Optimal. Leaf size=65 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3} \]
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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 = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6262, 673, 665}
\begin {gather*} \frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}+\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 665
Rule 673
Rule 6262
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{(c-a c x)^3} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {1}{5} \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{5 a c^3 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{15 a c^3 (1-a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 0.54 \begin {gather*} \frac {(4-a x) (1+a x)^{3/2}}{15 a c^3 (1-a x)^{5/2}} \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. \(220\) vs.
\(2(57)=114\).
time = 0.84, size = 221, normalized size = 3.40
method | result | size |
gosper | \(-\frac {\left (a x -4\right ) \left (a x +1\right )^{2}}{15 \left (a x -1\right )^{2} c^{3} a \sqrt {-a^{2} x^{2}+1}}\) | \(40\) |
trager | \(\frac {\left (a^{2} x^{2}-3 a x -4\right ) \sqrt {-a^{2} x^{2}+1}}{15 c^{3} \left (a x -1\right )^{3} a}\) | \(41\) |
default | \(-\frac {\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}}{a^{2}}+\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {4 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}}{a^{3}}}{c^{3}}\) | \(221\) |
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. 126 vs.
\(2 (55) = 110\).
time = 0.48, size = 126, normalized size = 1.94 \begin {gather*} -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{5 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, a^{2} c^{3} x - a c^{3}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (a^{3} c^{3} x^{2} - 2 \, a^{2} c^{3} x + a c^{3}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{15 \, {\left (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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Fricas [A]
time = 0.36, size = 89, normalized size = 1.37 \begin {gather*} \frac {4 \, a^{3} x^{3} - 12 \, a^{2} x^{2} + 12 \, a x + {\left (a^{2} x^{2} - 3 \, a x - 4\right )} \sqrt {-a^{2} x^{2} + 1} - 4}{15 \, {\left (a^{4} c^{3} x^{3} - 3 \, a^{3} c^{3} x^{2} + 3 \, 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 {a x}{a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 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^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 3 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. 145 vs.
\(2 (55) = 110\).
time = 0.43, size = 145, normalized size = 2.23 \begin {gather*} -\frac {2 \, {\left (\frac {5 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}}{a^{2} x} - \frac {25 \, {\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )}^{2}}{a^{4} x^{2}} + \frac {15 \, {\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}} - 4\right )}}{15 \, c^{3} {\left (\frac {\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a}{a^{2} x} - 1\right )}^{5} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.83, size = 183, normalized size = 2.82 \begin {gather*} \frac {2\,\sqrt {1-a^2\,x^2}}{5\,\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 )}-\frac {\sqrt {1-a^2\,x^2}}{15\,\sqrt {-a^2}\,\left (c^3\,x\,\sqrt {-a^2}-\frac {c^3\,\sqrt {-a^2}}{a}\right )}-\frac {a\,\sqrt {1-a^2\,x^2}}{15\,\left (a^4\,c^3\,x^2-2\,a^3\,c^3\,x+a^2\,c^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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