Optimal. Leaf size=66 \[ -\frac {c 2^{\frac {n}{2}+1} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (2-\frac {n}{2},-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]
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Rubi [A] time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6129, 69} \[ -\frac {c 2^{\frac {n}{2}+1} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (2-\frac {n}{2},-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 6129
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} (c-a c x) \, dx &=c \int (1-a x)^{1-\frac {n}{2}} (1+a x)^{n/2} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} c (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (2-\frac {n}{2},-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (4-n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 63, normalized size = 0.95 \[ \frac {c 2^{\frac {n}{2}+1} (1-a x)^{2-\frac {n}{2}} \, _2F_1\left (2-\frac {n}{2},-\frac {n}{2};3-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int -{\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.05, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a c x +c \right )\, dx \]
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 \[ -\int {\left (a c x - c\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,\left (c-a\,c\,x\right ) \,d x \]
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 \[ - c \left (\int a x e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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