Optimal. Leaf size=91 \[ -\frac{12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3} \]
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Rubi [A] time = 0.100999, 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} \[ -\frac{12 (2 a x+3) e^{-3 \coth ^{-1}(a x)}}{35 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+3) e^{-3 \coth ^{-1}(a x)}}{7 a c^3 \left (1-a^2 x^2\right )^2}+\frac{8 e^{-3 \coth ^{-1}(a x)}}{35 a c^3} \]
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 = 0.185284, size = 66, normalized size = 0.73 \[ \frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4+24 a^3 x^3+20 a^2 x^2-4 a x-13\right )}{35 c^3 (a x-1) (a x+1)^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.047, 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\, \left ({a}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}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 = 1.03213, size = 139, normalized size = 1.53 \begin{align*} -\frac{1}{560} \, a{\left (\frac{5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 28 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 70 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 140 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac{35}{a^{2} c^{3} \sqrt{\frac{a x - 1}{a x + 1}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53686, size = 182, normalized size = 2. \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} + 2 \, a^{4} c^{3} x^{3} - 2 \, 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 [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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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