Optimal. Leaf size=129 \[ \frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5} \]
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Rubi [A]
time = 0.07, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8} \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)^5} \, dx &=c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^8} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {1}{11} \left (3 c^2\right ) \int \frac {\left (1-a^2 x^2\right )^{3/2}}{(c-a c x)^7} \, dx\\ &=\frac {\left (1-a^2 x^2\right )^{5/2}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {1}{33} (2 c) \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}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2}{231} \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}}{11 a c^5 (1-a x)^8}+\frac {\left (1-a^2 x^2\right )^{5/2}}{33 a c^5 (1-a x)^7}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{231 a c^5 (1-a x)^6}+\frac {2 \left (1-a^2 x^2\right )^{5/2}}{1155 a c^5 (1-a x)^5}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 51, normalized size = 0.40 \begin {gather*} -\frac {(1+a x)^{5/2} \left (-152+61 a x-16 a^2 x^2+2 a^3 x^3\right )}{1155 a c^5 (1-a x)^{11/2}} \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. \(835\) vs.
\(2(113)=226\).
time = 1.14, size = 836, normalized size = 6.48
method | result | size |
gosper | \(-\frac {\left (2 a^{3} x^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )^{4}}{1155 \left (a x -1\right )^{4} c^{5} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}} a}\) | \(57\) |
trager | \(-\frac {\left (2 a^{5} x^{5}-12 a^{4} x^{4}+31 a^{3} x^{3}-46 a^{2} x^{2}-243 a x -152\right ) \sqrt {-a^{2} x^{2}+1}}{1155 c^{5} \left (a x -1\right )^{6} a}\) | \(66\) |
default | \(-\frac {\frac {\frac {8}{11 a \left (x -\frac {1}{a}\right )^{5} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {48 a \left (\frac {1}{9 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {5 a \left (\frac {1}{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 {4 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}\right )}{9}\right )}{11}}{a^{5}}+\frac {\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}}{a^{2}}+\frac {\frac {6}{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 {24 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 {4}{3 a \left (x -\frac {1}{a}\right )^{4} \sqrt {-a^{2} \left (x -\frac {1}{a}\right )^{2}-2 a \left (x -\frac {1}{a}\right )}}-\frac {20 a \left (\frac {1}{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 {4 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}\right )}{3}}{a^{4}}}{c^{5}}\) | \(836\) |
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. 462 vs.
\(2 (109) = 218\).
time = 0.27, size = 462, normalized size = 3.58 \begin {gather*} -\frac {8}{11 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{6} c^{5} x^{5} - 5 \, \sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} + 10 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 10 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 5 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {28}{33 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{5} c^{5} x^{4} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} + 6 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {58}{231 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{4} c^{5} x^{3} - 3 \, \sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} + 3 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{3} c^{5} x^{2} - 2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x + \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} - \frac {1}{1155 \, {\left (\sqrt {-a^{2} x^{2} + 1} a^{2} c^{5} x - \sqrt {-a^{2} x^{2} + 1} a c^{5}\right )}} + \frac {2 \, x}{1155 \, \sqrt {-a^{2} x^{2} + 1} c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 171, normalized size = 1.33 \begin {gather*} \frac {152 \, a^{6} x^{6} - 912 \, a^{5} x^{5} + 2280 \, a^{4} x^{4} - 3040 \, a^{3} x^{3} + 2280 \, a^{2} x^{2} - 912 \, a x - {\left (2 \, a^{5} x^{5} - 12 \, a^{4} x^{4} + 31 \, a^{3} x^{3} - 46 \, a^{2} x^{2} - 243 \, a x - 152\right )} \sqrt {-a^{2} x^{2} + 1} + 152}{1155 \, {\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, 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 {3 a x}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 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 {3 a^{2} x^{2}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 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 {a^{3} x^{3}}{- a^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 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^{7} x^{7} \sqrt {- a^{2} x^{2} + 1} + 5 a^{6} x^{6} \sqrt {- a^{2} x^{2} + 1} - 9 a^{5} x^{5} \sqrt {- a^{2} x^{2} + 1} + 5 a^{4} x^{4} \sqrt {- a^{2} x^{2} + 1} + 5 a^{3} x^{3} \sqrt {- a^{2} x^{2} + 1} - 9 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 = 0.46, size = 509, normalized size = 3.95 \begin {gather*} \frac {\frac {48 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 {5 \, {\left (63 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{5} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 385 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{4} \sqrt {-\frac {2 \, c}{a c x - c} - 1} + 990 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{3} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1386 \, {\left (\frac {2 \, c}{a c x - c} + 1\right )}^{2} \sqrt {-\frac {2 \, c}{a c x - c} - 1} - 1155 \, {\left (-\frac {2 \, c}{a c x - c} - 1\right )}^{\frac {3}{2}} - 693 \, \sqrt {-\frac {2 \, c}{a c x - c} - 1}\right )}}{c^{3}} + \frac {22 \, {\left (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}\right )}}{c^{3}} + \frac {99 \, {\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 )}}{27720 \, 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 = 604, normalized size = 4.68 \begin {gather*} \frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^6}{693\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^6}{231\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {20\,a^6}{99\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}+\frac {4\,a^7}{11\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^6\,\sqrt {-a^2}}+\frac {80\,a^9}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,{\left (-a^2\right )}^{3/2}}+\frac {32\,a^{11}}{693\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{5/2}}\right )}{a^6\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {32\,a^5}{315\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}-\frac {16\,a^5}{105\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}+\frac {4\,a^5}{9\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^5}-\frac {16\,a^2\,{\left (-a^2\right )}^{3/2}}{63\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4}+\frac {32\,a^6}{315\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,\sqrt {-a^2}}\right )}{a^5\,\sqrt {-a^2}}-\frac {\sqrt {1-a^2\,x^2}\,\left (\frac {3\,a^4}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^3}-\frac {2\,a^4}{35\,c^5\,\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}+\frac {a^5}{7\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^4\,\sqrt {-a^2}}+\frac {2\,a^7}{35\,c^5\,{\left (x\,\sqrt {-a^2}-\frac {\sqrt {-a^2}}{a}\right )}^2\,{\left (-a^2\right )}^{3/2}}\right )}{a^4\,\sqrt {-a^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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