Optimal. Leaf size=92 \[ -\frac {4 b (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^2 (2-n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6298, 133}
\begin {gather*} -\frac {4 b (-a-b x+1)^{1-\frac {n}{2}} (a+b x+1)^{\frac {n-2}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(a+1) (-a-b x+1)}{(1-a) (a+b x+1)}\right )}{(1-a)^2 (2-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 133
Rule 6298
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a+b x)}}{x^2} \, dx &=\int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x^2} \, dx\\ &=-\frac {4 b (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^2 (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 83, normalized size = 0.90 \begin {gather*} \frac {4 b (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{-1+\frac {n}{2}} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(1+a) (-1+a+b x)}{(-1+a) (1+a+b x)}\right )}{(-1+a)^2 (-2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (b x +a \right )}}{x^{2}}\, dx\]
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 + b x \right )}}}{x^{2}}\, 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+b\,x\right )}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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