Optimal. Leaf size=131 \[ \frac {e^{n \text {ArcTan}(a x)} \left (2 i+n-i n^2\right )}{2 a^4 c n}-\frac {e^{n \text {ArcTan}(a x)} n x}{2 a^3 c}+\frac {e^{n \text {ArcTan}(a x)} x^2}{2 a^2 c}+\frac {i e^{n \text {ArcTan}(a x)} \left (-2+n^2\right ) \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};-e^{2 i \text {ArcTan}(a x)}\right )}{a^4 c n} \]
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Rubi [A]
time = 0.17, antiderivative size = 206, normalized size of antiderivative = 1.57, 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, 102, 148,
71} \begin {gather*} \frac {2^{-1-\frac {i n}{2}} \left (2-n^2\right ) (1-i a x)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {1}{2} (1-i a x)\right )}{a^4 c (2+i n)}+\frac {i (1+i a x)^{-\frac {i n}{2}} \left (i a n^2 x-n^2-i n+2\right ) (1-i a x)^{\frac {i n}{2}}}{2 a^4 c n}+\frac {x^2 (1+i a x)^{-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}}}{2 a^2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 102
Rule 148
Rule 5190
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)} x^3}{c+a^2 c x^2} \, dx &=\frac {\int x^3 (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, dx}{c}\\ &=\frac {x^2 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 a^2 c}+\frac {\int x (1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} (-2-a n x) \, dx}{2 a^2 c}\\ &=\frac {x^2 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 a^2 c}+\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2-i n-n^2+i a n^2 x\right )}{2 a^4 c n}-\frac {\left (i \left (2-n^2\right )\right ) \int (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, dx}{2 a^3 c}\\ &=\frac {x^2 (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 a^2 c}+\frac {i (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2-i n-n^2+i a n^2 x\right )}{2 a^4 c n}+\frac {2^{-1-\frac {i n}{2}} \left (2-n^2\right ) (1-i a x)^{1+\frac {i n}{2}} \, _2F_1\left (1+\frac {i n}{2},1+\frac {i n}{2};2+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{a^4 c (2+i n)}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 141, normalized size = 1.08 \begin {gather*} \frac {(1-i a x)^{\frac {i n}{2}} \left (\frac {(1+i a x)^{-\frac {i n}{2}} \left (2 i+n+a^2 n x^2-n^2 (i+a x)\right )}{n}+\frac {2^{-\frac {i n}{2}} \left (-2+n^2\right ) (i+a x) \, _2F_1\left (1+\frac {i n}{2},1+\frac {i n}{2};2+\frac {i n}{2};\frac {1}{2} (1-i a x)\right )}{-2 i+n}\right )}{2 a^4 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )} x^{3}}{a^{2} c \,x^{2}+c}\, dx\]
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^{3} 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^3\,{\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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