Optimal. Leaf size=97 \[ \frac{16 x}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}} \]
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Rubi [A] time = 0.0566229, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {6139, 639, 192, 191} \[ \frac{16 x}{35 c^4 \sqrt{1-a^2 x^2}}+\frac{8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}} \]
Antiderivative was successfully verified.
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Rule 6139
Rule 639
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac{\int \frac{1-a x}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 \int \frac{1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{24 \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^4}\\ &=-\frac{1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac{6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac{8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{35 c^4 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0344725, size = 75, normalized size = 0.77 \[ \frac{16 a^6 x^6+16 a^5 x^5-40 a^4 x^4-40 a^3 x^3+30 a^2 x^2+30 a x-5}{35 a c^4 (1-a x)^{5/2} (a x+1)^{7/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.03, size = 74, normalized size = 0.8 \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}{ \left ( 35\,ax+35 \right ){c}^{4}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4}{\left (a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.72434, size = 413, normalized size = 4.26 \begin{align*} -\frac{5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x +{\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{-a^{2} x^{2} + 1} - 5}{35 \,{\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{- a^{7} x^{7} \sqrt{- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt{- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt{- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt{- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt{- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt{- a^{2} x^{2} + 1} + a x \sqrt{- a^{2} x^{2} + 1} + \sqrt{- a^{2} x^{2} + 1}}\, dx}{c^{4}} \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{-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4}{\left (a x + 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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