Optimal. Leaf size=77 \[ \frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2} \]
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Rubi [A] time = 0.12, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {6131, 6128, 266, 47, 50, 63, 208} \[ \frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a} \]
Antiderivative was successfully verified.
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Rule 47
Rule 50
Rule 63
Rule 208
Rule 266
Rule 6128
Rule 6131
Rubi steps
\begin {align*} \int e^{3 \tanh ^{-1}(a x)} \left (c-\frac {c}{a x}\right )^3 \, dx &=-\frac {c^3 \int \frac {e^{3 \tanh ^{-1}(a x)} (1-a x)^3}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \int \frac {\left (1-a^2 x^2\right )^{3/2}}{x^3} \, dx}{a^3}\\ &=-\frac {c^3 \operatorname {Subst}\left (\int \frac {\left (1-a^2 x\right )^{3/2}}{x^2} \, dx,x,x^2\right )}{2 a^3}\\ &=\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-a^2 x}}{x} \, dx,x,x^2\right )}{4 a}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}+\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-a^2 x}} \, dx,x,x^2\right )}{4 a}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {\left (3 c^3\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{a^2}-\frac {x^2}{a^2}} \, dx,x,\sqrt {1-a^2 x^2}\right )}{2 a^3}\\ &=\frac {3 c^3 \sqrt {1-a^2 x^2}}{2 a}+\frac {c^3 \left (1-a^2 x^2\right )^{3/2}}{2 a^3 x^2}-\frac {3 c^3 \tanh ^{-1}\left (\sqrt {1-a^2 x^2}\right )}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 77, normalized size = 1.00 \[ \frac {\sqrt {1-a^2 x^2} \left (\frac {c^3}{2 a^2 x^2}+c^3\right )}{a}-\frac {3 c^3 \log \left (\sqrt {1-a^2 x^2}+1\right )}{2 a}+\frac {3 c^3 \log (a x)}{2 a} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 78, normalized size = 1.01 \[ \frac {3 \, a^{2} c^{3} x^{2} \log \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{x}\right ) + 2 \, a^{2} c^{3} x^{2} + {\left (2 \, a^{2} c^{3} x^{2} + c^{3}\right )} \sqrt {-a^{2} x^{2} + 1}}{2 \, a^{3} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 97, normalized size = 1.26 \[ -\frac {3 \, a^{4} c^{3} \log \left (\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 3 \, a^{4} c^{3} \log \left (-\sqrt {-a^{2} x^{2} + 1} + 1\right ) - 4 \, \sqrt {-a^{2} x^{2} + 1} a^{4} c^{3} - \frac {2 \, \sqrt {-a^{2} x^{2} + 1} a^{2} c^{3}}{x^{2}}}{4 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 118, normalized size = 1.53 \[ \frac {c^{3} \left (a^{6} \left (-\frac {x^{2}}{a^{2} \sqrt {-a^{2} x^{2}+1}}+\frac {2}{a^{4} \sqrt {-a^{2} x^{2}+1}}\right )-\frac {3 a^{2}}{\sqrt {-a^{2} x^{2}+1}}+\frac {3 a^{2} \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}-\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{2}+\frac {1}{2 x^{2} \sqrt {-a^{2} x^{2}+1}}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 188, normalized size = 2.44 \[ -a^{3} c^{3} {\left (\frac {x^{2}}{\sqrt {-a^{2} x^{2} + 1} a^{2}} - \frac {2}{\sqrt {-a^{2} x^{2} + 1} a^{4}}\right )} + \frac {3 \, c^{3} {\left (\frac {1}{\sqrt {-a^{2} x^{2} + 1}} - \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right )\right )}}{a} - \frac {3 \, c^{3}}{\sqrt {-a^{2} x^{2} + 1} a} + \frac {{\left (3 \, a^{2} \log \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) - \frac {3 \, a^{2}}{\sqrt {-a^{2} x^{2} + 1}} + \frac {1}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\right )} c^{3}}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 64, normalized size = 0.83 \[ \frac {c^3\,\sqrt {1-a^2\,x^2}}{a}-\frac {3\,c^3\,\mathrm {atanh}\left (\sqrt {1-a^2\,x^2}\right )}{2\,a}+\frac {c^3\,\sqrt {1-a^2\,x^2}}{2\,a^3\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 35.55, size = 104, normalized size = 1.35 \[ \frac {2 c^{3} \sqrt {- a^{2} x^{2} + 1} + \frac {3 c^{3} \log {\left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}}{2} - \frac {3 c^{3} \log {\left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}} \right )}}{2} + \frac {c^{3}}{2 \left (1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}}\right )} + \frac {c^{3}}{2 \left (-1 + \frac {1}{\sqrt {- a^{2} x^{2} + 1}}\right )}}{2 a} \]
Verification of antiderivative is not currently implemented for this CAS.
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