Optimal. Leaf size=233 \[ \frac {1}{5} a^4 c^2 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{4} a^3 c^2 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {1}{4} a^2 c^2 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{8} a c^2 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {3}{8} c^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{8 a} \]
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Rubi [A] time = 0.20, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6191, 6195, 94, 92, 208} \[ \frac {1}{5} a^4 c^2 x^5 \left (1-\frac {1}{a x}\right )^{5/2} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{4} a^3 c^2 x^4 \left (1-\frac {1}{a x}\right )^{3/2} \left (\frac {1}{a x}+1\right )^{5/2}+\frac {1}{4} a^2 c^2 x^3 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{5/2}-\frac {1}{8} a c^2 x^2 \sqrt {1-\frac {1}{a x}} \left (\frac {1}{a x}+1\right )^{3/2}-\frac {3}{8} c^2 x \sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}-\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {\frac {1}{a x}+1}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 92
Rule 94
Rule 208
Rule 6191
Rule 6195
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-a^2 c x^2\right )^2 \, dx &=\left (a^4 c^2\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a^2 x^2}\right )^2 x^4 \, dx\\ &=-\left (\left (a^4 c^2\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{5/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^6} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\left (a^3 c^2\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {1}{4} \left (3 a^2 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a}} \left (1+\frac {x}{a}\right )^{3/2}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {1}{4} \left (a c^2\right ) \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^{3/2}}{x^3 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {1}{8} \left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a}}}{x^2 \sqrt {1-\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a}} \sqrt {1+\frac {x}{a}}} \, dx,x,\frac {1}{x}\right )}{8 a}\\ &=-\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {\left (3 c^2\right ) \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 )}{8 a^2}\\ &=-\frac {3}{8} c^2 \sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}} x-\frac {1}{8} a c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{3/2} x^2+\frac {1}{4} a^2 c^2 \sqrt {1-\frac {1}{a x}} \left (1+\frac {1}{a x}\right )^{5/2} x^3-\frac {1}{4} a^3 c^2 \left (1-\frac {1}{a x}\right )^{3/2} \left (1+\frac {1}{a x}\right )^{5/2} x^4+\frac {1}{5} a^4 c^2 \left (1-\frac {1}{a x}\right )^{5/2} \left (1+\frac {1}{a x}\right )^{5/2} x^5-\frac {3 c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a x}} \sqrt {1+\frac {1}{a x}}\right )}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 79, normalized size = 0.34 \[ \frac {c^2 \left (a x \sqrt {1-\frac {1}{a^2 x^2}} \left (8 a^4 x^4-10 a^3 x^3-16 a^2 x^2+25 a x+8\right )-15 \log \left (x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{40 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.24, size = 126, normalized size = 0.54 \[ -\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (8 \, a^{5} c^{2} x^{5} - 2 \, a^{4} c^{2} x^{4} - 26 \, a^{3} c^{2} x^{3} + 9 \, a^{2} c^{2} x^{2} + 33 \, a c^{2} x + 8 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{40 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 126, normalized size = 0.54 \[ \frac {3 \, c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{8 \, {\left | a \right |}} + \frac {1}{40} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (25 \, c^{2} \mathrm {sgn}\left (a x + 1\right ) - 2 \, {\left (8 \, a c^{2} \mathrm {sgn}\left (a x + 1\right ) - {\left (4 \, a^{3} c^{2} x \mathrm {sgn}\left (a x + 1\right ) - 5 \, a^{2} c^{2} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} x + \frac {8 \, c^{2} \mathrm {sgn}\left (a x + 1\right )}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 183, normalized size = 0.79 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{2} \left (24 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-30 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +16 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}+45 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -40 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-45 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{120 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 259, normalized size = 1.11 \[ -\frac {1}{40} \, a {\left (\frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (15 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {9}{2}} - 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 128 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 70 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 15 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {5 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {10 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {10 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {5 \, {\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + \frac {{\left (a x - 1\right )}^{5} a^{2}}{{\left (a x + 1\right )}^{5}} - a^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 214, normalized size = 0.92 \[ \frac {\frac {3\,c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{2}+\frac {32\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{5}+\frac {7\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{2}-\frac {3\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{9/2}}{4}}{a-\frac {5\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {10\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {10\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {5\,a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}-\frac {a\,{\left (a\,x-1\right )}^5}{{\left (a\,x+1\right )}^5}}-\frac {3\,c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
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 \[ c^{2} \left (\int \left (- 2 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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