Optimal. Leaf size=255 \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]
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Rubi [A] time = 0.17, antiderivative size = 255, 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} \[ \frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {\frac {1}{a x}+1}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{3/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}}-\frac {\tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a c^3} \]
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^{-\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 )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {\frac {1}{a}-\frac {5 x}{a^2}}{x \left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {a \operatorname {Subst}\left (\int \frac {-\frac {3}{a^2}+\frac {16 x}{a^3}}{x \left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{3 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\frac {3}{a^3}-\frac {39 x}{a^4}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{3 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^3 \operatorname {Subst}\left (\int \frac {\frac {15}{a^4}-\frac {84 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{15 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^4 \operatorname {Subst}\left (\int \frac {\frac {45}{a^5}-\frac {99 x}{a^6}}{x \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{45 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {a^5 \operatorname {Subst}\left (\int \frac {45}{a^6 x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{45 c^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}+\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^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\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^3}\\ &=-\frac {4}{3 a c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {13}{3 a c^3 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {14 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{5/2}}+\frac {11 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \left (1+\frac {1}{a x}\right )^{3/2}}+\frac {16 \sqrt {1-\frac {1}{a x}}}{5 a c^3 \sqrt {1+\frac {1}{a x}}}+\frac {x}{c^3 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2}}-\frac {\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 = 0.23, size = 101, normalized size = 0.40 \[ \frac {\frac {a x \sqrt {1-\frac {1}{a^2 x^2}} \left (15 a^5 x^5+38 a^4 x^4-52 a^3 x^3-87 a^2 x^2+33 a x+48\right )}{15 (a x-1)^2 (a x+1)^3}-\log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )}{a c^3} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.79, size = 161, normalized size = 0.63 \[ -\frac {15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (15 \, a^{5} x^{5} + 38 \, a^{4} x^{4} - 52 \, a^{3} x^{3} - 87 \, a^{2} x^{2} + 33 \, a x + 48\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{15 \, {\left (a^{5} c^{3} x^{4} - 2 \, a^{3} c^{3} x^{2} + a c^{3}\right )}} \]
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 \[ \int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{{\left (c - \frac {c}{a^{2} x^{2}}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 714, normalized size = 2.80 \[ -\frac {\left (-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{7} a^{7}+240 \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}+285 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{5} a^{5}-525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{6} a^{6}+240 \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}-83 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{4} a^{4}+1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{5} a^{5}-720 \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}-218 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{3} a^{3}+1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{4} a^{4}-720 \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}+342 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x^{2} a^{2}-1575 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+720 \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}-3 \sqrt {a^{2}}\, \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} x a -1575 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x^{2} a^{2}+720 \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}-243 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}+525 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}\, x a -240 \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}}-240 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 ) \sqrt {\frac {a x -1}{a x +1}}}{240 a \left (a x -1\right )^{3} \left (a x +1\right )^{3} \sqrt {a^{2}}\, c^{3} \sqrt {\left (a x -1\right ) \left (a x +1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 197, normalized size = 0.77 \[ \frac {1}{240} \, a {\left (\frac {5 \, {\left (\frac {23 \, {\left (a x - 1\right )}}{a x + 1} - \frac {120 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + 1\right )}}{a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - a^{2} c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}}} + \frac {3 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 40 \, \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 450 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} c^{3}} - \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2} c^{3}} + \frac {240 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2} c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 178, normalized size = 0.70 \[ \frac {15\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{8\,a\,c^3}-\frac {\frac {23\,\left (a\,x-1\right )}{3\,\left (a\,x+1\right )}-\frac {40\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}+\frac {1}{3}}{16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}-16\,a\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{6\,a\,c^3}+\frac {{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{80\,a\,c^3}+\frac {\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{a\,c^3} \]
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 \[ \frac {a^{6} \int \frac {x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a^{6} x^{6} - 3 a^{4} x^{4} + 3 a^{2} x^{2} - 1}\, dx}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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