Optimal. Leaf size=96 \[ \frac{2 i c (1-i a x)^3}{3 a \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{a^2 c x^2+c}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a \sqrt{c}} \]
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Rubi [A] time = 0.0775118, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5074, 669, 653, 217, 206} \[ \frac{2 i c (1-i a x)^3}{3 a \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{a^2 c x^2+c}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )}{a \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 5074
Rule 669
Rule 653
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{e^{-4 i \tan ^{-1}(a x)}}{\sqrt{c+a^2 c x^2}} \, dx &=c^2 \int \frac{(1-i a x)^4}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\\ &=\frac{2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-c \int \frac{(1-i a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\\ &=\frac{2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{c+a^2 c x^2}}+\int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\\ &=\frac{2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{c+a^2 c x^2}}+\operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )\\ &=\frac{2 i c (1-i a x)^3}{3 a \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 i (1-i a x)}{a \sqrt{c+a^2 c x^2}}+\frac{\tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )}{a \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0796571, size = 132, normalized size = 1.38 \[ \frac{2 \sqrt{a^2 x^2+1} \left (2 i \sqrt{1+i a x} \left (2 a^2 x^2+i a x+1\right )+3 i \sqrt{1-i a x} (a x-i)^2 \sin ^{-1}\left (\frac{\sqrt{1-i a x}}{\sqrt{2}}\right )\right )}{3 a \sqrt{1-i a x} (a x-i)^2 \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.172, size = 136, normalized size = 1.4 \begin{align*}{\ln \left ({{a}^{2}cx{\frac{1}{\sqrt{{a}^{2}c}}}}+\sqrt{{a}^{2}c{x}^{2}+c} \right ){\frac{1}{\sqrt{{a}^{2}c}}}}-{\frac{8}{3\,{a}^{2}c}\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) } \left ( x-{\frac{i}{a}} \right ) ^{-1}}-{\frac{{\frac{4\,i}{3}}}{{a}^{3}c}\sqrt{ \left ( x-{\frac{i}{a}} \right ) ^{2}{a}^{2}c+2\,iac \left ( x-{\frac{i}{a}} \right ) } \left ( x-{\frac{i}{a}} \right ) ^{-2}} \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 [B] time = 2.16369, size = 417, normalized size = 4.34 \begin{align*} \frac{{\left (3 \, a^{3} c x^{2} - 6 i \, a^{2} c x - 3 \, a c\right )} \sqrt{\frac{1}{a^{2} c}} \log \left (\frac{2 \,{\left (a^{2} c x + \sqrt{a^{2} c x^{2} + c} a^{2} c \sqrt{\frac{1}{a^{2} c}}\right )}}{x}\right ) -{\left (3 \, a^{3} c x^{2} - 6 i \, a^{2} c x - 3 \, a c\right )} \sqrt{\frac{1}{a^{2} c}} \log \left (\frac{2 \,{\left (a^{2} c x - \sqrt{a^{2} c x^{2} + c} a^{2} c \sqrt{\frac{1}{a^{2} c}}\right )}}{x}\right ) - \sqrt{a^{2} c x^{2} + c}{\left (16 \, a x - 8 i\right )}}{6 \, a^{3} c x^{2} - 12 i \, a^{2} c x - 6 \, a 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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