Optimal. Leaf size=123 \[ \frac {a (1+n) (1-a x)^{-n/2} (1+a x)^{n/2}}{c n}-\frac {(1-a x)^{-n/2} (1+a x)^{n/2}}{c x}-\frac {2 a (1-a x)^{-n/2} (1+a x)^{n/2} \, _2F_1\left (1,\frac {n}{2};\frac {2+n}{2};\frac {1+a x}{1-a x}\right )}{c} \]
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Rubi [A]
time = 0.09, antiderivative size = 123, normalized size of antiderivative = 1.00, 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 = {6285, 105, 160,
12, 133} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 105
Rule 133
Rule 160
Rule 6285
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 = 0.03, size = 103, normalized size = 0.84 \begin {gather*} \frac {(1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}} \left ((-2+n) (1+a x) (a x+n (-1+a x))-2 a n^2 x (-1+a x) \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{1+a x}\right )\right )}{c (-2+n) n x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{2} x^{4} - x^{2}}\, dx}{c} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,\left (c-a^2\,c\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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