Optimal. Leaf size=129 \[ \frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3} \]
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Rubi [A]
time = 0.06, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (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^{\tanh ^{-1}(a x)}}{(c-a c x)^5} \, dx &=c \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^6} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {1}{3} \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^5} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^4} \, dx}{21 c}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \int \frac {\sqrt {1-a^2 x^2}}{(c-a c x)^3} \, dx}{105 c^2}\\ &=\frac {\left (1-a^2 x^2\right )^{3/2}}{9 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{3/2}}{21 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{105 a c^5 (1-a x)^4}+\frac {2 \left (1-a^2 x^2\right )^{3/2}}{315 a c^5 (1-a x)^3}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 0.40 \begin {gather*} \frac {(1+a x)^{3/2} \left (58-33 a x+12 a^2 x^2-2 a^3 x^3\right )}{315 a c^5 (1-a x)^{9/2}} \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. \(400\) vs.
\(2(113)=226\).
time = 0.77, size = 401, normalized size = 3.11
method | result | size |
gosper | \(-\frac {\left (2 a^{3} x^{3}-12 a^{2} x^{2}+33 a x -58\right ) \left (a x +1\right )^{2}}{315 \left (a x -1\right )^{4} c^{5} a \sqrt {-a^{2} x^{2}+1}}\) | \(57\) |
trager | \(\frac {\left (2 a^{4} x^{4}-10 a^{3} x^{3}+21 a^{2} x^{2}-25 a x -58\right ) \sqrt {-a^{2} x^{2}+1}}{315 c^{5} \left (a x -1\right )^{5} a}\) | \(58\) |
default | \(-\frac {\frac {\frac {2 \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{9 a \left (x -\frac {1}{a}\right )^{5}}-\frac {8 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}\right )}{9}}{a^{5}}+\frac {\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{7 a \left (x -\frac {1}{a}\right )^{4}}-\frac {3 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{5 a \left (x -\frac {1}{a}\right )^{3}}-\frac {2 a \left (\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 a \left (x -\frac {1}{a}\right )^{2}}-\frac {\sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}{3 \left (x -\frac {1}{a}\right )}\right )}{5}\right )}{7}}{a^{4}}}{c^{5}}\) | \(401\) |
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. 264 vs.
\(2 (109) = 218\).
time = 0.47, size = 264, normalized size = 2.05 \begin {gather*} -\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{9 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{63 \, {\left (a^{5} c^{5} x^{4} - 4 \, a^{4} c^{5} x^{3} + 6 \, a^{3} c^{5} x^{2} - 4 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {\sqrt {-a^{2} x^{2} + 1}}{105 \, {\left (a^{4} c^{5} x^{3} - 3 \, a^{3} c^{5} x^{2} + 3 \, a^{2} c^{5} x - a c^{5}\right )}} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{3} c^{5} x^{2} - 2 \, a^{2} c^{5} x + a c^{5}\right )}} + \frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{315 \, {\left (a^{2} c^{5} x - a c^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 144, normalized size = 1.12 \begin {gather*} \frac {58 \, a^{5} x^{5} - 290 \, a^{4} x^{4} + 580 \, a^{3} x^{3} - 580 \, a^{2} x^{2} + 290 \, a x + {\left (2 \, a^{4} x^{4} - 10 \, a^{3} x^{3} + 21 \, a^{2} x^{2} - 25 \, a x - 58\right )} \sqrt {-a^{2} x^{2} + 1} - 58}{315 \, {\left (a^{6} c^{5} x^{5} - 5 \, a^{5} c^{5} x^{4} + 10 \, a^{4} c^{5} x^{3} - 10 \, a^{3} c^{5} x^{2} + 5 \, a^{2} c^{5} x - a c^{5}\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^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 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} - 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 10 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 10 a^{2} x^{2} \sqrt {- a^{2} x^{2} + 1} + 5 a x \sqrt {- a^{2} x^{2} + 1} - \sqrt {- a^{2} x^{2} + 1}}\, dx}{c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.44, size = 321, normalized size = 2.49 \begin {gather*} \frac {-\frac {16 i \, \mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}{c^{3}} - \frac {\frac {35 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 180 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 378 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 420 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} + 315 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}}{c^{3}} + \frac {9 \, {\left (5 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 21 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 35 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 35 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}}}{\mathrm {sgn}\left (\frac {1}{a c x - c}\right ) \mathrm {sgn}\left (a\right ) \mathrm {sgn}\left (c\right )}}{2520 \, c^{2} {\left | a \right |}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.87, size = 57, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}\,\left (-2\,a^4\,x^4+10\,a^3\,x^3-21\,a^2\,x^2+25\,a\,x+58\right )}{315\,a\,c^5\,{\left (a\,x-1\right )}^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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