Optimal. Leaf size=138 \[ \frac{16 x}{63 c^4 \sqrt{1-a^2 x^2}}+\frac{8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (a x+1) \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{1}{9 a c^4 (a x+1)^3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.0930441, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {6139, 655, 659, 192, 191} \[ \frac{16 x}{63 c^4 \sqrt{1-a^2 x^2}}+\frac{8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (a x+1) \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (a x+1)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{1}{9 a c^4 (a x+1)^3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 6139
Rule 655
Rule 659
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{e^{-3 \tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac{\int \frac{(1-a x)^3}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^4}\\ &=\frac{\int \frac{1}{(1+a x)^3 \left (1-a^2 x^2\right )^{5/2}} \, dx}{c^4}\\ &=-\frac{1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}+\frac{2 \int \frac{1}{(1+a x)^2 \left (1-a^2 x^2\right )^{5/2}} \, dx}{3 c^4}\\ &=-\frac{1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}+\frac{10 \int \frac{1}{(1+a x) \left (1-a^2 x^2\right )^{5/2}} \, dx}{21 c^4}\\ &=-\frac{1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac{8 \int \frac{1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{21 c^4}\\ &=\frac{8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac{16 \int \frac{1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{63 c^4}\\ &=\frac{8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}-\frac{1}{9 a c^4 (1+a x)^3 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x)^2 \left (1-a^2 x^2\right )^{3/2}}-\frac{2}{21 a c^4 (1+a x) \left (1-a^2 x^2\right )^{3/2}}+\frac{16 x}{63 c^4 \sqrt{1-a^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.0323553, size = 75, normalized size = 0.54 \[ -\frac{16 a^6 x^6+48 a^5 x^5+24 a^4 x^4-56 a^3 x^3-66 a^2 x^2-6 a x+19}{63 a c^4 (1-a x)^{3/2} (a x+1)^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 74, normalized size = 0.5 \begin{align*} -{\frac{16\,{x}^{6}{a}^{6}+48\,{x}^{5}{a}^{5}+24\,{x}^{4}{a}^{4}-56\,{x}^{3}{a}^{3}-66\,{a}^{2}{x}^{2}-6\,ax+19}{63\, \left ( ax+1 \right ) ^{3}{c}^{4}a} \left ( -{a}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{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{{\left (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.07722, size = 419, normalized size = 3.04 \begin{align*} -\frac{19 \, a^{7} x^{7} + 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} - 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 19 \, a^{2} x^{2} + 57 \, a x +{\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt{-a^{2} x^{2} + 1} + 19}{63 \,{\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, 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(-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 (-a^{2} x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4}{\left (a x + 1\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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