Optimal. Leaf size=91 \[ \frac{4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3} \]
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Rubi [A] time = 0.0969696, 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} \[ \frac{4 (2 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{(4 a x+1) e^{-\coth ^{-1}(a x)}}{15 a c^3 \left (1-a^2 x^2\right )^2}-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3} \]
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 )^3} \, dx &=\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 \int \frac{e^{-\coth ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^2} \, dx}{5 c}\\ &=\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 e^{-\coth ^{-1}(a x)} (1+2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}+\frac{8 \int \frac{e^{-\coth ^{-1}(a x)}}{c-a^2 c x^2} \, dx}{15 c^2}\\ &=-\frac{8 e^{-\coth ^{-1}(a x)}}{15 a c^3}+\frac{e^{-\coth ^{-1}(a x)} (1+4 a x)}{15 a c^3 \left (1-a^2 x^2\right )^2}+\frac{4 e^{-\coth ^{-1}(a x)} (1+2 a x)}{15 a c^3 \left (1-a^2 x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.188673, size = 64, normalized size = 0.7 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (8 a^4 x^4+8 a^3 x^3-12 a^2 x^2-12 a x+3\right )}{15 (a x-1)^2 (a c x+c)^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 65, normalized size = 0.7 \begin{align*} -{\frac{8\,{x}^{4}{a}^{4}+8\,{x}^{3}{a}^{3}-12\,{a}^{2}{x}^{2}-12\,ax+3}{15\, \left ({a}^{2}{x}^{2}-1 \right ) ^{2}{c}^{3}a}\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 = 0.978571, size = 138, normalized size = 1.52 \begin{align*} -\frac{1}{240} \, a{\left (\frac{3 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} - 20 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} + 90 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac{5 \,{\left (\frac{12 \,{\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \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 = 1.51358, size = 163, normalized size = 1.79 \begin{align*} -\frac{{\left (8 \, a^{4} x^{4} + 8 \, a^{3} x^{3} - 12 \, a^{2} x^{2} - 12 \, a x + 3\right )} \sqrt{\frac{a x - 1}{a x + 1}}}{15 \,{\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \frac{\int \frac{\sqrt{\frac{a x}{a x + 1} - \frac{1}{a x + 1}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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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