Optimal. Leaf size=126 \[ \frac {i a^2 e^{n \text {ArcTan}(a x)} \left (-2+i n+n^2\right )}{2 c n}-\frac {e^{n \text {ArcTan}(a x)}}{2 c x^2}-\frac {a e^{n \text {ArcTan}(a x)} n}{2 c x}-\frac {i a^2 e^{n \text {ArcTan}(a x)} \left (-2+n^2\right ) \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};e^{2 i \text {ArcTan}(a x)}\right )}{c n} \]
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Rubi [A]
time = 0.13, antiderivative size = 233, normalized size of antiderivative = 1.85, 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 = {5190, 105, 156,
160, 12, 133} \begin {gather*} \frac {i a^2 \left (2-n^2\right ) (1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \, _2F_1\left (1,-\frac {i n}{2};1-\frac {i n}{2};\frac {i a x+1}{1-i a x}\right )}{c 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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 156
Rule 160
Rule 5190
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 = 0.06, size = 174, normalized size = 1.38 \begin {gather*} \frac {(1-i a x)^{\frac {i n}{2}} (1+i a x)^{-\frac {i n}{2}} \left (i (-2 i+n) (-i+a x) \left (-2 a^2 x^2+a n^2 x (i+a x)+i n \left (1+a^2 x^2\right )\right )+2 a^2 n \left (-2+n^2\right ) x^2 (1-i a x) \, _2F_1\left (1,1+\frac {i n}{2};2+\frac {i n}{2};\frac {i+a x}{i-a x}\right )\right )}{2 c n (-2 i+n) x^2 (-i+a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctan \left (a x \right )}}{x^{3} \left (a^{2} c \,x^{2}+c \right )}\, dx\]
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 \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 {e^{n \operatorname {atan}{\left (a x \right )}}}{a^{2} x^{5} + x^{3}}\, 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 {{\mathrm {e}}^{n\,\mathrm {atan}\left (a\,x\right )}}{x^3\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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