Optimal. Leaf size=71 \[ -\frac {2^{\frac {n}{2}+1} (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{b (2-n)} \]
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Rubi [A] time = 0.01, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6161, 69} \[ -\frac {2^{\frac {n}{2}+1} (-a-b x+1)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (-a-b x+1)\right )}{b (2-n)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 6161
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a+b x)} \, dx &=\int (1-a-b x)^{-n/2} (1+a+b x)^{n/2} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} (1-a-b x)^{1-\frac {n}{2}} \, _2F_1\left (1-\frac {n}{2},-\frac {n}{2};2-\frac {n}{2};\frac {1}{2} (1-a-b x)\right )}{b (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 0.70 \[ \frac {4 e^{(n+2) \tanh ^{-1}(a+b x)} \, _2F_1\left (2,\frac {n}{2}+1;\frac {n}{2}+2;-e^{2 \tanh ^{-1}(a+b x)}\right )}{b (n+2)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (b x +a \right )}\, dx \]
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 \[ \int \left (\frac {b x + a + 1}{b x + a - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a+b\,x\right )} \,d x \]
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 \[ \int e^{n \operatorname {atanh}{\left (a + b x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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