Optimal. Leaf size=164 \[ \frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (1-i a x)\right )}{a^3 c}-\frac {(1+i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^3 c n}+\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c} \]
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Rubi [A] time = 0.13, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5082, 90, 79, 69} \[ \frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (1-i a x)\right )}{a^3 c}+\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c}-\frac {(1+i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^3 c n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 79
Rule 90
Rule 5082
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^2}{c+a^2 c x^2} \, dx &=\frac {\int x^2 (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, dx}{c}\\ &=\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c}+\frac {\int (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} (-1-a n x) \, dx}{a^2 c}\\ &=-\frac {(1+i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^3 c n}+\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c}+\frac {(i n) \int (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \, dx}{a^2 c}\\ &=-\frac {(1+i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^3 c n}+\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c}+\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};1+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 121, normalized size = 0.74 \[ \frac {(1-i a x)^{\frac {i n}{2}} (2+2 i a x)^{-\frac {i n}{2}} \left (2 i n (1+i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};\frac {i n}{2}+1;\frac {1}{2} (1-i a x)\right )+2^{\frac {i n}{2}} (-1+n (a x-i))\right )}{a^3 c n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, x\right ) \]
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 = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{2}}{a^{2} c \,x^{2}+c}\, dx \]
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 \[ \int \frac {x^{2} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x^{2} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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