Optimal. Leaf size=253 \[ \frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]
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Rubi [A] time = 0.17, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 152
Rule 208
Rule 6194
Rubi steps
\begin {align*} \int \frac {e^{-3 \coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^3} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {3}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {3}{a^2}+\frac {8 x}{a^3}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {21}{a^3}+\frac {33 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{7 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^3 \operatorname {Subst}\left (\int \frac {-\frac {105}{a^4}+\frac {108 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{35 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^4 \operatorname {Subst}\left (\int \frac {-\frac {315}{a^5}+\frac {213 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {a^5 \operatorname {Subst}\left (\int -\frac {315}{a^6 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{105 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {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 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {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 c^3}\\ &=-\frac {2}{a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{7 a c^3 \left (1+\frac {1}{a x}\right )^{7/2}}+\frac {54 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {71 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {176 \sqrt {1-\frac {1}{a x}}}{35 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{7/2}}-\frac {3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^3}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 101, normalized size = 0.40 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (35 a^5 x^5+286 a^4 x^4+368 a^3 x^3-125 a^2 x^2-423 a x-176\right )}{35 (a x-1) (a x+1)^4}-3 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a c^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.46, size = 179, normalized size = 0.71 \[ -\frac {105 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, {\left (a^{4} x^{4} + 2 \, a^{3} x^{3} - 2 \, a x - 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (35 \, a^{5} x^{5} + 286 \, a^{4} x^{4} + 368 \, a^{3} x^{3} - 125 \, a^{2} x^{2} - 423 \, a x - 176\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{35 \, {\left (a^{5} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{3} - 2 \, a^{2} c^{3} x - a c^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 714, normalized size = 2.82 \[ -\frac {\left (-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{7} a^{7}+3360 \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^{7} a^{8}+2555 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}+10080 \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^{6} a^{7}+1873 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}-3675 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}+3360 \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}-4426 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-16800 \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}-3350 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+18375 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}-16800 \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}+2511 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+3360 \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}+1957 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-11025 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a +10080 \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}-3675 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+3360 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 )\right ) \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}}}{1120 a \left (a x +1\right )^{3} \sqrt {a^{2}}\, c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \left (a x -1\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 199, normalized size = 0.79 \[ -\frac {1}{560} \, a {\left (\frac {35 \, {\left (\frac {33 \, {\left (a x - 1\right )}}{a x + 1} - 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - a^{2} c^{3} \sqrt {\frac {a x - 1}{a x + 1}}} - \frac {5 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 56 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 350 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 2520 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} + \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} - \frac {1680 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 183, normalized size = 0.72 \[ \frac {\frac {33\,\left (a\,x-1\right )}{a\,x+1}-1}{16\,a\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}+\frac {9\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{2\,a\,c^3}+\frac {5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{8\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{10\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{112\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,6{}\mathrm {i}}{a\,c^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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