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 = 0.06, antiderivative size = 97, normalized size of antiderivative = 1.00, 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 191
Rule 192
Rule 639
Rule 6139
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*}
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Mathematica [A] time = 0.04, 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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fricas [B] time = 0.59, size = 197, normalized size = 2.03 \[ -\frac {5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x + {\left (16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1} - 5}{35 \, {\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 74, normalized size = 0.76 \[ \frac {16 x^{6} a^{6}+16 x^{5} a^{5}-40 x^{4} a^{4}-40 x^{3} a^{3}+30 a^{2} x^{2}+30 a x -5}{35 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a x +1\right ) c^{4} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.18, size = 145, normalized size = 1.49 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^4}+\frac {1}{56\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {17\,x}{70\,c^4}-\frac {1}{7\,a\,c^4}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{56\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{35\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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