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 = 0.0549454, 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 = {5075, 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 5075
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 = 0.0267966, size = 78, normalized size = 1.44 \[ \frac{(2-i a x) \sqrt{1+i a x} \sqrt{a^2 x^2+1}}{3 a c \sqrt{1-i a x} (a x+i) \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.074, size = 398, normalized size = 7.4 \begin{align*} -{\frac{x}{c}{\frac{1}{\sqrt{{a}^{2}c{x}^{2}+c}}}}+{\frac{1}{a} \left ( i\sqrt{-{a}^{2}}+a \right ) \left ( -{\frac{1}{3\,c}{\frac{1}{\sqrt{-{a}^{2}}}} \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c+2\,c\sqrt{-{a}^{2}} \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}}-{\frac{1}{3\,{c}^{2}} \left ( 2\, \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ){a}^{2}c+2\,c\sqrt{-{a}^{2}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}{\frac{1}{\sqrt{ \left ( x-{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c+2\,c\sqrt{-{a}^{2}} \left ( x-{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}}+{\frac{1}{a} \left ( i\sqrt{-{a}^{2}}-a \right ) \left ({\frac{1}{3\,c}{\frac{1}{\sqrt{-{a}^{2}}}} \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{-1}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c-2\,c\sqrt{-{a}^{2}} \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}}+{\frac{1}{3\,{c}^{2}} \left ( 2\, \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ){a}^{2}c-2\,c\sqrt{-{a}^{2}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}{\frac{1}{\sqrt{ \left ( x+{\frac{1}{{a}^{2}}\sqrt{-{a}^{2}}} \right ) ^{2}{a}^{2}c-2\,c\sqrt{-{a}^{2}} \left ( x+{\frac{\sqrt{-{a}^{2}}}{{a}^{2}}} \right ) }}}} \right ){\frac{1}{\sqrt{-{a}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, a x + 1\right )}^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{3}{2}}{\left (a^{2} x^{2} + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37992, 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{\left (i a x + 1\right )^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}} \left (a^{2} x^{2} + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14482, size = 103, normalized size = 1.91 \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 )}}{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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