Optimal. Leaf size=68 \[ -\frac {c^3 2^{\frac {n}{2}+1} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \]
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Rubi [A] time = 0.05, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6129, 69} \[ -\frac {c^3 2^{\frac {n}{2}+1} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \]
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)^3 \, dx &=c^3 \int (1-a x)^{3-\frac {n}{2}} (1+a x)^{n/2} \, dx\\ &=-\frac {2^{1+\frac {n}{2}} c^3 (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 65, normalized size = 0.96 \[ \frac {c^3 2^{\frac {n}{2}+1} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (4-\frac {n}{2},-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-8)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{3} c^{3} x^{3} - 3 \, a^{2} c^{3} x^{2} + 3 \, a c^{3} x - c^{3}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int -{\left (a c x - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.24, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a c x +c \right )^{3}\, dx \]
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 \[ -\int {\left (a c x - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a\,c\,x\right )}^3 \,d x \]
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 \[ - c^{3} \left (\int 3 a x e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- 3 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{3} x^{3} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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