Optimal. Leaf size=43 \[ -\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0658911, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5076, 5073, 31} \[ -\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 31
Rubi steps
\begin{align*} \int \frac{e^{-i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{-i \tan ^{-1}(a x)}}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int \frac{1}{1+i a x} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{i \sqrt{1+a^2 x^2} \log (i-a x)}{a \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0157528, size = 43, normalized size = 1. \[ -\frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.109, size = 42, normalized size = 1. \begin{align*}{\frac{-i\ln \left ( 1+iax \right ) }{ac}\sqrt{c \left ({a}^{2}{x}^{2}+1 \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998187, size = 20, normalized size = 0.47 \begin{align*} -\frac{i \, \log \left (i \, a x + 1\right )}{a \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25501, size = 578, normalized size = 13.44 \begin{align*} \frac{1}{2} i \, \sqrt{\frac{1}{a^{2} c}} \log \left (\frac{{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} +{\left (i \, a^{9} c x^{4} + 2 \, a^{8} c x^{3} + i \, a^{7} c x^{2} + 2 \, a^{6} c x\right )} \sqrt{\frac{1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\right ) - \frac{1}{2} i \, \sqrt{\frac{1}{a^{2} c}} \log \left (\frac{{\left (-i \, a^{6} x^{2} - 2 \, a^{5} x + 2 i \, a^{4}\right )} \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} +{\left (-i \, a^{9} c x^{4} - 2 \, a^{8} c x^{3} - i \, a^{7} c x^{2} - 2 \, a^{6} c x\right )} \sqrt{\frac{1}{a^{2} c}}}{8 \, a^{3} x^{3} - 8 i \, a^{2} x^{2} + 8 \, a x - 8 i}\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{\sqrt{a^{2} x^{2} + 1}}{\sqrt{c \left (a^{2} x^{2} + 1\right )} \left (i a x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a^{2} x^{2} + 1}}{\sqrt{a^{2} c x^{2} + c}{\left (i \, a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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