Optimal. Leaf size=54 \[ \frac{x}{3 c \sqrt{a^2 c x^2+c}}+\frac{2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.0548632, antiderivative size = 54, 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 = {5074, 653, 191} \[ \frac{x}{3 c \sqrt{a^2 c x^2+c}}+\frac{2 i (1-i a x)}{3 a \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 5074
Rule 653
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-2 i \tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=c \int \frac{(1-i a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{3} \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 i (1-i a x)}{3 a \left (c+a^2 c x^2\right )^{3/2}}+\frac{x}{3 c \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.0333535, size = 78, normalized size = 1.44 \[ \frac{\sqrt{1-i a x} (2+i a x) \sqrt{a^2 x^2+1}}{3 a c \sqrt{1+i a x} (a x-i) \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.07, size = 137, normalized size = 2.5 \begin{align*} -{\frac{x}{c}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}-{\frac{2\,i}{a} \left ({\frac{{\frac{i}{3}}}{ac} \left ( x-{\frac{i}{a}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) }}}}+{\frac{{\frac{i}{3}}}{a{c}^{2}} \left ( 2\, \left ( x-{\frac{i}{a}} \right ){a}^{2}c+2\,iac \right ){\frac{1}{\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) }}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.38089, size = 101, normalized size = 1.87 \begin{align*} \frac{\sqrt{a^{2} c x^{2} + c}{\left (a x - 2 i\right )}}{3 \, a^{3} c^{2} x^{2} - 6 i \, a^{2} c^{2} x - 3 \, a c^{2}} \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{a^{2} x^{2} + 1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (i a x + 1\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22282, size = 105, normalized size = 1.94 \begin{align*} \frac{2 \, \sqrt{a^{2} c}{\left (\sqrt{c} i - 3 \, \sqrt{a^{2} c} x + 3 \, \sqrt{a^{2} c x^{2} + c}\right )} i^{2}}{3 \,{\left (\sqrt{c} i - \sqrt{a^{2} c} x + \sqrt{a^{2} c x^{2} + c}\right )}^{3} a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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