Optimal. Leaf size=268 \[ \frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}{5 a}+c^3 x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{20 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{60 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{24 a}-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{8 a}+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}+\frac {c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.19, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6194, 97, 154, 157, 41, 216, 92, 208} \[ \frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{7/2}}{5 a}+c^3 x \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{7/2}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}{20 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{60 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{24 a}-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{8 a}+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}+\frac {c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 41
Rule 92
Rule 97
Rule 154
Rule 157
Rule 208
Rule 216
Rule 6194
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} \left (c-\frac {c}{a^2 x^2}\right )^3 \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-c^3 \operatorname {Subst}\left (\int \frac {\left (\frac {1}{a}-\frac {6 x}{a^2}\right ) \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{5} \left (a c^3\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {5}{a^2}-\frac {23 x}{a^3}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{20} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {20}{a^3}-\frac {43 x}{a^4}\right ) \left (1+\frac {x}{a}\right )^{5/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {1}{60} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {60}{a^4}+\frac {155 x}{a^5}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x-\frac {1}{120} \left (a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {120}{a^5}-\frac {345 x}{a^6}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {1}{120} \left (a^5 c^3\right ) \operatorname {Subst}\left (\int \frac {-\frac {120}{a^6}+\frac {225 x}{a^7}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}-\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {c^3 \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a}-\frac {x^2}{a}} \, dx,x,\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a^2}+\frac {\left (15 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a^2}\\ &=-\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{8 a}-\frac {31 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{24 a}-\frac {43 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{60 a}+\frac {23 c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}{20 a}+\frac {6 c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{7/2}}{5 a}+c^3 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{7/2} x+\frac {15 c^3 \csc ^{-1}(a x)}{8 a}+\frac {c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 104, normalized size = 0.39 \[ \frac {c^3 \left (225 a^4 \sin ^{-1}\left (\frac {1}{a x}\right )+120 a^4 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {\sqrt {1-\frac {1}{a^2 x^2}} \left (120 a^5 x^5-184 a^4 x^4+135 a^3 x^3+88 a^2 x^2-30 a x-24\right )}{x^4}\right )}{120 a^5} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 179, normalized size = 0.67 \[ -\frac {450 \, a^{5} c^{3} x^{5} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 120 \, a^{5} c^{3} x^{5} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (120 \, a^{6} c^{3} x^{6} - 64 \, a^{5} c^{3} x^{5} - 49 \, a^{4} c^{3} x^{4} + 223 \, a^{3} c^{3} x^{3} + 58 \, a^{2} c^{3} x^{2} - 54 \, a c^{3} x - 24 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{120 \, a^{6} x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 273, normalized size = 1.02 \[ -\frac {1}{60} \, a c^{3} {\left (\frac {225 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {60 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {60 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {120 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {\frac {310 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {1424 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + \frac {970 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} + \frac {225 \, {\left (a x - 1\right )}^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{4}} + 15 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{5}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 272, normalized size = 1.01 \[ \frac {\left (a x -1\right ) c^{3} \left (-120 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{6} a^{6}+120 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+225 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{5} a^{5}+225 \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}\, x^{5} a^{5}+120 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}-105 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{3} a^{3}-64 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{120 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{6} x^{5} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 302, normalized size = 1.13 \[ -\frac {1}{60} \, {\left (\frac {225 \, c^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {60 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {345 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}} + 1345 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} + 1654 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 86 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 305 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} + \frac {5 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - \frac {4 \, {\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - \frac {{\left (a x - 1\right )}^{6} a^{2}}{{\left (a x + 1\right )}^{6}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.32, size = 258, normalized size = 0.96 \[ \frac {\frac {7\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}+\frac {61\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {43\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{30}+\frac {827\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{30}+\frac {269\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{12}+\frac {23\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}}{4}}{a+\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {5\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {4\,a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}-\frac {a\,{\left (a\,x-1\right )}^6}{{\left (a\,x+1\right )}^6}}-\frac {15\,c^3\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}+\frac {2\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{3} \left (\int \frac {a^{6}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {3 a^{2}}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{4}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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