Optimal. Leaf size=239 \[ -\frac {2 a (a x+1)^{n/2} (1-a x)^{-n/2} \, _2F_1\left (1,\frac {n}{2};\frac {n+2}{2};\frac {a x+1}{1-a x}\right )}{c^2}+\frac {a \left (n^2+4 n+6\right ) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)}-\frac {a \left (-n^3-n^2+4 n+6\right ) (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{c^2 n \left (4-n^2\right )}+\frac {a (n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2 (n+2)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2 x} \]
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Rubi [A] time = 0.25, antiderivative size = 253, normalized size of antiderivative = 1.06, number of steps used = 7, 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} \[ -\frac {2 a n (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}} \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )}{c^2 (2-n)}-\frac {a \left (-n^3-n^2+4 n+6\right ) (a x+1)^{\frac {n-2}{2}} (1-a x)^{1-\frac {n}{2}}}{c^2 n \left (4-n^2\right )}+\frac {a \left (n^2+4 n+6\right ) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-n/2}}{c^2 n (n+2)}+\frac {a (n+3) (a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2 (n+2)}-\frac {(a x+1)^{\frac {n-2}{2}} (1-a x)^{-\frac {n}{2}-1}}{c^2 x} \]
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 )^2} \, dx &=\frac {\int \frac {(1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}}}{x^2} \, dx}{c^2}\\ &=-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}-\frac {\int \frac {(1-a x)^{-2-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \left (-a n-3 a^2 x\right )}{x} \, dx}{c^2}\\ &=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}+\frac {\int \frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{-2+\frac {n}{2}} \left (a^2 n (2+n)+2 a^3 (3+n) x\right )}{x} \, dx}{a c^2 (2+n)}\\ &=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}+\frac {a \left (6+4 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n (2+n)}-\frac {\int \frac {(1-a x)^{-n/2} (1+a x)^{-2+\frac {n}{2}} \left (-a^3 n^2 (2+n)-a^4 \left (6+4 n+n^2\right ) x\right )}{x} \, dx}{a^2 c^2 n (2+n)}\\ &=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}-\frac {a \left (6+4 n-n^2-n^3\right ) (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac {a \left (6+4 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n (2+n)}+\frac {\int \frac {a^4 (2-n) n^2 (2+n) (1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{a^3 c^2 n \left (4-n^2\right )}\\ &=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}-\frac {a \left (6+4 n-n^2-n^3\right ) (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac {a \left (6+4 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n (2+n)}+\frac {(a n) \int \frac {(1-a x)^{-n/2} (1+a x)^{-1+\frac {n}{2}}}{x} \, dx}{c^2}\\ &=\frac {a (3+n) (1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 (2+n)}-\frac {(1-a x)^{-1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 x}-\frac {a \left (6+4 n-n^2-n^3\right ) (1-a x)^{1-\frac {n}{2}} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n \left (4-n^2\right )}+\frac {a \left (6+4 n+n^2\right ) (1-a x)^{-n/2} (1+a x)^{\frac {1}{2} (-2+n)}}{c^2 n (2+n)}-\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 (2-n)}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 163, normalized size = 0.68 \[ -\frac {(1-a x)^{-\frac {n}{2}-1} (a x+1)^{\frac {n}{2}-1} \left (-6 a^3 x^3+a n^2 x \left (a^2 x^2-2\right )+n \left (-4 a^3 x^3+6 a^2 x^2+4 a x-4\right )-2 a (n+2) n^2 x (a x-1)^2 \, _2F_1\left (1,1-\frac {n}{2};2-\frac {n}{2};\frac {1-a x}{a x+1}\right )+n^3 (a x-1)^2 (a x+1)+6 a x\right )}{c^2 (n-2) n (n+2) x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{a^{4} c^{2} x^{6} - 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm e}^{n \arctanh \left (a x \right )}}{x^{2} \left (-a^{2} c \,x^{2}+c \right )^{2}}\, dx \]
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 \[ \int \frac {\left (\frac {a x + 1}{a x - 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {e}}^{n\,\mathrm {atanh}\left (a\,x\right )}}{x^2\,{\left (c-a^2\,c\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {e^{n \operatorname {atanh}{\left (a x \right )}}}{a^{4} x^{6} - 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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