Optimal. Leaf size=119 \[ \frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}} \]
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Rubi [A] time = 0.09, 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 = {6138, 655, 659, 191} \[ \frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 191
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 )^3} \, dx &=\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^3}\\ &=\frac {\int \frac {1}{(1-a x)^3 \left (1-a^2 x^2\right )^{3/2}} \, dx}{c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4 \int \frac {1}{(1-a x)^2 \left (1-a^2 x^2\right )^{3/2}} \, dx}{7 c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {12 \int \frac {1}{(1-a x) \left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}+\frac {8 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^3}\\ &=\frac {8 x}{35 c^3 \sqrt {1-a^2 x^2}}+\frac {1}{7 a c^3 (1-a x)^3 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x)^2 \sqrt {1-a^2 x^2}}+\frac {4}{35 a c^3 (1-a x) \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 61, normalized size = 0.51 \[ \frac {8 a^4 x^4-24 a^3 x^3+20 a^2 x^2+4 a x-13}{35 a c^3 (a x-1)^3 \sqrt {1-a^2 x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 144, normalized size = 1.21 \[ \frac {13 \, a^{5} x^{5} - 39 \, a^{4} x^{4} + 26 \, a^{3} x^{3} + 26 \, a^{2} x^{2} - 39 \, a x - {\left (8 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 20 \, a^{2} x^{2} + 4 \, a x - 13\right )} \sqrt {-a^{2} x^{2} + 1} + 13}{35 \, {\left (a^{6} c^{3} x^{5} - 3 \, a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} + 2 \, a^{3} c^{3} x^{2} - 3 \, a^{2} c^{3} x + a c^{3}\right )}} \]
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 \[ \int -\frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 63, normalized size = 0.53 \[ -\frac {\left (8 x^{4} a^{4}-24 x^{3} a^{3}+20 a^{2} x^{2}+4 a x -13\right ) \left (a x +1\right )}{35 \left (a x -1\right )^{2} c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]
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 \[ -\int \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{3} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 125, normalized size = 1.05 \[ \frac {29\,\sqrt {1-a^2\,x^2}}{280\,a\,c^3\,{\left (a\,x-1\right )}^2}-\frac {13\,\sqrt {1-a^2\,x^2}}{140\,a\,c^3\,{\left (a\,x-1\right )}^3}+\frac {\sqrt {1-a^2\,x^2}}{14\,a\,c^3\,{\left (a\,x-1\right )}^4}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^3}+\frac {29}{280\,a\,c^3}\right )}{\left (a\,x-1\right )\,\left (a\,x+1\right )} \]
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 \[ \frac {\int \frac {3 a x}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{8} x^{8} \sqrt {- a^{2} x^{2} + 1} - 4 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 6 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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