Optimal. Leaf size=49 \[ \frac{i \sqrt{a^2 x^2+1}}{2 a c (1+i a x)^2 \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.0720498, antiderivative size = 49, 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, 32} \[ \frac{i \sqrt{a^2 x^2+1}}{2 a c (1+i a x)^2 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 5076
Rule 5073
Rule 32
Rubi steps
\begin{align*} \int \frac{e^{-3 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=\frac{\sqrt{1+a^2 x^2} \int \frac{e^{-3 i \tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{\sqrt{1+a^2 x^2} \int \frac{1}{(1+i a x)^3} \, dx}{c \sqrt{c+a^2 c x^2}}\\ &=\frac{i \sqrt{1+a^2 x^2}}{2 a c (1+i a x)^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0405942, size = 57, normalized size = 1.16 \[ -\frac{i \sqrt{a^2 x^2+1} \sqrt{a^2 c x^2+c}}{2 a c^2 (a x-i)^3 (a x+i)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 45, normalized size = 0.9 \begin{align*}{\frac{-ax+i}{2\,a \left ( 1+iax \right ) ^{3}} \left ({a}^{2}{x}^{2}+1 \right ) ^{{\frac{3}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02254, size = 39, normalized size = 0.8 \begin{align*} \frac{1}{2 i \, a^{3} c^{\frac{3}{2}} x^{2} + 4 \, a^{2} c^{\frac{3}{2}} x - 2 i \, a c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23682, size = 151, normalized size = 3.08 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c} \sqrt{a^{2} x^{2} + 1}{\left (-i \, a x^{2} - 2 \, x\right )}}{2 \, a^{4} c^{2} x^{4} - 4 i \, a^{3} c^{2} x^{3} - 4 i \, a c^{2} x - 2 \, c^{2}} \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}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\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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