Optimal. Leaf size=97 \[ \frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)}+\frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac {\sqrt {1-a^2 x^2}}{5 a c^2 (1-a x)^3} \]
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Rubi [A] time = 0.07, 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 = {6138, 655, 659, 651} \[ \frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)}+\frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac {\sqrt {1-a^2 x^2}}{5 a c^2 (1-a x)^3} \]
Antiderivative was successfully verified.
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Rule 651
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 )^2} \, dx &=\frac {\int \frac {(1+a x)^3}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{c^2}\\ &=\frac {\int \frac {1}{(1-a x)^3 \sqrt {1-a^2 x^2}} \, dx}{c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac {2 \int \frac {1}{(1-a x)^2 \sqrt {1-a^2 x^2}} \, dx}{5 c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac {2 \int \frac {1}{(1-a x) \sqrt {1-a^2 x^2}} \, dx}{15 c^2}\\ &=\frac {\sqrt {1-a^2 x^2}}{5 a c^2 (1-a x)^3}+\frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)^2}+\frac {2 \sqrt {1-a^2 x^2}}{15 a c^2 (1-a x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.44 \[ \frac {\sqrt {a x+1} \left (2 a^2 x^2-6 a x+7\right )}{15 a c^2 (1-a x)^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.55, size = 91, normalized size = 0.94 \[ \frac {7 \, a^{3} x^{3} - 21 \, a^{2} x^{2} + 21 \, a x - {\left (2 \, a^{2} x^{2} - 6 \, a x + 7\right )} \sqrt {-a^{2} x^{2} + 1} - 7}{15 \, {\left (a^{4} c^{2} x^{3} - 3 \, a^{3} c^{2} x^{2} + 3 \, a^{2} c^{2} x - a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 145, normalized size = 1.49 \[ -\frac {2 \, {\left (\frac {20 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {40 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {30 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {15 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} - 7\right )}}{15 \, c^{2} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 49, normalized size = 0.51 \[ -\frac {\left (2 a^{2} x^{2}-6 a x +7\right ) \left (a x +1\right )^{2}}{15 \left (a x -1\right ) c^{2} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a} \]
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 \[ \int \frac {{\left (a x + 1\right )}^{3}}{{\left (a^{2} c x^{2} - c\right )}^{2} {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.91, size = 127, normalized size = 1.31 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {2\,a^3}{15\,c^2\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {a^3}{5\,c^2\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {2\,a^4}{15\,c^2\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^3\,\sqrt {-a^2}} \]
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 \[ \frac {\int \frac {3 a x}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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