Optimal. Leaf size=123 \[ -\frac {2 a (a x+1)^{n/2} (1-a x)^{-n/2} \, _2F_1\left (1,\frac {n}{2};\frac {n+2}{2};\frac {a x+1}{1-a x}\right )}{c}+\frac {a (n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{c n}-\frac {(a x+1)^{n/2} (1-a x)^{-n/2}}{c x} \]
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Rubi [A] time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6150, 129, 155, 12, 131} \[ -\frac {2 a n (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{c (2-n)}+\frac {a (n+1) (a x+1)^{n/2} (1-a x)^{-n/2}}{c n}-\frac {(a x+1)^{n/2} (1-a x)^{-n/2}}{c x} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 129
Rule 131
Rule 155
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{n \tanh ^{-1}(a x)}}{x^2 \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^2} \, dx}{c}\\ &=-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac {\int \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-1+\frac {n}{2}} \left (-a n-a^2 x\right )}{x} \, dx}{c}\\ &=\frac {a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}+\frac {\int \frac {a^2 n^2 (1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{a c n}\\ &=\frac {a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}+\frac {(a n) \int \frac {(1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{c}\\ &=\frac {a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac {2 a n (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 = 0.06, size = 103, normalized size = 0.84 \[ \frac {(1-a x)^{-n/2} (a x+1)^{\frac {n}{2}-1} \left ((n-2) (a x+1) (n (a x-1)+a x)-2 a n^2 x (a x-1) \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )\right )}{c (n-2) n x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{2} c x^{4} - c x^{2}}, x\right ) \]
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 \[ \int -\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )}\, dx \]
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 \[ -\int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,\left (c-a^2\,c\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{4} - x^{2}}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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