Optimal. Leaf size=65 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5} \]
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Rubi [A]
time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6262, 673, 665}
\begin {gather*} \frac {\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{3 \tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^6} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6}+\frac {1}{7} c^2 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{7 a c^3 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{35 a c^3 (1-a x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 34, normalized size = 0.52 \begin {gather*} -\frac {(-6+a x) (1+a x)^{5/2}}{35 a c^3 (1-a x)^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(440\) vs.
\(2(57)=114\).
time = 1.09, size = 441, normalized size = 6.78
method | result | size |
gosper | \(-\frac {\left (a x -6\right ) \left (a x +1\right )^{4}}{35 \left (a x -1\right )^{2} c^{3} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(40\) |
trager | \(-\frac {\left (a^{3} x^{3}-4 a^{2} x^{2}-11 a x -6\right ) \sqrt {-a^{2} x^{2}+1}}{35 c^{3} \left (a x -1\right )^{4} a}\) | \(49\) |
default | \(-\frac {\frac {x}{\sqrt {-a^{2} x^{2}+1}}+\frac {\frac {12}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {36 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}}{a^{2}}+\frac {\frac {8}{7 a \left (x -\frac {1}{a}\right )^{3} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {32 a \left (\frac {1}{5 a \left (x -\frac {1}{a}\right )^{2} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {3 a \left (\frac {1}{3 a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {-2 a^{2} \left (x -\frac {1}{a}\right )-2 a}{3 a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}\right )}{5}\right )}{7}}{a^{3}}+\frac {\frac {2}{a \left (x -\frac {1}{a}\right ) \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}+\frac {2 \left (-2 a^{2} \left (x -\frac {1}{a}\right )-2 a \right )}{a \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}}{a}}{c^{3}}\) | \(441\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 216 vs.
\(2 (55) = 110\).
time = 0.26, size = 216, normalized size = 3.32 \begin {gather*} -\frac {8}{7 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac {52}{35 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{3} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x + \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} - \frac {18}{35 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{3} x - \sqrt {-a^{2} x^{2} + 1} a c^{3}\right )}} + \frac {x}{35 \, \sqrt {-a^{2} x^{2} + 1} c^{3}} \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. 116 vs.
\(2 (55) = 110\).
time = 0.35, size = 116, normalized size = 1.78 \begin {gather*} \frac {6 \, a^{4} x^{4} - 24 \, a^{3} x^{3} + 36 \, a^{2} x^{2} - 24 \, a x - {\left (a^{3} x^{3} - 4 \, a^{2} x^{2} - 11 \, a x - 6\right )} \sqrt {-a^{2} x^{2} + 1} + 6}{35 \, {\left (a^{5} c^{3} x^{4} - 4 \, a^{4} c^{3} x^{3} + 6 \, a^{3} c^{3} x^{2} - 4 \, a^{2} c^{3} x + a c^{3}\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 {3 a x}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {3 a^{2} x^{2}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {a^{3} x^{3}}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx + \int \frac {1}{- a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 3 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} - 2 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 2 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 3 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 199 vs.
\(2 (55) = 110\).
time = 0.43, size = 199, normalized size = 3.06 \begin {gather*} -\frac {2 \, {\left (\frac {7 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {91 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {70 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{3}}{a^{6} x^{3}} - \frac {140 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{4}}{a^{8} x^{4}} + \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{5}}{a^{10} x^{5}} - \frac {35 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{6}}{a^{12} x^{6}} - 6\right )}}{35 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{7} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.81, size = 299, normalized size = 4.60 \begin {gather*} \frac {8\,a^3\,\sqrt {1-a^2\,x^2}}{35\,\left (a^6\,c^3\,x^2-2\,a^5\,c^3\,x+a^4\,c^3\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{5\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {4\,a\,\sqrt {1-a^2\,x^2}}{7\,\left (a^6\,c^3\,x^4-4\,a^5\,c^3\,x^3+6\,a^4\,c^3\,x^2-4\,a^3\,c^3\,x+a^2\,c^3\right )}+\frac {\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {16\,\sqrt {1-a^2\,x^2}}{35\,\sqrt {-a^2}\,\left (3\,c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}+a^2\,c^3\,x^3\,\sqrt {-a^2}-3\,a\,c^3\,x^2\,\sqrt {-a^2}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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