Optimal. Leaf size=164 \[ -\frac {(1+i n) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^3 c n}+\frac {x (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{a^2 c}+\frac {i 2^{1-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2},\frac {i n}{2};1+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a^3 c} \]
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Rubi [A]
time = 0.10, 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 = {5190, 92, 80,
71} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 80
Rule 92
Rule 5190
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 = 0.10, size = 77, normalized size = 0.47 \begin {gather*} \frac {e^{n \text {ArcTan}(a x)} \left (-2 i-n+4 e^{2 i \text {ArcTan}(a x)} n \, _2F_1\left (2,1-\frac {i n}{2};2-\frac {i n}{2};-e^{2 i \text {ArcTan}(a x)}\right )\right )}{a^3 c n (2 i+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \frac {\int \frac {x^{2} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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