Optimal. Leaf size=105 \[ -\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 x^2}-\frac {2 a^2 n (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )}{2-n} \]
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Rubi [A]
time = 0.03, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6261, 98, 133}
\begin {gather*} -\frac {2 a^2 n (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{2-n}-\frac {(a x+1)^{\frac {n+2}{2}} (1-a x)^{1-\frac {n}{2}}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rule 6261
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^3} \, dx &=\int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{x^3} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 x^2}+\frac {1}{2} (a n) \int \frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{x^2} \, dx\\ &=-\frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {2+n}{2}}}{2 x^2}-\frac {2 a^2 n (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )}{2-n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 91, normalized size = 0.87 \begin {gather*} \frac {(1-a x)^{1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (-\left ((-2+n) (1+a x)^2\right )+4 a^2 n x^2 \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )\right )}{2 (-2+n) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{3}}\, dx\]
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*} \int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{x^{3}}\, dx \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 {atanh}\left (a\,x\right )}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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