Optimal. Leaf size=91 \[ -\frac{(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac{8 e^{3 \coth ^{-1}(a x)}}{35 a c^3} \]
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Rubi [A] time = 0.100469, antiderivative size = 91, normalized size of antiderivative = 1., 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} \[ -\frac{(3-4 a x) e^{3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 (3-2 a x) e^{3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}-\frac{8 e^{3 \coth ^{-1}(a x)}}{35 a c^3} \]
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 )^3} \, dx &=-\frac{e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 \int \frac{e^{3 \coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{7 c}\\ &=-\frac{e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}-\frac{24 \int \frac{e^{3 \coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{35 c^2}\\ &=-\frac{8 e^{3 \coth ^{-1}(a x)}}{35 a c^3}-\frac{e^{3 \coth ^{-1}(a x)} (3-4 a x)}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{12 e^{3 \coth ^{-1}(a x)} (3-2 a x)}{35 a c^3 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.204774, size = 66, normalized size = 0.73 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13\right )}{35 c^3 (a x-1)^4 (a x+1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.122, size = 65, normalized size = 0.7 \begin{align*} -{\frac{8\,{x}^{4}{a}^{4}-24\,{x}^{3}{a}^{3}+20\,{a}^{2}{x}^{2}+4\,ax-13}{35\,{c}^{3} \left ({a}^{2}{x}^{2}-1 \right ) ^{2}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 = 1.2443, size = 131, normalized size = 1.44 \begin{align*} -\frac{1}{560} \, a{\left (\frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{\frac{28 \,{\left (a x - 1\right )}}{a x + 1} - \frac{70 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5}{a^{2} c^{3} \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 = 1.60854, size = 205, normalized size = 2.25 \begin{align*} -\frac{{\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{35 \,{\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\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 = 1.17203, size = 150, normalized size = 1.65 \begin{align*} -\frac{1}{560} \, a{\left (\frac{{\left (a x + 1\right )}^{3}{\left (\frac{28 \,{\left (a x - 1\right )}}{a x + 1} - \frac{70 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac{140 \,{\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 5\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}} + \frac{35 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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