Optimal. Leaf size=103 \[ \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right )}{3 a}-\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.13, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6177, 813, 815, 844, 216, 266, 63, 208} \[ \frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 x \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right )}{3 a}-\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 813
Rule 815
Rule 844
Rule 6177
Rubi steps
\begin {align*} \int e^{3 \coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\left (c^3 \operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right ) \left (1-\frac {x^2}{a^2}\right )^{3/2}}{x^2} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {1}{2} c^3 \operatorname {Subst}\left (\int \frac {\left (\frac {2 c}{a}+\frac {6 c x}{a^2}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}-\frac {1}{4} \left (a^2 c^3\right ) \operatorname {Subst}\left (\int \frac {-\frac {4 c}{a^3}-\frac {6 c x}{a^4}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {\left (3 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{2 a^2}+\frac {c^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}+\frac {c^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\left (a c^4\right ) \operatorname {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}} \left (2 a+\frac {3}{x}\right )}{2 a^2}+\frac {c^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \left (3 a+\frac {1}{x}\right ) x}{3 a}+\frac {3 c^4 \csc ^{-1}(a x)}{2 a}-\frac {c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.23, size = 175, normalized size = 1.70 \[ -\frac {c^4 \left (-24 a^5 x^5-32 a^4 x^4+12 a^3 x^3+40 a^2 x^2+42 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-15 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {1}{a x}\right )+24 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )+12 a x-8\right )}{24 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.75, size = 156, normalized size = 1.51 \[ -\frac {18 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 6 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{4} x^{4} + 14 \, a^{3} c^{4} x^{3} + 11 \, a^{2} c^{4} x^{2} + a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 212, normalized size = 2.06 \[ -\frac {1}{3} \, a c^{4} {\left (\frac {9 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {6 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}} - \frac {\frac {20 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} + \frac {3 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} + 9 \, \sqrt {\frac {a x - 1}{a x + 1}}}{a^{2} {\left (\frac {a x - 1}{a x + 1} + 1\right )}^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 233, normalized size = 2.26 \[ -\frac {\left (a x -1\right )^{2} c^{4} \left (-6 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-9 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-9 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+6 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}+3 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a -2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} \left (a x +1\right ) \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 223, normalized size = 2.17 \[ -\frac {1}{3} \, {\left (\frac {9 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} + \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {3 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 29 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 183, normalized size = 1.78 \[ \frac {5\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {29\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}+\frac {c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}+c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {3\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {2\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{4} \left (\int \left (- \frac {4 a}{\frac {a x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {6 a^{2}}{\frac {a x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \left (- \frac {4 a^{3}}{\frac {a x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\right )\, dx + \int \frac {a^{4}}{\frac {a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx + \int \frac {1}{\frac {a x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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