Optimal. Leaf size=165 \[ -\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {6 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \]
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Rubi [A]
time = 0.35, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6312, 866,
1819, 821, 272, 65, 214} \begin {gather*} -\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^3}-\frac {16}{7 a^2 c^3 x \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {6 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 866
Rule 1819
Rule 6312
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^3} \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2 \left (c-\frac {c x}{a}\right )^6} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^6}{x^2 \left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^9}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {\text {Subst}\left (\int \frac {-7 c^6-\frac {42 c^6 x}{a}-\frac {80 c^6 x^2}{a^2}+\frac {42 c^6 x^3}{a^3}+\frac {7 c^6 x^4}{a^4}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^9}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}-\frac {\text {Subst}\left (\int \frac {35 c^6+\frac {210 c^6 x}{a}+\frac {355 c^6 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 c^9}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\text {Subst}\left (\int \frac {-105 c^6-\frac {630 c^6 x}{a}-\frac {780 c^6 x^2}{a^2}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^9}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}-\frac {\text {Subst}\left (\int \frac {105 c^6+\frac {630 c^6 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{105 c^9}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {6 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^3}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}-\frac {3 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{a c^3}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {(6 a) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^3}\\ &=-\frac {32 \left (a+\frac {1}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {2 \left (7 a+\frac {13}{x}\right )}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {42 a+\frac {59}{x}}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {16}{7 a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{5/2} x}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^3}+\frac {6 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^3}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 112, normalized size = 0.68 \begin {gather*} \frac {66-156 a x+39 a^2 x^2+145 a^3 x^3-109 a^4 x^4+7 a^5 x^5+42 a \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{7 a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x (-1+a x)^3} \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. \(529\) vs.
\(2(145)=290\).
time = 0.12, size = 530, normalized size = 3.21
method | result | size |
risch | \(\frac {a x -1}{a \,c^{3} \sqrt {\frac {a x -1}{a x +1}}}+\frac {\left (\frac {6 \ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right )}{a^{3} \sqrt {a^{2}}}-\frac {88 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{5} \left (x -\frac {1}{a}\right )}-\frac {4 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{8} \left (x -\frac {1}{a}\right )^{4}}-\frac {20 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{7} \left (x -\frac {1}{a}\right )^{3}}-\frac {45 \sqrt {a^{2} \left (x -\frac {1}{a}\right )^{2}+2 a \left (x -\frac {1}{a}\right )}}{7 a^{6} \left (x -\frac {1}{a}\right )^{2}}\right ) a^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{c^{3} \left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}\) | \(265\) |
default | \(-\frac {-42 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{5} x^{5}-42 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{6} x^{5}+35 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, a^{3} x^{3}+210 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{4} x^{4}+210 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{5} x^{4}-87 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a^{2} x^{2}-420 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a^{3} x^{3}-420 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{4} x^{3}+78 \sqrt {a^{2}}\, \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} a x +420 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a^{2} x^{2}+420 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{3} x^{2}-24 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-210 \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \sqrt {a^{2}}\, a x -210 \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x +42 \sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}+42 a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{\sqrt {a^{2}}}\right )}{7 a \sqrt {a^{2}}\, \left (a x -1\right )^{3} c^{3} \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, \left (a x +1\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}\) | \(530\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 169, normalized size = 1.02 \begin {gather*} \frac {1}{14} \, a {\left (\frac {\frac {6 \, {\left (a x - 1\right )}}{a x + 1} + \frac {21 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {112 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {168 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 1}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {84 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {84 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 204, normalized size = 1.24 \begin {gather*} \frac {42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 42 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (7 \, a^{5} x^{5} - 109 \, a^{4} x^{4} + 145 \, a^{3} x^{3} + 39 \, a^{2} x^{2} - 156 \, a x + 66\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{7 \, {\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 {a^{3} \int \frac {x^{3}}{\frac {a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 137, normalized size = 0.83 \begin {gather*} \frac {12\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^3}-\frac {\frac {3\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {16\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {24\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {6\,\left (a\,x-1\right )}{7\,\left (a\,x+1\right )}+\frac {1}{7}}{2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-2\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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