Optimal. Leaf size=65 \[ -\frac {4 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{m \log (a)}-\frac {4 a^{3 m x}}{3 m \log (a)}+\frac {a^{4 m x}}{4 m \log (a)}+x \]
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Rubi [A] time = 0.02, antiderivative size = 65, 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} \[ -\frac {4 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{m \log (a)}-\frac {4 a^{3 m x}}{3 m \log (a)}+\frac {a^{4 m x}}{4 m \log (a)}+x \]
Antiderivative was successfully verified.
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Rule 43
Rule 2282
Rubi steps
\begin {align*} \int \left (1-a^{m x}\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(1-x)^4}{x} \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=\frac {\operatorname {Subst}\left (\int \left (-4+\frac {1}{x}+6 x-4 x^2+x^3\right ) \, dx,x,a^{m x}\right )}{m \log (a)}\\ &=x-\frac {4 a^{m x}}{m \log (a)}+\frac {3 a^{2 m x}}{m \log (a)}-\frac {4 a^{3 m x}}{3 m \log (a)}+\frac {a^{4 m x}}{4 m \log (a)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.75 \[ \frac {-4 a^{m x}+3 a^{2 m x}-\frac {4}{3} a^{3 m x}+\frac {1}{4} a^{4 m x}+m x \log (a)}{m \log (a)} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (1-a^{m x}\right )^4 \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.18, size = 47, normalized size = 0.72 \[ \frac {12 \, m x \log \relax (a) + 3 \, a^{4 \, m x} - 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} - 48 \, a^{m x}}{12 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 48, normalized size = 0.74 \[ \frac {12 \, m x \log \left ({\left | a \right |}\right ) + 3 \, a^{4 \, m x} - 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} - 48 \, a^{m x}}{12 \, m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 50, normalized size = 0.77
method | result | size |
derivativedivides | \(\frac {\frac {a^{4 m x}}{4}-\frac {4 a^{3 m x}}{3}+3 a^{2 m x}-4 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(50\) |
default | \(\frac {\frac {a^{4 m x}}{4}-\frac {4 a^{3 m x}}{3}+3 a^{2 m x}-4 a^{m x}+\ln \left (a^{m x}\right )}{m \ln \relax (a )}\) | \(50\) |
risch | \(x -\frac {4 a^{m x}}{m \ln \relax (a )}+\frac {3 a^{2 m x}}{m \ln \relax (a )}-\frac {4 a^{3 m x}}{3 m \ln \relax (a )}+\frac {a^{4 m x}}{4 m \ln \relax (a )}\) | \(65\) |
norman | \(x -\frac {4 \,{\mathrm e}^{m x \ln \relax (a )}}{m \ln \relax (a )}+\frac {3 \,{\mathrm e}^{2 m x \ln \relax (a )}}{m \ln \relax (a )}-\frac {4 \,{\mathrm e}^{3 m x \ln \relax (a )}}{3 m \ln \relax (a )}+\frac {{\mathrm e}^{4 m x \ln \relax (a )}}{4 m \ln \relax (a )}\) | \(69\) |
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 = 0.51, size = 61, normalized size = 0.94 \[ x + \frac {a^{4 \, m x}}{4 \, m \log \relax (a)} - \frac {4 \, a^{3 \, m x}}{3 \, m \log \relax (a)} + \frac {3 \, a^{2 \, m x}}{m \log \relax (a)} - \frac {4 \, a^{m x}}{m \log \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.31, size = 43, normalized size = 0.66 \[ x-\frac {4\,a^{m\,x}-3\,a^{2\,m\,x}+\frac {4\,a^{3\,m\,x}}{3}-\frac {a^{4\,m\,x}}{4}}{m\,\ln \relax (a)} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 88, normalized size = 1.35 \[ x + \begin {cases} \frac {3 a^{4 m x} m^{3} \log {\relax (a )}^{3} - 16 a^{3 m x} m^{3} \log {\relax (a )}^{3} + 36 a^{2 m x} m^{3} \log {\relax (a )}^{3} - 48 a^{m x} m^{3} \log {\relax (a )}^{3}}{12 m^{4} \log {\relax (a )}^{4}} & \text {for}\: 12 m^{4} \log {\relax (a )}^{4} \neq 0 \\- x & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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