Optimal. Leaf size=127 \[ \frac{8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{-\coth ^{-1}(a x)}}{35 a c^4} \]
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Rubi [A] time = 0.135755, 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} \[ \frac{8 (2 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )}+\frac{2 (4 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^2}+\frac{(6 a x+1) e^{-\coth ^{-1}(a x)}}{35 a c^4 \left (1-a^2 x^2\right )^3}-\frac{16 e^{-\coth ^{-1}(a x)}}{35 a c^4} \]
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 = 0.254699, size = 80, normalized size = 0.63 \[ -\frac{x \sqrt{1-\frac{1}{a^2 x^2}} \left (16 a^6 x^6+16 a^5 x^5-40 a^4 x^4-40 a^3 x^3+30 a^2 x^2+30 a x-5\right )}{35 (a x-1)^3 (a c x+c)^4} \]
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}+16\,{x}^{5}{a}^{5}-40\,{x}^{4}{a}^{4}-40\,{x}^{3}{a}^{3}+30\,{a}^{2}{x}^{2}+30\,ax-5}{35\, \left ({a}^{2}{x}^{2}-1 \right ) ^{3}{c}^{4}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 = 1.04753, size = 182, normalized size = 1.43 \begin{align*} \frac{1}{2240} \, a{\left (\frac{5 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{7}{2}} - 42 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}} + 175 \, \left (\frac{a x - 1}{a x + 1}\right )^{\frac{3}{2}} - 700 \, \sqrt{\frac{a x - 1}{a x + 1}}}{a^{2} c^{4}} + \frac{7 \,{\left (\frac{10 \,{\left (a x - 1\right )}}{a x + 1} - \frac{75 \,{\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} - 1\right )}}{a^{2} c^{4} \left (\frac{a x - 1}{a x + 1}\right )^{\frac{5}{2}}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59774, size = 223, normalized size = 1.76 \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} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{3} c^{4} x^{2} - 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{\sqrt{\frac{a x - 1}{a x + 1}}}{{\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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