Optimal. Leaf size=119 \[ -\frac{(1-6 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (1-2 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}-\frac{2 (1-4 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4} \]
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Rubi [A] time = 0.127988, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {6185, 6183} \[ -\frac{(1-6 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{8 (1-2 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}-\frac{2 (1-4 a x) e^{\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4} \]
Antiderivative was successfully verified.
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Rule 6185
Rule 6183
Rubi steps
\begin{align*} \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}+\frac{6 \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^3} \, dx}{7 c}\\ &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{24 \int \frac{e^{\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{35 c^2}\\ &=-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{\coth ^{-1}(a x)} (1-2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{16 \int \frac{e^{\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^3}\\ &=\frac{16 e^{\coth ^{-1}(a x)}}{35 a c^4}-\frac{e^{\coth ^{-1}(a x)} (1-6 a x)}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{2 e^{\coth ^{-1}(a x)} (1-4 a x)}{35 a c^4 \left (1-a^2 x^2\right )^2}-\frac{8 e^{\coth ^{-1}(a x)} (1-2 a x)}{35 a c^4 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.247455, size = 82, normalized size = 0.69 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6-16 a^5 x^5-40 a^4 x^4+40 a^3 x^3+30 a^2 x^2-30 a x-5\right )}{35 c^4 (a x-1)^4 (a x+1)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.044, size = 81, normalized size = 0.7 \begin{align*}{\frac{16\,{x}^{6}{a}^{6}-16\,{x}^{5}{a}^{5}-40\,{x}^{4}{a}^{4}+40\,{x}^{3}{a}^{3}+30\,{a}^{2}{x}^{2}-30\,ax-5}{35\,{c}^{4} \left ({a}^{2}{x}^{2}-1 \right ) ^{3}a}{\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10038, size = 178, normalized size = 1.5 \begin{align*} \frac{1}{2240} \, a{\left (\frac{7 \,{\left (\left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 10 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 75 \, \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{2} c^{4}} + \frac{\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{175 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{700 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55682, size = 278, normalized size = 2.34 \begin{align*} \frac{{\left (16 \, a^{6} x^{6} - 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} + 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} - 30 \, a x - 5\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\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 [A] time = 1.12541, size = 259, normalized size = 2.18 \begin{align*} \frac{1}{2240} \, a{\left (\frac{{\left (a x + 1\right )}^{3}{\left (\frac{42 \,{\left (a x - 1\right )}}{a x + 1} - \frac{175 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{700 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt{\frac{a x - 1}{a x + 1}}} - \frac{7 \,{\left (\frac{10 \,{\left (a x - 1\right )} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{a x + 1} - \frac{{\left (a x - 1\right )}^{2} a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - 75 \, a^{8} c^{16} \sqrt{\frac{a x - 1}{a x + 1}}\right )}}{a^{10} c^{20}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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