Optimal. Leaf size=33 \[ -\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}+x \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2282, 43} \[ -\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2282
Rubi steps
\begin {align*} \int \left (1-a^{m x}\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^2}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2+\frac {1}{x}+x\right ) \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=x-\frac {2 a^{m x}}{m \log (a)}+\frac {a^{2 m x}}{2 m \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 0.76 \[ \frac {\left (a^{m x}-4\right ) a^{m x}}{2 m \log (a)}+x \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1-a^{m x}\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.43, size = 29, normalized size = 0.88 \[ \frac {2 \, m x \log \relax (a) + a^{2 \, m x} - 4 \, a^{m x}}{2 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 30, normalized size = 0.91 \[ \frac {2 \, m x \log \left ({\left | a \right |}\right ) + a^{2 \, m x} - 4 \, a^{m x}}{2 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 32, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {a^{2 m x}}{2}-2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(32\) |
default | \(\frac {\frac {a^{2 m x}}{2}-2 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(32\) |
risch | \(x -\frac {2 a^{m x}}{m \ln \relax (a )}+\frac {a^{2 m x}}{2 m \ln \relax (a )}\) | \(33\) |
norman | \(x -\frac {2 \,{\mathrm e}^{m x \ln \relax (a )}}{m \ln \relax (a )}+\frac {{\mathrm e}^{2 m x \ln \relax (a )}}{2 m \ln \relax (a )}\) | \(35\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 31, normalized size = 0.94 \[ x + \frac {a^{2 \, m x}}{2 \, m \log \relax (a)} - \frac {2 \, a^{m x}}{m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 27, normalized size = 0.82 \[ x-\frac {2\,a^{m\,x}-\frac {a^{2\,m\,x}}{2}}{m\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.12, size = 46, normalized size = 1.39 \[ x + \begin {cases} \frac {a^{2 m x} m \log {\relax (a )} - 4 a^{m x} m \log {\relax (a )}}{2 m^{2} \log {\relax (a )}^{2}} & \text {for}\: 2 m^{2} \log {\relax (a )}^{2} \neq 0 \\- x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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