Optimal. Leaf size=126 \[ -\frac {i a^2 \left (n^2-2\right ) e^{n \tan ^{-1}(a x)} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};e^{2 i \tan ^{-1}(a x)}\right )}{c n}+\frac {i a^2 \left (n^2+i n-2\right ) e^{n \tan ^{-1}(a x)}}{2 c n}-\frac {e^{n \tan ^{-1}(a x)}}{2 c x^2}-\frac {a n e^{n \tan ^{-1}(a x)}}{2 c x} \]
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Rubi [A] time = 0.18, antiderivative size = 242, normalized size of antiderivative = 1.92, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5082, 129, 151, 155, 12, 131} \[ \frac {a^2 \left (2-n^2\right ) (1-i a x)^{1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \, _2F_1\left (1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {1-i a x}{i a x+1}\right )}{c (2+i n)}-\frac {a^2 \left (-i n^2+n+2 i\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 129
Rule 131
Rule 151
Rule 155
Rule 5082
Rubi steps
\begin {align*} \int \frac {e^{n \tan ^{-1}(a x)}}{x^3 \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^3} \, dx}{c}\\ &=-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {\int \frac {(1-i a x)^{-1+\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}} \left (-a n+2 a^2 x\right )}{x^2} \, dx}{2 c}\\ &=-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{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^2 \left (2-n^2\right )-a^3 n x\right )}{x} \, dx}{2 c}\\ &=-\frac {a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x}-\frac {\int \frac {a^3 n \left (2-n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{2 a c n}\\ &=-\frac {a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x}-\frac {\left (a^2 \left (2-n^2\right )\right ) \int \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-1-\frac {i n}{2}}}{x} \, dx}{2 c}\\ &=-\frac {a^2 \left (2 i+n-i n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c n}-\frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x^2}-\frac {a n (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}}}{2 c x}+\frac {a^2 \left (2-n^2\right ) (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 = 0.08, size = 174, normalized size = 1.38 \[ \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (2 a^2 n \left (n^2-2\right ) x^2 (1-i a x) \, _2F_1\left (1,\frac {i n}{2}+1;\frac {i n}{2}+2;\frac {a x+i}{i-a x}\right )+i (n-2 i) (a x-i) \left (i n \left (a^2 x^2+1\right )-2 a^2 x^2+a n^2 x (a x+i)\right )\right )}{2 c n (n-2 i) x^2 (a x-i)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (n \arctan \left (a x\right )\right )}}{a^{2} c x^{5} + c x^{3}}, x\right ) \]
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 \[ \int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ \int \frac {e^{\left (n \arctan \left (a x\right )\right )}}{{\left (a^{2} c x^{2} + c\right )} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \]
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 \[ \frac {\int \frac {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{5} + x^{3}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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