Optimal. Leaf size=171 \[ -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4}+\frac {5 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]
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Rubi [A] time = 0.50, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {6177, 852, 1805, 807, 266, 63, 208} \[ -\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {x \sqrt {1-\frac {1}{a^2 x^2}}}{c^4}+\frac {5 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 852
Rule 1805
Rule 6177
Rubi steps
\begin {align*} \int \frac {e^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a x}\right )^4} \, dx &=-\left (c \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^2 \left (c-\frac {c x}{a}\right )^5} \, dx,x,\frac {1}{x}\right )\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c+\frac {c x}{a}\right )^5}{x^2 \left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^9}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {-7 c^5-\frac {35 c^5 x}{a}-\frac {61 c^5 x^2}{a^2}+\frac {7 c^5 x^3}{a^3}}{x^2 \left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^9}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {\operatorname {Subst}\left (\int \frac {35 c^5+\frac {175 c^5 x}{a}+\frac {272 c^5 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 {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {-105 c^5-\frac {525 c^5 x}{a}-\frac {614 c^5 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 {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}-\frac {\operatorname {Subst}\left (\int \frac {105 c^5+\frac {525 c^5 x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{105 c^9}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a c^4}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a c^4}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {(5 a) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )}{c^4}\\ &=-\frac {16 \left (a+\frac {1}{x}\right )}{7 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {4 \left (7 a+\frac {17}{x}\right )}{35 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}-\frac {175 a+\frac {307}{x}}{105 a^2 c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}}-\frac {525 a+\frac {719}{x}}{105 a^2 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}+\frac {\sqrt {1-\frac {1}{a^2 x^2}} x}{c^4}+\frac {5 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a c^4}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 112, normalized size = 0.65 \[ \frac {105 a^5 x^5-1339 a^4 x^4+1812 a^3 x^3+485 a^2 x^2+525 a x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-1947 a x+824}{105 a^2 c^4 x \sqrt {1-\frac {1}{a^2 x^2}} (a x-1)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 204, normalized size = 1.19 \[ \frac {525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 525 \, {\left (a^{4} x^{4} - 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} - 4 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (105 \, a^{5} x^{5} - 1339 \, a^{4} x^{4} + 1812 \, a^{3} x^{3} + 485 \, a^{2} x^{2} - 1947 \, a x + 824\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{105 \, {\left (a^{5} c^{4} x^{4} - 4 \, a^{4} c^{4} x^{3} + 6 \, a^{3} c^{4} x^{2} - 4 \, a^{2} c^{4} x + a c^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 182, normalized size = 1.06 \[ \frac {1}{420} \, a {\left (\frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {2100 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{4}} - \frac {{\left (a x + 1\right )}^{3} {\left (\frac {126 \, {\left (a x - 1\right )}}{a x + 1} + \frac {595 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {3360 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} + 15\right )}}{{\left (a x - 1\right )}^{3} a^{2} c^{4} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {840 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{4} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 523, normalized size = 3.06 \[ -\frac {-525 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-525 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+420 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+2625 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}+2625 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}-1076 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-5250 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-5250 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+970 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a +5250 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+5250 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-299 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-2625 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -2625 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+525 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{105 a \left (a x -1\right )^{4} \sqrt {a^{2}}\, c^{4} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 169, normalized size = 0.99 \[ \frac {1}{420} \, a {\left (\frac {\frac {111 \, {\left (a x - 1\right )}}{a x + 1} + \frac {469 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {2765 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - \frac {4200 \, {\left (a x - 1\right )}^{4}}{{\left (a x + 1\right )}^{4}} + 15}{a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - a^{2} c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}}} + \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{4}} - \frac {2100 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.11, size = 137, normalized size = 0.80 \[ \frac {10\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^4}-\frac {\frac {67\,{\left (a\,x-1\right )}^2}{15\,{\left (a\,x+1\right )}^2}+\frac {79\,{\left (a\,x-1\right )}^3}{3\,{\left (a\,x+1\right )}^3}-\frac {40\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}+\frac {37\,\left (a\,x-1\right )}{35\,\left (a\,x+1\right )}+\frac {1}{7}}{4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}-4\,a\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}} \]
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 \[ \frac {a^{4} \int \frac {x^{4}}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + 6 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 4 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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