Optimal. Leaf size=70 \[ -\frac {c^3 2^{\frac {n}{2}+4} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-3,4-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \]
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Rubi [A] time = 0.06, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6140, 69} \[ -\frac {c^3 2^{\frac {n}{2}+4} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-3,4-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (8-n)} \]
Antiderivative was successfully verified.
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Rule 69
Rule 6140
Rubi steps
\begin {align*} \int e^{n \tanh ^{-1}(a x)} \left (c-a^2 c x^2\right )^3 \, dx &=c^3 \int (1-a x)^{3-\frac {n}{2}} (1+a x)^{3+\frac {n}{2}} \, dx\\ &=-\frac {2^{4+\frac {n}{2}} c^3 (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (-3-\frac {n}{2},4-\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 = 67, normalized size = 0.96 \[ \frac {c^3 2^{\frac {n}{2}+4} (1-a x)^{4-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-3,4-\frac {n}{2};5-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-8)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int -{\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.27, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{3}\, dx \]
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 \[ -\int {\left (a^{2} c x^{2} - c\right )}^{3} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^3 \,d x \]
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 \[ - c^{3} \left (\int 3 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- 3 a^{4} x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{6} x^{6} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int \left (- e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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