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 = 0.14, antiderivative size = 180, normalized size of antiderivative = 2.00, 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 12
Rule 129
Rule 131
Rule 155
Rule 5082
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*}
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Mathematica [A] time = 0.06, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{4} + c x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{2} \left (a^{2} c \,x^{2}+c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^2\,\left (c\,a^2\,x^2+c\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{4} + x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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