Optimal. Leaf size=135 \[ c^4 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}-\frac {5 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {25 c^4 \csc ^{-1}(a x)}{2 a} \]
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Rubi [A] time = 0.44, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6177, 1807, 1809, 844, 216, 266, 63, 208} \[ c^4 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}-\frac {5 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}-\frac {25 c^4 \csc ^{-1}(a x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 216
Rule 266
Rule 844
Rule 1807
Rule 1809
Rule 6177
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^4 \, dx &=-\frac {\operatorname {Subst}\left (\int \frac {\left (c-\frac {c x}{a}\right )^5}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\operatorname {Subst}\left (\int \frac {\frac {5 c^5}{a}-\frac {10 c^5 x}{a^2}+\frac {10 c^5 x^2}{a^3}-\frac {5 c^5 x^3}{a^4}+\frac {c^5 x^4}{a^5}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {a^2 \operatorname {Subst}\left (\int \frac {-\frac {15 c^5}{a^3}+\frac {30 c^5 x}{a^4}-\frac {32 c^5 x^2}{a^5}+\frac {15 c^5 x^3}{a^6}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{3 c}\\ &=-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {a^4 \operatorname {Subst}\left (\int \frac {\frac {30 c^5}{a^5}-\frac {75 c^5 x}{a^6}+\frac {64 c^5 x^2}{a^7}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 c}\\ &=-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {a^6 \operatorname {Subst}\left (\int \frac {-\frac {30 c^5}{a^7}+\frac {75 c^5 x}{a^8}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{6 c}\\ &=-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\left (25 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 {\left (5 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}\\ &=-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {25 c^4 \csc ^{-1}(a x)}{2 a}+\frac {\left (5 c^4\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a}\\ &=-\frac {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {25 c^4 \csc ^{-1}(a x)}{2 a}-\left (5 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 {32 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a}-\frac {c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{3 a^3 x^2}+\frac {5 c^4 \sqrt {1-\frac {1}{a^2 x^2}}}{2 a^2 x}+c^4 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {25 c^4 \csc ^{-1}(a x)}{2 a}-\frac {5 c^4 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 175, normalized size = 1.30 \[ \frac {c^4 \left (6 a^5 x^5-64 a^4 x^4+9 a^3 x^3+62 a^2 x^2+90 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {\sqrt {1-\frac {1}{a x}}}{\sqrt {2}}\right )-30 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \sin ^{-1}\left (\frac {1}{a x}\right )-30 a^4 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-15 a x+2\right )}{6 a^5 x^4 \sqrt {1-\frac {1}{a^2 x^2}}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.58, size = 156, normalized size = 1.16 \[ \frac {150 \, a^{3} c^{4} x^{3} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - 30 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + 30 \, a^{3} c^{4} x^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (6 \, a^{4} c^{4} x^{4} - 58 \, a^{3} c^{4} x^{3} - 49 \, a^{2} c^{4} x^{2} + 13 \, a c^{4} x - 2 \, c^{4}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a^{4} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 265, normalized size = 1.96 \[ \frac {25 \, c^{4} \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right ) \mathrm {sgn}\left (a x + 1\right )}{a} + \frac {5 \, c^{4} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} c^{4} \mathrm {sgn}\left (a x + 1\right )}{a} - \frac {15 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{5} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 60 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{4} a c^{4} \mathrm {sgn}\left (a x + 1\right ) + 132 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} a c^{4} \mathrm {sgn}\left (a x + 1\right ) - 15 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )} c^{4} {\left | a \right |} \mathrm {sgn}\left (a x + 1\right ) + 64 \, a c^{4} \mathrm {sgn}\left (a x + 1\right )}{3 \, {\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3} a {\left | a \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 290, normalized size = 2.15 \[ \frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{4} \left (-66 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{4} a^{4}+96 \sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x^{3} a^{3}+66 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x^{2} a^{2}-75 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x^{3} a^{3}-75 a^{3} x^{3} \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+66 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-96 \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) x^{3} a^{4}-15 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +2 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\right )}{6 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a^{4} x^{3} \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 223, normalized size = 1.65 \[ \frac {1}{3} \, {\left (\frac {75 \, c^{4} \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a^{2}} - \frac {15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} + \frac {15 \, c^{4} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {87 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 61 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 55 \, c^{4} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 45 \, c^{4} \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {2 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {2 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} + a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 185, normalized size = 1.37 \[ \frac {25\,c^4\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a}-\frac {15\,c^4\,\sqrt {\frac {a\,x-1}{a\,x+1}}+\frac {55\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-\frac {61\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{3}-29\,c^4\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{a+\frac {2\,a\,\left (a\,x-1\right )}{a\,x+1}-\frac {2\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}-\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {10\,c^4\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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 \[ \frac {c^{4} \left (\int a^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{4}}\, dx + \int \left (- \frac {4 a \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{3}}\right )\, dx + \int \frac {6 a^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x^{2}}\, dx + \int \left (- \frac {4 a^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{x}\right )\, dx\right )}{a^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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