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 = 0.44, 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 = 0.48, size = 51, normalized size = 0.98 \[ \frac {\sqrt {{\left (b x + a\right )}^{2} + 1} i}{b} - \frac {\log \left (-a b - {\left (x {\left | b \right |} - \sqrt {{\left (b x + a\right )}^{2} + 1}\right )} {\left | b \right |}\right )}{{\left | b \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 69, normalized size = 1.33 \[ \frac {i \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}}{b}+\frac {\ln \left (\frac {b^{2} x +a b}{\sqrt {b^{2}}}+\sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\right )}{\sqrt {b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 62, normalized size = 1.19 \[ \frac {\operatorname {arsinh}\left (\frac {2 \, {\left (b^{2} x + a b\right )}}{\sqrt {-4 \, a^{2} b^{2} + 4 \, {\left (a^{2} + 1\right )} b^{2}}}\right )}{b} + \frac {i \, \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.09, size = 97, normalized size = 1.87 \[ \frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}\,1{}\mathrm {i}}{b}+\frac {\mathrm {asinh}\left (a+b\,x\right )}{b}+\frac {a\,\mathrm {asinh}\left (a+b\,x\right )\,1{}\mathrm {i}}{b}-\frac {a\,b^2\,\ln \left (\sqrt {a^2+2\,a\,b\,x+b^2\,x^2+1}+\frac {x\,b^2+a\,b}{\sqrt {b^2}}\right )\,1{}\mathrm {i}}{{\left (b^2\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.41, size = 36, normalized size = 0.69 \[ \begin {cases} \frac {i \sqrt {\left (a + b x\right )^{2} + 1} + \operatorname {asinh}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\\frac {x \left (i a + 1\right )}{\sqrt {a^{2} + 1}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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