Optimal. Leaf size=141 \[ \frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}}+\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{21 a c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{21 a c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{9 a c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 141, 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 = {6138, 655, 659, 192, 191} \[ \frac {16 x}{63 c^4 \sqrt {1-a^2 x^2}}+\frac {8 x}{63 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{21 a c^4 (1-a x) \left (1-a^2 x^2\right )^{3/2}}+\frac {2}{21 a c^4 (1-a x)^2 \left (1-a^2 x^2\right )^{3/2}}+\frac {1}{9 a c^4 (1-a x)^3 \left (1-a^2 x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 192
Rule 655
Rule 659
Rule 6138
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 = 0.03, size = 75, normalized size = 0.53 \[ \frac {16 a^6 x^6-48 a^5 x^5+24 a^4 x^4+56 a^3 x^3-66 a^2 x^2+6 a x+19}{63 a c^4 (1-a x)^{9/2} (a x+1)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 198, normalized size = 1.40 \[ \frac {19 \, a^{7} x^{7} - 57 \, a^{6} x^{6} + 19 \, a^{5} x^{5} + 95 \, a^{4} x^{4} - 95 \, a^{3} x^{3} - 19 \, a^{2} x^{2} + 57 \, a x - {\left (16 \, a^{6} x^{6} - 48 \, a^{5} x^{5} + 24 \, a^{4} x^{4} + 56 \, a^{3} x^{3} - 66 \, a^{2} x^{2} + 6 \, a x + 19\right )} \sqrt {-a^{2} x^{2} + 1} - 19}{63 \, {\left (a^{8} c^{4} x^{7} - 3 \, a^{7} c^{4} x^{6} + a^{6} c^{4} x^{5} + 5 \, a^{5} c^{4} x^{4} - 5 \, a^{4} c^{4} x^{3} - a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 74, normalized size = 0.52 \[ -\frac {16 x^{6} a^{6}-48 x^{5} a^{5}+24 x^{4} a^{4}+56 x^{3} a^{3}-66 a^{2} x^{2}+6 a x +19}{63 \left (a x -1\right )^{3} c^{4} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{4} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.26, size = 156, normalized size = 1.11 \[ \frac {13\,\sqrt {1-a^2\,x^2}}{252\,a\,c^4\,{\left (a\,x-1\right )}^4}-\frac {23\,\sqrt {1-a^2\,x^2}}{336\,a\,c^4\,{\left (a\,x-1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{36\,a\,c^4\,{\left (a\,x-1\right )}^5}+\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {197\,x}{1008\,c^4}+\frac {155}{1008\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{63\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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 \[ \frac {\int \frac {3 a x}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{10} x^{10} \sqrt {- a^{2} x^{2} + 1} + 5 a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 10 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 10 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 5 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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