sqrt [4]{-1} (-i b)^{3/2} \sinh ^{-1}\left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {b} \sqrt {-i (a+b x+i)}}{\sqrt {-i b}}\right )}{b^{3/2}}-\frac {2 \sqrt {-1+i a} \tanh ^{-1}\left (\frac {\sqrt {-1-i a} \sqrt {-i (a+b x+i)}}{\sqrt {-1+i a} \sqrt {i a+i b x+1}}\right )}{\sqrt {-1-i a}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.47, size = 155, normalized size = 1.74 \[ -\frac {1}{2} \, \sqrt {-\frac {4 \, a + 4 i}{a - i}} \log \left (-b x + \frac {1}{2} \, {\left (i \, a + 1\right )} \sqrt {-\frac {4 \, a + 4 i}{a - i}} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + \frac {1}{2} \, \sqrt {-\frac {4 \, a + 4 i}{a - i}} \log \left (-b x + \frac {1}{2} \, {\left (-i \, a - 1\right )} \sqrt {-\frac {4 \, a + 4 i}{a - i}} + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + i \, \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 77, normalized size = 0.87 \[ \frac {b i \log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} + \frac {2 \, {\left (a + i\right )} \arctan \left (-\frac {{\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} i}{\sqrt {a^{2} + 1}}\right )}{\sqrt {a^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 283, normalized size = 3.18 \[ \frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{i-a}+\frac {i a b \ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\left (i-a \right ) \sqrt {b^{2}}}-\frac {i \sqrt {a^{2}+1}\, \ln \left (\frac {2 a^{2}+2+2 a b x +2 \sqrt {a^{2}+1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{x}\right )}{i-a}-\frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{i-a}+\frac {b \ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{\left (i-a \right ) \sqrt {b^{2}}} \]
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 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {\sqrt {{\left (a+b\,x\right )}^2+1}}{x\,\left (1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - i \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a x + b x^{2} - i x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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