Optimal. Leaf size=127 \[ -\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^4-20 n^2+64\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ -\frac {(n-4 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 (n-2 a x) e^{n \coth ^{-1}(a x)}}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^4-20 n^2+64\right )} \]
Antiderivative was successfully verified.
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Rule 6183
Rule 6185
Rubi steps
\begin {align*} \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx &=-\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}+\frac {12 \int \frac {e^{n \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{c \left (16-n^2\right )}\\ &=-\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}+\frac {24 \int \frac {e^{n \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{c^2 \left (64-20 n^2+n^4\right )}\\ &=\frac {24 e^{n \coth ^{-1}(a x)}}{a c^3 n \left (64-20 n^2+n^4\right )}-\frac {e^{n \coth ^{-1}(a x)} (n-4 a x)}{a c^3 \left (16-n^2\right ) \left (1-a^2 x^2\right )^2}-\frac {12 e^{n \coth ^{-1}(a x)} (n-2 a x)}{a c^3 \left (4-n^2\right ) \left (16-n^2\right ) \left (1-a^2 x^2\right )}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 97, normalized size = 0.76 \[ \frac {\left (4 n^2 \left (3 a^2 x^2-4\right )-8 a n x \left (3 a^2 x^2-5\right )+24 \left (a^2 x^2-1\right )^2-4 a n^3 x+n^4\right ) e^{n \coth ^{-1}(a x)}}{a c^3 n \left (n^2-16\right ) \left (n^2-4\right ) \left (a^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 175, normalized size = 1.38 \[ -\frac {{\left (24 \, a^{4} x^{4} + 24 \, a^{3} n x^{3} + n^{4} + 12 \, {\left (a^{2} n^{2} - 4 \, a^{2}\right )} x^{2} - 16 \, n^{2} + 4 \, {\left (a n^{3} - 10 \, a n\right )} x + 24\right )} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{a c^{3} n^{5} - 20 \, a c^{3} n^{3} + 64 \, a c^{3} n + {\left (a^{5} c^{3} n^{5} - 20 \, a^{5} c^{3} n^{3} + 64 \, a^{5} c^{3} n\right )} x^{4} - 2 \, {\left (a^{3} c^{3} n^{5} - 20 \, a^{3} c^{3} n^{3} + 64 \, a^{3} c^{3} n\right )} x^{2}} \]
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 \[ \int -\frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 101, normalized size = 0.80 \[ \frac {\left (24 x^{4} a^{4}-24 x^{3} a^{3} n +12 a^{2} n^{2} x^{2}-4 a \,n^{3} x -48 a^{2} x^{2}+n^{4}+40 x a n -16 n^{2}+24\right ) {\mathrm e}^{n \,\mathrm {arccoth}\left (a x \right )}}{\left (a^{2} x^{2}-1\right )^{2} c^{3} a \left (n^{2}-16\right ) \left (n^{2}-4\right ) n} \]
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 \[ -\int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {1}{2} \, n}}{{\left (a^{2} c x^{2} - c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.73, size = 192, normalized size = 1.51 \[ \frac {{\left (\frac {a\,x+1}{a\,x}\right )}^{n/2}\,\left (\frac {24\,x^4}{a\,c^3\,n\,\left (n^4-20\,n^2+64\right )}-\frac {4\,x\,\left (n^2-10\right )}{a^4\,c^3\,\left (n^4-20\,n^2+64\right )}-\frac {24\,x^3}{a^2\,c^3\,\left (n^4-20\,n^2+64\right )}+\frac {n^4-16\,n^2+24}{a^5\,c^3\,n\,\left (n^4-20\,n^2+64\right )}+\frac {x^2\,\left (12\,n^2-48\right )}{a^3\,c^3\,n\,\left (n^4-20\,n^2+64\right )}\right )}{{\left (\frac {a\,x-1}{a\,x}\right )}^{n/2}\,\left (\frac {1}{a^4}+x^4-\frac {2\,x^2}{a^2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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