Optimal. Leaf size=70 \[ -\frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)} \]
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Rubi [A] time = 0.05, 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} \[ -\frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)} \]
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 )^2 \, dx &=c^2 \int (1-a x)^{2-\frac {n}{2}} (1+a x)^{2+\frac {n}{2}} \, dx\\ &=-\frac {2^{3+\frac {n}{2}} c^2 (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-2-\frac {n}{2},3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (6-n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 67, normalized size = 0.96 \[ \frac {c^2 2^{\frac {n}{2}+3} (1-a x)^{3-\frac {n}{2}} \, _2F_1\left (-\frac {n}{2}-2,3-\frac {n}{2};4-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a (n-6)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.21, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}, x\right ) \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm e}^{n \arctanh \left (a x \right )} \left (-a^{2} c \,x^{2}+c \right )^{2}\, dx \]
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 \[ \int {\left (a^{2} c x^{2} - c\right )}^{2} \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}\,{d x} \]
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 \[ \int {\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}\,{\left (c-a^2\,c\,x^2\right )}^2 \,d x \]
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 \[ c^{2} \left (\int \left (- 2 a^{2} x^{2} e^{n \operatorname {atanh}{\left (a x \right )}}\right )\, dx + \int a^{4} x^{4} e^{n \operatorname {atanh}{\left (a x \right )}}\, dx + \int e^{n \operatorname {atanh}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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