Optimal. Leaf size=135 \[ \frac {2 (-a-b x+1)^{-n/2} (a+b x+1)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {(a+1) (-a-b x+1)}{(1-a) (a+b x+1)}\right )}{n}-\frac {2^{\frac {n}{2}+1} (-a-b x+1)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{n} \]
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Rubi [A] time = 0.07, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6163, 105, 69, 131} \[ \frac {2 (-a-b x+1)^{-n/2} (a+b x+1)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {(a+1) (-a-b x+1)}{(1-a) (a+b x+1)}\right )}{n}-\frac {2^{\frac {n}{2}+1} (-a-b x+1)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 105
Rule 131
Rule 6163
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a+b x)}}{x} \, dx &=\int \frac {(1-a-b x)^{-n/2} (1+a+b x)^{n/2}}{x} \, dx\\ &=-\left ((-1+a) \int \frac {(1-a-b x)^{-1-\frac {n}{2}} (1+a+b x)^{n/2}}{x} \, dx\right )-b \int (1-a-b x)^{-1-\frac {n}{2}} (1+a+b x)^{n/2} \, dx\\ &=\frac {2 (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {(1+a) (1-a-b x)}{(1-a) (1+a+b x)}\right )}{n}-\frac {2^{1+\frac {n}{2}} (1-a-b x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 111, normalized size = 0.82 \[ \frac {2 (-a-b x+1)^{-n/2} \left ((a+b x+1)^{n/2} \, _2F_1\left (1,-\frac {n}{2};1-\frac {n}{2};\frac {(a+1) (a+b x-1)}{(a-1) (a+b x+1)}\right )-2^{n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x}, x\right ) \]
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 \[ \int \frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (b x +a \right )}}{x}\, dx \]
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 \[ \int \frac {\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )}}{x} \,d x \]
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 \[ \int \frac {e^{n \operatorname {atanh}{\left (a + b x \right )}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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