Optimal. Leaf size=90 \[ -\frac{2 i a \, _2F_1\left (1,-\frac{i n}{2};1-\frac{i n}{2};\frac{2 i}{a x+i}-1\right ) e^{n \tan ^{-1}(a x)}}{c}+\frac{i a (n+i) e^{n \tan ^{-1}(a x)}}{c n}-\frac{e^{n \tan ^{-1}(a x)}}{c x} \]
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Rubi [A] time = 0.137301, antiderivative size = 180, normalized size of antiderivative = 2., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {5082, 129, 155, 12, 131} \[ -\frac{2 a n (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{1-i a x}{i a x+1}\right )}{c (2+i n)}-\frac{a (1-i n) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c x} \]
Warning: Unable to verify antiderivative.
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Rule 5082
Rule 129
Rule 155
Rule 12
Rule 131
Rubi steps
\begin{align*} \int \frac{e^{n \tan ^{-1}(a x)}}{x^2 \left (c+a^2 c x^2\right )} \, dx &=\frac{\int \frac{(1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}}}{x^2} \, dx}{c}\\ &=-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c x}-\frac{\int \frac{(1-i a x)^{-1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \left (-a n+a^2 x\right )}{x} \, dx}{c}\\ &=-\frac{a (1-i n) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c x}+\frac{\int \frac{a^2 n^2 (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}}}{x} \, dx}{a c n}\\ &=-\frac{a (1-i n) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c x}+\frac{(a n) \int \frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}}}{x} \, dx}{c}\\ &=-\frac{a (1-i n) (1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c n}-\frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}}}{c x}-\frac{2 a n (1-i a x)^{1+\frac{i n}{2}} (1+i a x)^{-1-\frac{i n}{2}} \, _2F_1\left (1,1+\frac{i n}{2};2+\frac{i n}{2};\frac{1-i a x}{1+i a x}\right )}{c (2+i n)}\\ \end{align*}
Mathematica [A] time = 0.0458572, size = 142, normalized size = 1.58 \[ \frac{(1-i a x)^{\frac{i n}{2}} (1+i a x)^{-\frac{i n}{2}} \left (2 a n^2 x (1-i a x) \, _2F_1\left (1,\frac{i n}{2}+1;\frac{i n}{2}+2;\frac{a x+i}{i-a x}\right )+(n-2 i) (1+i a x) (n (a x+i)+i a x)\right )}{c n (n-2 i) x (a x-i)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.326, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{n\arctan \left ( ax \right ) }}}{{x}^{2} \left ({a}^{2}c{x}^{2}+c \right ) }}\, dx \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{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{4} + c x^{2}}, x\right ) \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*} \frac{\int \frac{e^{n \operatorname{atan}{\left (a x \right )}}}{a^{2} x^{4} + x^{2}}\, dx}{c} \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{e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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