Optimal. Leaf size=125 \[ \frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
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Rubi [A] time = 0.39, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6175, 6178, 852, 1635, 789, 651} \[ \frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 651
Rule 789
Rule 852
Rule 1635
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int \frac {e^{3 \coth ^{-1}(a x)}}{(c-a c x)^5} \, dx &=-\frac {\int \frac {e^{3 \coth ^{-1}(a x)}}{\left (1-\frac {1}{a x}\right )^5 x^5} \, dx}{a^5 c^5}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (1-\frac {x^2}{a^2}\right )^{3/2}}{\left (1-\frac {x}{a}\right )^8} \, dx,x,\frac {1}{x}\right )}{a^5 c^5}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3 \left (1+\frac {x}{a}\right )^8}{\left (1-\frac {x^2}{a^2}\right )^{13/2}} \, dx,x,\frac {1}{x}\right )}{a^5 c^5}\\ &=\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^7 \left (8 a^3+11 a^2 x+11 a x^2\right )}{\left (1-\frac {x^2}{a^2}\right )^{11/2}} \, dx,x,\frac {1}{x}\right )}{11 a^5 c^5}\\ &=-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}+\frac {\operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^6 \left (138 a^3+99 a^2 x\right )}{\left (1-\frac {x^2}{a^2}\right )^{9/2}} \, dx,x,\frac {1}{x}\right )}{99 a^5 c^5}\\ &=\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}-\frac {152 \operatorname {Subst}\left (\int \frac {\left (1+\frac {x}{a}\right )^5}{\left (1-\frac {x^2}{a^2}\right )^{7/2}} \, dx,x,\frac {1}{x}\right )}{231 a^2 c^5}\\ &=-\frac {152 \left (a+\frac {1}{x}\right )^5}{1155 a^6 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{5/2}}+\frac {79 \left (a+\frac {1}{x}\right )^6}{231 a^7 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{7/2}}-\frac {10 \left (a+\frac {1}{x}\right )^7}{33 a^8 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{9/2}}+\frac {\left (a+\frac {1}{x}\right )^8}{11 a^9 c^5 \left (1-\frac {1}{a^2 x^2}\right )^{11/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 58, normalized size = 0.46 \[ -\frac {x \sqrt {1-\frac {1}{a^2 x^2}} (a x+1)^2 \left (2 a^3 x^3-16 a^2 x^2+61 a x-152\right )}{1155 c^5 (a x-1)^6} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 134, normalized size = 1.07 \[ -\frac {{\left (2 \, a^{6} x^{6} - 10 \, a^{5} x^{5} + 19 \, a^{4} x^{4} - 15 \, a^{3} x^{3} - 289 \, a^{2} x^{2} - 395 \, a x - 152\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{1155 \, {\left (a^{7} c^{5} x^{6} - 6 \, a^{6} c^{5} x^{5} + 15 \, a^{5} c^{5} x^{4} - 20 \, a^{4} c^{5} x^{3} + 15 \, a^{3} c^{5} x^{2} - 6 \, a^{2} c^{5} x + a c^{5}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 85, normalized size = 0.68 \[ -\frac {{\left (a x + 1\right )}^{5} {\left (\frac {385 \, {\left (a x - 1\right )}}{a x + 1} - \frac {495 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {231 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 105\right )}}{9240 \, {\left (a x - 1\right )}^{5} a c^{5} \sqrt {\frac {a x - 1}{a x + 1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 58, normalized size = 0.46 \[ -\frac {\left (2 x^{3} a^{3}-16 a^{2} x^{2}+61 a x -152\right ) \left (a x +1\right )}{1155 \left (a x -1\right )^{4} c^{5} \left (\frac {a x -1}{a x +1}\right )^{\frac {3}{2}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 71, normalized size = 0.57 \[ -\frac {\frac {385 \, {\left (a x - 1\right )}}{a x + 1} - \frac {495 \, {\left (a x - 1\right )}^{2}}{{\left (a x + 1\right )}^{2}} + \frac {231 \, {\left (a x - 1\right )}^{3}}{{\left (a x + 1\right )}^{3}} - 105}{9240 \, a c^{5} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {11}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 72, normalized size = 0.58 \[ \frac {\frac {3\,{\left (a\,x-1\right )}^2}{7\,{\left (a\,x+1\right )}^2}-\frac {{\left (a\,x-1\right )}^3}{5\,{\left (a\,x+1\right )}^3}-\frac {a\,x-1}{3\,\left (a\,x+1\right )}+\frac {1}{11}}{8\,a\,c^5\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{11/2}} \]
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 \[ - \frac {\int \frac {1}{\frac {a^{6} x^{6} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {6 a^{5} x^{5} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {15 a^{4} x^{4} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {20 a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {15 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} - \frac {6 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1} + \frac {\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}{a x + 1}}\, dx}{c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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