Optimal. Leaf size=152 \[ -\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (2,1-\frac {n}{2};2-\frac {n}{2};\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{(1-a)^3 (1+a) (2-n)} \]
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Rubi [A]
time = 0.07, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6298, 98, 133}
\begin {gather*} -\frac {(a+b x+1)^{\frac {n+2}{2}} (-a-b x+1)^{1-\frac {n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (a+b x+1)^{\frac {n-2}{2}} (-a-b x+1)^{1-\frac {n}{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)^3 (a+1) (2-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 98
Rule 133
Rule 6298
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a+b x)}}{x^3} \, dx &=\int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x^3} \, dx\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}+\frac {(b (2 a+n)) \int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x^2} \, dx}{2 \left (1-a^2\right )}\\ &=-\frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{\frac {2+n}{2}}}{2 \left (1-a^2\right ) x^2}-\frac {2 b^2 (2 a+n) (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)^3 (1+a) (2-n)}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 123, normalized size = 0.81 \begin {gather*} \frac {(1-a-b x)^{1-\frac {n}{2}} (1+a+b x)^{-1+\frac {n}{2}} \left ((-1+a)^2 (-2+n) (1+a+b x)^2-4 b^2 (2 a+n) x^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 )\right )}{2 (-1+a)^3 (1+a) (-2+n) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{n \arctanh \left (b x +a \right )}}{x^{3}}\, dx\]
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 + b 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+b\,x\right )}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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