Optimal. Leaf size=52 \[ \frac {\sinh ^{-1}(a+b x)}{b}-\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b} \]
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Rubi [A] time = 0.03, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5093, 50, 53, 619, 215} \[ \frac {\sinh ^{-1}(a+b x)}{b}-\frac {i \sqrt {-i a-i b x+1} \sqrt {i a+i b x+1}}{b} \]
Antiderivative was successfully verified.
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Rule 50
Rule 53
Rule 215
Rule 619
Rule 5093
Rubi steps
\begin {align*} \int e^{-i \tan ^{-1}(a+b x)} \, dx &=\int \frac {\sqrt {1-i a-i b x}}{\sqrt {1+i a+i b x}} \, dx\\ &=-\frac {i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}+\int \frac {1}{\sqrt {1-i a-i b x} \sqrt {1+i a+i b x}} \, dx\\ &=-\frac {i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}+\int \frac {1}{\sqrt {(1-i a) (1+i a)+2 a b x+b^2 x^2}} \, dx\\ &=-\frac {i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}+\frac {\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^2}\\ &=-\frac {i \sqrt {1-i a-i b x} \sqrt {1+i a+i b x}}{b}+\frac {\sinh ^{-1}(a+b x)}{b}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.54 \[ \frac {\sinh ^{-1}(a+b x)-i \sqrt {(a+b x)^2+1}}{b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 60, normalized size = 1.15 \[ \frac {-i \, a - 2 i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 2 \, \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 52, normalized size = 1.00 \[ -\frac {\sqrt {{\left (b x + a\right )}^{2} + 1} i}{b} - \frac {\log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 122, normalized size = 2.35 \[ -\frac {i \sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}}{b}+\frac {\ln \left (\frac {i b +\left (x -\frac {i-a}{b}\right ) b^{2}}{\sqrt {b^{2}}}+\sqrt {\left (x -\frac {i-a}{b}\right )^{2} b^{2}+2 i b \left (x -\frac {i-a}{b}\right )}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 35, normalized size = 0.67 \[ \frac {\operatorname {arsinh}\left (b x + a\right )}{b} - \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {{\left (a+b\,x\right )}^2+1}}{1+a\,1{}\mathrm {i}+b\,x\,1{}\mathrm {i}} \,d x \]
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 \[ - i \int \frac {\sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{a + b x - i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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