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 = 0.09, antiderivative size = 138, normalized size of antiderivative = 1.00, 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 191
Rule 192
Rule 655
Rule 659
Rule 6139
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.65, size = 195, normalized size = 1.41 \[ -\frac {19 \, a^{7} x^{7} + 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} - 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} + 19 \, a^{2} x^{2} + 57 \, a x + {\left (16 \, a^{6} x^{6} + 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} - 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} - 6 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} + 19}{63 \, {\left (a^{8} c^{4} x^{7} + 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} + a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x + a c^{4}\right )}} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 74, normalized size = 0.54 \[ -\frac {16 x^{6} a^{6}+48 x^{5} a^{5}+24 x^{4} a^{4}-56 x^{3} a^{3}-66 a^{2} x^{2}-6 a x +19}{63 \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a x +1\right )^{3} c^{4} a} \]
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 \[ \int \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (a x + 1\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.27, size = 156, normalized size = 1.13 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {197\,x}{1008\,c^4}-\frac {155}{1008\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{252\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}}{36\,a\,c^4\,{\left (a\,x+1\right )}^5}-\frac {23\,\sqrt {1-a^2\,x^2}}{336\,a\,c^4\,{\left (a\,x+1\right )}^3}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{63\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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 \[ \frac {\int \frac {1}{a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 3 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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