Optimal. Leaf size=119 \[ \frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6139, 639, 192, 191} \[ \frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}} \]
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 )^5} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{11/2}} \, dx}{c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{9 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{21 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{105 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{315 c^5}\\ &=-\frac {1-a x}{9 a c^5 \left (1-a^2 x^2\right )^{9/2}}+\frac {8 x}{63 c^5 \left (1-a^2 x^2\right )^{7/2}}+\frac {16 x}{105 c^5 \left (1-a^2 x^2\right )^{5/2}}+\frac {64 x}{315 c^5 \left (1-a^2 x^2\right )^{3/2}}+\frac {128 x}{315 c^5 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 91, normalized size = 0.76 \[ -\frac {128 a^8 x^8+128 a^7 x^7-448 a^6 x^6-448 a^5 x^5+560 a^4 x^4+560 a^3 x^3-280 a^2 x^2-280 a x+35}{315 a c^5 (1-a x)^{7/2} (a x+1)^{9/2}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.75, size = 249, normalized size = 2.09 \[ -\frac {35 \, a^{9} x^{9} + 35 \, a^{8} x^{8} - 140 \, a^{7} x^{7} - 140 \, a^{6} x^{6} + 210 \, a^{5} x^{5} + 210 \, a^{4} x^{4} - 140 \, a^{3} x^{3} - 140 \, a^{2} x^{2} + 35 \, a x + {\left (128 \, a^{8} x^{8} + 128 \, a^{7} x^{7} - 448 \, a^{6} x^{6} - 448 \, a^{5} x^{5} + 560 \, a^{4} x^{4} + 560 \, a^{3} x^{3} - 280 \, a^{2} x^{2} - 280 \, a x + 35\right )} \sqrt {-a^{2} x^{2} + 1} + 35}{315 \, {\left (a^{10} c^{5} x^{9} + a^{9} c^{5} x^{8} - 4 \, a^{8} c^{5} x^{7} - 4 \, a^{7} c^{5} x^{6} + 6 \, a^{6} c^{5} x^{5} + 6 \, a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} - 4 \, a^{3} c^{5} x^{2} + a^{2} c^{5} x + a c^{5}\right )}} \]
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 \[ \int -\frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 90, normalized size = 0.76 \[ -\frac {128 x^{8} a^{8}+128 a^{7} x^{7}-448 x^{6} a^{6}-448 x^{5} a^{5}+560 x^{4} a^{4}+560 x^{3} a^{3}-280 a^{2} x^{2}-280 a x +35}{315 \left (-a^{2} x^{2}+1\right )^{\frac {7}{2}} \left (a x +1\right ) c^{5} a} \]
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 \[ -\int \frac {\sqrt {-a^{2} x^{2} + 1}}{{\left (a^{2} c x^{2} - c\right )}^{5} {\left (a x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 177, normalized size = 1.49 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {53\,x}{252\,c^5}-\frac {5}{36\,a\,c^5}\right )}{{\left (a\,x-1\right )}^4\,{\left (a\,x+1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}}{144\,a\,c^5\,{\left (a\,x+1\right )}^5}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {733\,x}{5040\,c^5}+\frac {5}{144\,a\,c^5}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {128\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,\left (a\,x-1\right )\,\left (a\,x+1\right )}+\frac {64\,x\,\sqrt {1-a^2\,x^2}}{315\,c^5\,{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2} \]
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 \[ \frac {\int \frac {1}{a^{9} x^{9} \sqrt {- a^{2} x^{2} + 1} + a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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