Optimal. Leaf size=97 \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5} \]
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Rubi [A] time = 0.07, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6127, 659, 651} \[ \frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5} \]
Antiderivative was successfully verified.
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Rule 651
Rule 659
Rule 6127
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac {2}{7} \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{35 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{7 a c^4 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{35 a c^4 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^4 (1-a x)^3}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.44 \[ -\frac {(a x+1)^{3/2} \left (-2 a^2 x^2+10 a x-23\right )}{105 a c^4 (1-a x)^{7/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.53, size = 116, normalized size = 1.20 \[ \frac {23 \, a^{4} x^{4} - 92 \, a^{3} x^{3} + 138 \, a^{2} x^{2} - 92 \, a x + {\left (2 \, a^{3} x^{3} - 8 \, a^{2} x^{2} + 13 \, a x + 23\right )} \sqrt {-a^{2} x^{2} + 1} + 23}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 199, normalized size = 2.05 \[ -\frac {2 \, {\left (\frac {56 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {273 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {350 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {455 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {210 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {105 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 23\right )}}{105 \, c^{4} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \]
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 \[ -\frac {\left (2 a^{2} x^{2}-10 a x +23\right ) \left (a x +1\right )^{2}}{105 \left (a x -1\right )^{3} c^{4} \sqrt {-a^{2} x^{2}+1}\, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 189, normalized size = 1.95 \[ \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{7 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{35 \, {\left (a^{4} c^{4} x^{3} - 3 \, a^{3} c^{4} x^{2} + 3 \, a^{2} c^{4} x - a c^{4}\right )}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{3} c^{4} x^{2} - 2 \, a^{2} c^{4} x + a c^{4}\right )}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{2} c^{4} x - a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.00, size = 49, normalized size = 0.51 \[ \frac {\sqrt {1-a^2\,x^2}\,\left (2\,a^3\,x^3-8\,a^2\,x^2+13\,a\,x+23\right )}{105\,a\,c^4\,{\left (a\,x-1\right )}^4} \]
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 \[ \frac {\int \frac {a x}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 4 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} + 6 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} - 4 a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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