Optimal. Leaf size=94 \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 c n}-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{a^2 c n} \]
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Rubi [A] time = 0.08, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {6150, 79, 69} \[ \frac {2^{\frac {n}{2}+1} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 c n}-\frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{a^2 c n} \]
Antiderivative was successfully verified.
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Rule 69
Rule 79
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)} x}{c-a^2 c x^2} \, dx &=\frac {\int x (1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \, dx}{c}\\ &=-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c n}+\frac {\int (1-a x)^{-1-\frac {n}{2}} (1+a x)^{n/2} \, dx}{a c}\\ &=-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{a^2 c n}+\frac {2^{1+\frac {n}{2}} (1-a x)^{-n/2} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )}{a^2 c n}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 74, normalized size = 0.79 \[ \frac {(1-a x)^{-n/2} \left (2^{\frac {n}{2}+1} \, _2F_1\left (-\frac {n}{2},-\frac {n}{2};1-\frac {n}{2};\frac {1}{2} (1-a x)\right )-(a x+1)^{n/2}\right )}{a^2 c n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}, x\right ) \]
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 \[ \int -\frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )} x}{-a^{2} c \,x^{2}+c}\, dx \]
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 \[ -\int \frac {x \left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{2} - c}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{c-a^2\,c\,x^2} \,d x \]
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 \[ - \frac {\int \frac {x e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{2} - 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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