Optimal. Leaf size=53 \[ \frac {4}{5 a c^4 (1-a x)^5}-\frac {1}{a c^4 (1-a x)^4}+\frac {1}{3 a c^4 (1-a x)^3} \]
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Rubi [A]
time = 0.03, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6264, 45}
\begin {gather*} \frac {1}{3 a c^4 (1-a x)^3}-\frac {1}{a c^4 (1-a x)^4}+\frac {4}{5 a c^4 (1-a x)^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 6264
Rubi steps
\begin {align*} \int \frac {e^{4 \tanh ^{-1}(a x)}}{(c-a c x)^4} \, dx &=\frac {\int \frac {(1+a x)^2}{(1-a x)^6} \, dx}{c^4}\\ &=\frac {\int \left (\frac {4}{(-1+a x)^6}+\frac {4}{(-1+a x)^5}+\frac {1}{(-1+a x)^4}\right ) \, dx}{c^4}\\ &=\frac {4}{5 a c^4 (1-a x)^5}-\frac {1}{a c^4 (1-a x)^4}+\frac {1}{3 a c^4 (1-a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.58 \begin {gather*} -\frac {2+5 a x+5 a^2 x^2}{15 a c^4 (-1+a x)^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.10, size = 42, normalized size = 0.79
method | result | size |
risch | \(\frac {-\frac {a \,x^{2}}{3}-\frac {x}{3}-\frac {2}{15 a}}{\left (a x -1\right )^{5} c^{4}}\) | \(27\) |
gosper | \(-\frac {5 a^{2} x^{2}+5 a x +2}{15 c^{4} \left (a x -1\right )^{5} a}\) | \(30\) |
default | \(\frac {-\frac {1}{a \left (a x -1\right )^{4}}-\frac {1}{3 a \left (a x -1\right )^{3}}-\frac {4}{5 a \left (a x -1\right )^{5}}}{c^{4}}\) | \(42\) |
norman | \(\frac {-\frac {a^{2} x^{3}}{3 c}-\frac {x}{c}-\frac {2 a^{3} x^{4}}{3 c}+\frac {8 a^{4} x^{5}}{15 c}-\frac {2 a^{5} x^{6}}{15 c}}{\left (a x -1\right )^{5} c^{3} \left (a x +1\right )}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 77, normalized size = 1.45 \begin {gather*} -\frac {5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 77, normalized size = 1.45 \begin {gather*} -\frac {5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left (a^{6} c^{4} x^{5} - 5 \, a^{5} c^{4} x^{4} + 10 \, a^{4} c^{4} x^{3} - 10 \, a^{3} c^{4} x^{2} + 5 \, a^{2} c^{4} x - a c^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 80, normalized size = 1.51 \begin {gather*} \frac {- 5 a^{2} x^{2} - 5 a x - 2}{15 a^{6} c^{4} x^{5} - 75 a^{5} c^{4} x^{4} + 150 a^{4} c^{4} x^{3} - 150 a^{3} c^{4} x^{2} + 75 a^{2} c^{4} x - 15 a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 29, normalized size = 0.55 \begin {gather*} -\frac {5 \, a^{2} x^{2} + 5 \, a x + 2}{15 \, {\left (a x - 1\right )}^{5} a c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.88, size = 29, normalized size = 0.55 \begin {gather*} -\frac {5\,a^2\,x^2+5\,a\,x+2}{15\,a\,c^4\,{\left (a\,x-1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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