Optimal. Leaf size=180 \[ \frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {\frac {1}{a x}+1}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6194, 103, 152, 12, 92, 208} \[ \frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {\frac {1}{a x}+1}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {\frac {1}{a x}+1}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^2} \]
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^{\coth ^{-1}(a x)}}{\left (c-\frac {c}{a^2 x^2}\right )^2} \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{a}-\frac {3 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{c^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {a \operatorname {Subst}\left (\int \frac {\frac {3}{a^2}+\frac {8 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {3}{a^3}-\frac {11 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {a^3 \operatorname {Subst}\left (\int -\frac {3}{a^4 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{3 c^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {\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^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\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^2}\\ &=-\frac {4}{3 a c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}-\frac {11}{3 a c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}+\frac {8 \sqrt {1-\frac {1}{a x}}}{3 a c^2 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^2 \left (1-\frac {1}{a x}\right )^{3/2} \sqrt {1+\frac {1}{a x}}}+\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{a c^2}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 83, normalized size = 0.46 \[ \frac {\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (3 a^3 x^3-7 a^2 x^2-5 a x+8\right )}{3 (a x-1)^2 (a x+1)}}{a c^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 134, normalized size = 0.74 \[ \frac {3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, {\left (a^{2} x^{2} - 2 \, a x + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (3 \, a^{3} x^{3} - 7 \, a^{2} x^{2} - 5 \, a x + 8\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{3 \, {\left (a^{3} c^{2} x^{2} - 2 \, a^{2} c^{2} x + a c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 171, normalized size = 0.95 \[ \frac {1}{12} \, a {\left (\frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2} c^{2}} - \frac {{\left (a x + 1\right )} {\left (\frac {18 \, {\left (a x - 1\right )}}{a x + 1} + 1\right )}}{{\left (a x - 1\right )} a^{2} c^{2} \sqrt {\frac {a x - 1}{a x + 1}}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}} - \frac {24 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 530, normalized size = 2.94 \[ -\frac {-45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{5} a^{6}+21 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+45 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}+24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{4} a^{5}+11 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}+90 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-5 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -90 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}-48 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{2} a^{3}-19 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -24 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x \,a^{2}+45 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+24 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )}{24 a \left (a x -1\right )^{2} \left (a x +1\right )^{2} \sqrt {a^{2}}\, c^{2} \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {\frac {a x -1}{a x +1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 160, normalized size = 0.89 \[ \frac {1}{12} \, a {\left (\frac {\frac {17 \, {\left (a x - 1\right )}}{a x + 1} - \frac {42 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1}{a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{2}} - \frac {12 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{2}} + \frac {3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 128, normalized size = 0.71 \[ \frac {\sqrt {\frac {a\,x-1}{a\,x+1}}}{4\,a\,c^2}-\frac {\frac {17\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {14\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-4\,a\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a\,c^2} \]
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 \[ \frac {a^{4} \int \frac {x^{4}}{a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} - 2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}} + \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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