Optimal. Leaf size=127 \[ -\frac {27}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {20}{3} c^3 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {35 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}+\frac {4}{3} a^2 c^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {1}{4} a^3 c^3 x^4 \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rubi [A] time = 0.35, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6175, 6178, 1807, 807, 266, 63, 208} \[ -\frac {1}{4} a^3 c^3 x^4 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {4}{3} a^2 c^3 x^3 \sqrt {1-\frac {1}{a^2 x^2}}-\frac {27}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {20}{3} c^3 x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {35 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 807
Rule 1807
Rule 6175
Rule 6178
Rubi steps
\begin {align*} \int e^{-\coth ^{-1}(a x)} (c-a c x)^3 \, dx &=-\left (\left (a^3 c^3\right ) \int e^{-\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^3 x^3 \, dx\right )\\ &=\left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right )^4}{x^5 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{4} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\frac {16}{a}-\frac {27 x}{a^2}+\frac {16 x^2}{a^3}-\frac {4 x^3}{a^4}}{x^4 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {1}{12} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\frac {81}{a^2}-\frac {80 x}{a^3}+\frac {12 x^2}{a^4}}{x^3 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {27}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{24} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\frac {160}{a^3}-\frac {105 x}{a^4}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )\\ &=\frac {20}{3} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {27}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{8 a}\\ &=\frac {20}{3} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {27}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4+\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{16 a}\\ &=\frac {20}{3} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {27}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {1}{8} \left (35 a c^3\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 {20}{3} c^3 \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {27}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {4}{3} a^2 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^3-\frac {1}{4} a^3 c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^4-\frac {35 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 72, normalized size = 0.57 \[ \frac {c^3 \left (a x \sqrt {1-\frac {1}{a^2 x^2}} \left (-6 a^3 x^3+32 a^2 x^2-81 a x+160\right )-105 \log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )\right )}{24 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.84, size = 114, normalized size = 0.90 \[ -\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (6 \, a^{4} c^{3} x^{4} - 26 \, a^{3} c^{3} x^{3} + 49 \, a^{2} c^{3} x^{2} - 79 \, a c^{3} x - 160 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 109, normalized size = 0.86 \[ \frac {35 \, c^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{8 \, {\left | a \right |}} + \frac {1}{24} \, \sqrt {a^{2} x^{2} - 1} {\left (\frac {160 \, c^{3} \mathrm {sgn}\left (a x + 1\right )}{a} - {\left (81 \, c^{3} \mathrm {sgn}\left (a x + 1\right ) + 2 \, {\left (3 \, a^{2} c^{3} x \mathrm {sgn}\left (a x + 1\right ) - 16 \, a c^{3} \mathrm {sgn}\left (a x + 1\right )\right )} x\right )} x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 196, normalized size = 1.54 \[ -\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) c^{3} \left (6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +87 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a -32 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-87 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a -192 \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}+192 a \ln \left (\frac {a^{2} x +\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right )\right )}{24 a \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 221, normalized size = 1.74 \[ -\frac {1}{24} \, {\left (\frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {105 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (279 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} - 511 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 385 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 105 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 176, normalized size = 1.39 \[ \frac {\frac {35\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {385\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {511\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{12}-\frac {93\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}}-\frac {35\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a} \]
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 \[ - c^{3} \left (\int 3 a x \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- 3 a^{2} x^{2} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx + \int a^{3} x^{3} \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\, dx + \int \left (- \sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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