Optimal. Leaf size=97 \[ -\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}} \]
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Rubi [A]
time = 0.04, 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 = {6274, 653, 198,
197} \begin {gather*} \frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 653
Rule 6274
Rubi steps
\begin {align*} \int \frac {e^{-\tanh ^{-1}(a x)}}{\left (c-a^2 c x^2\right )^4} \, dx &=\frac {\int \frac {1-a x}{\left (1-a^2 x^2\right )^{9/2}} \, dx}{c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 \int \frac {1}{\left (1-a^2 x^2\right )^{7/2}} \, dx}{7 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {24 \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{35 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{35 c^4}\\ &=-\frac {1-a x}{7 a c^4 \left (1-a^2 x^2\right )^{7/2}}+\frac {6 x}{35 c^4 \left (1-a^2 x^2\right )^{5/2}}+\frac {8 x}{35 c^4 \left (1-a^2 x^2\right )^{3/2}}+\frac {16 x}{35 c^4 \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 75, normalized size = 0.77 \begin {gather*} \frac {-5+30 a x+30 a^2 x^2-40 a^3 x^3-40 a^4 x^4+16 a^5 x^5+16 a^6 x^6}{35 a c^4 (1-a x)^{5/2} (1+a x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.06, size = 720, normalized size = 7.42
method | result | size |
gosper | \(\frac {16 x^{6} a^{6}+16 x^{5} a^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5}{35 \left (-a^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a x +1\right ) a \,c^{4}}\) | \(74\) |
trager | \(-\frac {\left (16 x^{6} a^{6}+16 x^{5} a^{5}-40 a^{4} x^{4}-40 a^{3} x^{3}+30 a^{2} x^{2}+30 a x -5\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{4} \left (a x +1\right )^{4} \left (a x -1\right )^{3} a}\) | \(81\) |
default | \(\frac {\frac {\frac {35 \sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}{256}+\frac {35 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{256 \sqrt {a^{2}}}}{a}+\frac {\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{5 a \left (x -\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{15 \left (x -\frac {1}{a}\right )^{3}}}{32 a^{4}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}}{8 a^{4}}+\frac {-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{32 a \left (x +\frac {1}{a}\right )^{2}}-\frac {5 a \left (\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}+\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )}}\right )}{\sqrt {a^{2}}}\right )}{32}}{a^{2}}+\frac {-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{7 a \left (x +\frac {1}{a}\right )^{5}}+\frac {2 a \left (-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{5 a \left (x +\frac {1}{a}\right )^{4}}-\frac {\left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{15 \left (x +\frac {1}{a}\right )^{3}}\right )}{7}}{16 a^{5}}-\frac {35 \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{256 a}-\frac {5 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{192 a^{4} \left (x -\frac {1}{a}\right )^{3}}-\frac {5 \left (-a^{2} \left (x +\frac {1}{a}\right )^{2}+2 a \left (x +\frac {1}{a}\right )\right )^{\frac {3}{2}}}{96 a^{4} \left (x +\frac {1}{a}\right )^{3}}+\frac {\frac {15 \left (-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a \right )^{\frac {3}{2}}}{128 a \left (x -\frac {1}{a}\right )^{2}}+\frac {15 a \left (\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}-\frac {a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 \left (x -\frac {1}{a}\right ) a}}\right )}{\sqrt {a^{2}}}\right )}{128}}{a^{2}}}{c^{4}}\) | \(720\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (80) = 160\).
time = 0.44, size = 197, normalized size = 2.03 \begin {gather*} -\frac {5 \, a^{7} x^{7} + 5 \, a^{6} x^{6} - 15 \, a^{5} x^{5} - 15 \, a^{4} x^{4} + 15 \, a^{3} x^{3} + 15 \, a^{2} x^{2} - 5 \, a x + {\left (16 \, a^{6} x^{6} + 16 \, a^{5} x^{5} - 40 \, a^{4} x^{4} - 40 \, a^{3} x^{3} + 30 \, a^{2} x^{2} + 30 \, a x - 5\right )} \sqrt {-a^{2} x^{2} + 1} - 5}{35 \, {\left (a^{8} c^{4} x^{7} + a^{7} c^{4} x^{6} - 3 \, a^{6} c^{4} x^{5} - 3 \, a^{5} c^{4} x^{4} + 3 \, a^{4} c^{4} x^{3} + 3 \, a^{3} c^{4} x^{2} - 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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} - a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} + 3 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 3 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + a x \sqrt {- a^{2} x^{2} + 1} + \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{4}} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.18, size = 145, normalized size = 1.49 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {8\,x}{35\,c^4}+\frac {1}{56\,a\,c^4}\right )}{{\left (a\,x-1\right )}^2\,{\left (a\,x+1\right )}^2}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {17\,x}{70\,c^4}-\frac {1}{7\,a\,c^4}\right )}{{\left (a\,x-1\right )}^3\,{\left (a\,x+1\right )}^3}-\frac {\sqrt {1-a^2\,x^2}}{56\,a\,c^4\,{\left (a\,x+1\right )}^4}-\frac {16\,x\,\sqrt {1-a^2\,x^2}}{35\,c^4\,\left (a\,x-1\right )\,\left (a\,x+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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