Optimal. Leaf size=90 \[ \frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{c n}-\frac {2 (1-a x)^{-n/2} (a x+1)^{n/2} \, _2F_1\left (1,\frac {n}{2};\frac {n+2}{2};\frac {a x+1}{1-a x}\right )}{c n} \]
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Rubi [A] time = 0.11, antiderivative size = 100, normalized size of antiderivative = 1.11, number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {6150, 96, 131} \[ \frac {(1-a x)^{-n/2} (a x+1)^{n/2}}{c n}-\frac {2 (1-a x)^{1-\frac {n}{2}} (a x+1)^{\frac {n-2}{2}} \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{c (2-n)} \]
Warning: Unable to verify antiderivative.
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Rule 96
Rule 131
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x \left (c-a^2 c x^2\right )} \, dx &=\frac {\int \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{c}\\ &=\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c n}+\frac {\int \frac {(1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{c}\\ &=\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {2 (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)} \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )}{c (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.94 \[ \frac {(1-a x)^{-n/2} (a x+1)^{\frac {n}{2}-1} \left ((n-2) (a x+1)-2 n (a x-1) \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )\right )}{c (n-2) n} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{3} - c x}, x\right ) \]
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 \[ \int -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x \left (-a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ -\int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x\,\left (c-a^2\,c\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{3} - x}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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