Optimal. Leaf size=131 \[ \frac {i \left (n^2-2\right ) e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};-e^{2 i \tan ^{-1}(a x)}\right )}{a^4 c n}+\frac {\left (-i n^2+n+2 i\right ) e^{n \tan ^{-1}(a x)}}{2 a^4 c n}-\frac {n x e^{n \tan ^{-1}(a x)}}{2 a^3 c}+\frac {x^2 e^{n \tan ^{-1}(a x)}}{2 a^2 c} \]
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Rubi [A] time = 0.23, 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 = {5082, 100, 143, 69} \[ \frac {2^{-1-\frac {i n}{2}} \left (2-n^2\right ) (1-i a x)^{1+\frac {i n}{2}} \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {1}{2} (1-i a x)\right )}{a^4 c (2+i n)}+\frac {i (1+i a x)^{-\frac {i n}{2}} \left (i a n^2 x-n^2-i n+2\right ) (1-i a x)^{\frac {i n}{2}}}{2 a^4 c n}+\frac {x^2 (1+i a x)^{-\frac {i n}{2}} (1-i a x)^{\frac {i n}{2}}}{2 a^2 c} \]
Warning: Unable to verify antiderivative.
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Rule 69
Rule 100
Rule 143
Rule 5082
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 = 0.13, size = 141, normalized size = 1.08 \[ \frac {(1-i a x)^{\frac {i n}{2}} \left (\frac {\left (a^2 n x^2-\left (n^2 (a x+i)\right )+n+2 i\right ) (1+i a x)^{-\frac {i n}{2}}}{n}+\frac {2^{-\frac {i n}{2}} \left (n^2-2\right ) (a x+i) \, _2F_1\left (\frac {i n}{2}+1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {1}{2} (1-i a x)\right )}{n-2 i}\right )}{2 a^4 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}, x\right ) \]
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 = 0.31, 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 \[ \int \frac {x^{3} e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{2} + c}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x^{3} e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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