Optimal. Leaf size=105 \[ -\frac {5}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a}+\frac {2}{3} a^2 c^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {1}{4} a^3 c^3 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \]
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Rubi [A] time = 0.23, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6175, 6178, 1807, 807, 266, 47, 63, 208} \[ -\frac {1}{4} a^3 c^3 x^4 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}+\frac {2}{3} a^2 c^3 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2}-\frac {5}{8} a c^3 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {5 c^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{8 a} \]
Antiderivative was successfully verified.
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Rule 47
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 )^2 \sqrt {1-\frac {x^2}{a^2}}}{x^5} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4-\frac {1}{4} \left (a^3 c^3\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {8}{a}-\frac {5 x}{a^2}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{4} \left (5 a c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{8} \left (5 a c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4-\frac {\left (5 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 {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {1}{8} \left (5 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 {5}{8} a c^3 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {2}{3} a^2 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {1}{4} a^3 c^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^4+\frac {5 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 = 0.17, size = 73, normalized size = 0.70 \[ \frac {c^3 \left (15 \log \left (a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+1\right )\right )-a x \sqrt {1-\frac {1}{a^2 x^2}} \left (6 a^3 x^3-16 a^2 x^2+9 a x+16\right )\right )}{24 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.50, size = 115, normalized size = 1.10 \[ \frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) - {\left (6 \, a^{4} c^{3} x^{4} - 10 \, a^{3} c^{3} x^{3} - 7 \, a^{2} c^{3} x^{2} + 25 \, a c^{3} x + 16 \, c^{3}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{24 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 185, normalized size = 1.76 \[ \frac {1}{24} \, a c^{3} {\left (\frac {15 \, \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, \log \left ({\left | \sqrt {\frac {a x - 1}{a x + 1}} - 1 \right |}\right )}{a^{2}} + \frac {2 \, {\left (\frac {55 \, {\left (a x - 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{a x + 1} - \frac {73 \, {\left (a x - 1\right )}^{2} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{2}} - \frac {15 \, {\left (a x - 1\right )}^{3} \sqrt {\frac {a x - 1}{a x + 1}}}{{\left (a x + 1\right )}^{3}} - 15 \, \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{a^{2} {\left (\frac {a x - 1}{a x + 1} - 1\right )}^{4}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 141, normalized size = 1.34 \[ \frac {\left (a x -1\right ) c^{3} \left (-6 \left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}\, x a +16 \left (\left (a x -1\right ) \left (a x +1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-15 \sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, x a +15 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{24 a \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 221, normalized size = 2.10 \[ \frac {1}{24} \, {\left (\frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {15 \, c^{3} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} + \frac {2 \, {\left (15 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {7}{2}} + 73 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} - 55 \, c^{3} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} + 15 \, c^{3} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {4 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {6 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {4 \, {\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - \frac {{\left (a x - 1\right )}^{4} a^{2}}{{\left (a x + 1\right )}^{4}} - a^{2}}\right )} a \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 177, normalized size = 1.69 \[ \frac {5\,c^3\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{4\,a}-\frac {\frac {5\,c^3\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{4}-\frac {55\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{12}+\frac {73\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{12}+\frac {5\,c^3\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{7/2}}{4}}{a-\frac {4\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {6\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {4\,a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}+\frac {a\,{\left (a\,x-1\right )}^4}{{\left (a\,x+1\right )}^4}} \]
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 \[ - c^{3} \left (\int \frac {3 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {3 a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{3} x^{3}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \left (- \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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