Optimal. Leaf size=194 \[ c^2 x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{3 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{6 a}-\frac {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{2 a}+\frac {3 c^2 \csc ^{-1}(a x)}{2 a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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Rubi [A] time = 0.13, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 10, 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} \[ c^2 x \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}}{3 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}}{6 a}-\frac {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}}{2 a}+\frac {3 c^2 \csc ^{-1}(a x)}{2 a}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{a} \]
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 )^2 \, dx &=-\left (c^2 \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{5/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x-c^2 \operatorname {Subst}\left (\int \frac {\left (\frac {1}{a}-\frac {4 x}{a^2}\right ) \sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{3} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {3}{a^2}-\frac {7 x}{a^3}\right ) \left (1+\frac {x}{a}\right )^{3/2}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{6 a}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {1}{6} \left (a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\left (-\frac {6}{a^3}+\frac {15 x}{a^4}\right ) \sqrt {1+\frac {x}{a}}}{x \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{6 a}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x-\frac {1}{6} \left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\frac {6}{a^4}-\frac {9 x}{a^5}}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{6 a}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}-\frac {c^2 \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 {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{6 a}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {c^2 \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 (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}\\ &=-\frac {5 c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}}{2 a}-\frac {7 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2}}{6 a}+\frac {4 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2}}{3 a}+c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x+\frac {3 c^2 \csc ^{-1}(a x)}{2 a}+\frac {c^2 \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 = 0.16, size = 94, normalized size = 0.48 \[ \frac {c^2 \left (6 a^2 x^2 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )+9 a^2 x^2 \sin ^{-1}\left (\frac {1}{a x}\right )+\sqrt {1-\frac {1}{a^2 x^2}} \left (6 a^3 x^3-8 a^2 x^2+3 a x+2\right )\right )}{6 a^3 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.52, size = 157, normalized size = 0.81 \[ -\frac {18 \, a^{3} c^{2} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 6 \, a^{3} c^{2} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{2} x^{4} - 2 \, a^{3} c^{2} x^{3} - 5 \, a^{2} c^{2} x^{2} + 5 \, a c^{2} x + 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 211, normalized size = 1.09 \[ -\frac {1}{3} \, a c^{2} {\left (\frac {9 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} + \frac {\frac {20 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {9 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 3 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 224, normalized size = 1.15 \[ \frac {\left (a x -1\right ) c^{2} \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}+9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}+9 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 223, normalized size = 1.15 \[ -\frac {1}{3} \, a {\left (\frac {9 \, c^{2} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {15 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 29 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 3 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 183, normalized size = 0.94 \[ \frac {c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {29\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}+5\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {3\,c^2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}+\frac {2\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{2} \left (\int \frac {a^{4}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {2 a^{2}}{x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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