Optimal. Leaf size=86 \[ \frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a (-a x+i) \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0755422, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {5076, 5073, 43} \[ \frac{i \sqrt{a^2 x^2+1} \log (-a x+i)}{a \sqrt{a^2 c x^2+c}}-\frac{2 \sqrt{a^2 x^2+1}}{a (-a x+i) \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{-3 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{-3 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-i a x}{(1+i a x)^2} \, dx}{\sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int \left (-\frac{2}{(-i+a x)^2}+\frac{i}{-i+a x}\right ) \, dx}{\sqrt{c+a^2 c x^2}}\\ &=-\frac{2 \sqrt{1+a^2 x^2}}{a (i-a x) \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 = 0.0275034, size = 60, normalized size = 0.7 \[ \frac{\sqrt{a^2 x^2+1} \left (\frac{i \log (-a x+i)}{a}-\frac{2}{a (-a x+i)}\right )}{\sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 66, normalized size = 0.8 \begin{align*}{\frac{-i\ln \left ( -ax+i \right ) xa-\ln \left ( -ax+i \right ) -2}{ac \left ( -ax+i \right ) }\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 = 0.998757, size = 47, normalized size = 0.55 \begin{align*} \frac{i \, \log \left (i \, a x + 1\right )}{a \sqrt{c}} + \frac{2}{a^{2} \sqrt{c} x - i \, a \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.25482, size = 801, normalized size = 9.31 \begin{align*} \frac{{\left (-i \, a^{3} c x^{3} - a^{2} c x^{2} - i \, a c x - c\right )} \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 ) +{\left (i \, a^{3} c x^{3} + a^{2} c x^{2} + i \, a c x + c\right )} \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 ) - 4 i \, \sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1} x}{2 \, a^{3} c x^{3} - 2 i \, a^{2} c x^{2} + 2 \, a c x - 2 i \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{\sqrt{a^{2} c x^{2} + c}{\left (i \, a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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