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 = 0.0845581, 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 = {5075, 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 5075
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 [C] time = 0.0216242, size = 71, normalized size = 0.74 \[ -\frac{4 i \sqrt{2 a^2 x^2+2} \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{1}{2} (1-i a x)\right )}{3 a (1-i a x)^{3/2} \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.229, size = 800, normalized size = 8.3 \begin{align*} \text{result too large to display} \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 )}^{4}}{\sqrt{a^{2} c x^{2} + c}{\left (a^{2} x^{2} + 1\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.22531, 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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (i a x + 1\right )^{4}}{\sqrt{c \left (a^{2} x^{2} + 1\right )} \left (a^{2} x^{2} + 1\right )^{2}}\, dx \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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