Optimal. Leaf size=127 \[ \frac{(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}+\frac{10 (4 a x+3) e^{-3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4} \]
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Rubi [A] time = 0.137822, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {6185, 6183} \[ \frac{(2 a x+1) e^{-3 \coth ^{-1}(a x)}}{9 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{21 a c^4 \left (1-a^2 x^2\right )}+\frac{10 (4 a x+3) e^{-3 \coth ^{-1}(a x)}}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac{e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{9 c}\\ &=\frac{e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}+\frac{40 \int \frac{e^{-3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{21 c^2}\\ &=\frac{e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}-\frac{16 \int \frac{e^{-3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{21 c^3}\\ &=\frac{16 e^{-3 \coth ^{-1}(a x)}}{63 a c^4}+\frac{e^{-3 \coth ^{-1}(a x)} (1+2 a x)}{9 a c^4 \left (1-a^2 x^2\right )^3}+\frac{10 e^{-3 \coth ^{-1}(a x)} (3+4 a x)}{63 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-3 \coth ^{-1}(a x)} (3+2 a x)}{21 a c^4 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.250705, size = 82, normalized size = 0.65 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6+48 a^5 x^5+24 a^4 x^4-56 a^3 x^3-66 a^2 x^2-6 a x+19\right )}{63 c^4 (a x-1)^2 (a x+1)^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 81, normalized size = 0.6 \begin{align*}{\frac{16\,{x}^{6}{a}^{6}+48\,{x}^{5}{a}^{5}+24\,{x}^{4}{a}^{4}-56\,{x}^{3}{a}^{3}-66\,{a}^{2}{x}^{2}-6\,ax+19}{63\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3}{c}^{4}a} \left ({\frac{ax-1}{ax+1}} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05721, size = 184, normalized size = 1.45 \begin{align*} \frac{1}{4032} \, a{\left (\frac{7 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{9}{2}} - 54 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} + 189 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 420 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 945 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac{21 \,{\left (\frac{18 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.63781, size = 278, normalized size = 2.19 \begin{align*} \frac{{\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{63 \,{\left (a^{7} c^{4} x^{6} + 2 \, a^{6} c^{4} x^{5} - a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 2 \, a^{2} c^{4} x + a c^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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