Optimal. Leaf size=78 \[ -\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a} \]
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Rubi [A]
time = 0.09, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {6310, 6313,
821, 272, 43, 65, 214} \begin {gather*} -\frac {1}{2} a c^2 x^2 \sqrt {1-\frac {1}{a^2 x^2}}+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}+\frac {1}{3} a^2 c^2 x^3 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 214
Rule 272
Rule 821
Rule 6310
Rule 6313
Rubi steps
\begin {align*} \int e^{\coth ^{-1}(a x)} (c-a c x)^2 \, dx &=\left (a^2 c^2\right ) \int e^{\coth ^{-1}(a x)} \left (1-\frac {1}{a x}\right )^2 x^2 \, dx\\ &=-\left (\left (a^2 c^2\right ) \text {Subst}\left (\int \frac {\left (1-\frac {x}{a}\right ) \sqrt {1-\frac {x^2}{a^2}}}{x^4} \, dx,x,\frac {1}{x}\right )\right )\\ &=\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\left (a c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x^2}{a^2}}}{x^3} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {1}{2} \left (a c^2\right ) \text {Subst}\left (\int \frac {\sqrt {1-\frac {x}{a^2}}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3-\frac {c^2 \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{4 a}\\ &=-\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {1}{2} \left (a c^2\right ) \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right )\\ &=-\frac {1}{2} a c^2 \sqrt {1-\frac {1}{a^2 x^2}} x^2+\frac {1}{3} a^2 c^2 \left (1-\frac {1}{a^2 x^2}\right )^{3/2} x^3+\frac {c^2 \tanh ^{-1}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 64, normalized size = 0.82 \begin {gather*} \frac {c^2 \left (a \sqrt {1-\frac {1}{a^2 x^2}} x \left (-2-3 a x+2 a^2 x^2\right )+3 \log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )\right )}{6 a} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.09, size = 121, normalized size = 1.55
method | result | size |
risch | \(\frac {\left (2 a^{2} x^{2}-3 a x -2\right ) \left (a x -1\right ) c^{2}}{6 a \sqrt {\frac {a x -1}{a x +1}}}+\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) c^{2} \sqrt {\left (a x +1\right ) \left (a x -1\right )}}{2 \sqrt {a^{2}}\, \sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right )}\) | \(112\) |
default | \(-\frac {\left (a x -1\right ) c^{2} \left (3 \sqrt {a^{2}}\, \sqrt {a^{2} x^{2}-1}\, a x -2 \left (\left (a x +1\right ) \left (a x -1\right )\right )^{\frac {3}{2}} \sqrt {a^{2}}-3 \ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a \right )}{6 \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x +1\right ) \left (a x -1\right )}\, a \sqrt {a^{2}}}\) | \(121\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 181 vs.
\(2 (66) = 132\).
time = 0.27, size = 181, normalized size = 2.32 \begin {gather*} \frac {1}{6} \, a {\left (\frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}} - \frac {2 \, {\left (3 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {5}{2}} + 8 \, c^{2} \left (\frac {a x - 1}{a x + 1}\right )^{\frac {3}{2}} - 3 \, c^{2} \sqrt {\frac {a x - 1}{a x + 1}}\right )}}{\frac {3 \, {\left (a x - 1\right )} a^{2}}{a x + 1} - \frac {3 \, {\left (a x - 1\right )}^{2} a^{2}}{{\left (a x + 1\right )}^{2}} + \frac {{\left (a x - 1\right )}^{3} a^{2}}{{\left (a x + 1\right )}^{3}} - a^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 103, normalized size = 1.32 \begin {gather*} \frac {3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - 3 \, c^{2} \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) + {\left (2 \, a^{3} c^{2} x^{3} - a^{2} c^{2} x^{2} - 5 \, a c^{2} x - 2 \, c^{2}\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{6 \, a} \end {gather*}
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 \begin {gather*} c^{2} \left (\int \left (- \frac {2 a x}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\right )\, dx + \int \frac {a^{2} x^{2}}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx + \int \frac {1}{\sqrt {\frac {a x}{a x + 1} - \frac {1}{a x + 1}}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 98, normalized size = 1.26 \begin {gather*} \frac {1}{6} \, \sqrt {a^{2} x^{2} - 1} {\left ({\left (\frac {2 \, a c^{2} x}{\mathrm {sgn}\left (a x + 1\right )} - \frac {3 \, c^{2}}{\mathrm {sgn}\left (a x + 1\right )}\right )} x - \frac {2 \, c^{2}}{a \mathrm {sgn}\left (a x + 1\right )}\right )} - \frac {c^{2} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{2 \, {\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 138, normalized size = 1.77 \begin {gather*} \frac {\frac {8\,c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{3/2}}{3}-c^2\,\sqrt {\frac {a\,x-1}{a\,x+1}}+c^2\,{\left (\frac {a\,x-1}{a\,x+1}\right )}^{5/2}}{a-\frac {3\,a\,\left (a\,x-1\right )}{a\,x+1}+\frac {3\,a\,{\left (a\,x-1\right )}^2}{{\left (a\,x+1\right )}^2}-\frac {a\,{\left (a\,x-1\right )}^3}{{\left (a\,x+1\right )}^3}}+\frac {c^2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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