3.520 \(\int (1-a^{m x})^4 \, dx\)

Optimal. Leaf size=65 \[ -\frac{4 a^{m x}}{m \log (a)}+\frac{3 a^{2 m x}}{m \log (a)}-\frac{4 a^{3 m x}}{3 m \log (a)}+\frac{a^{4 m x}}{4 m \log (a)}+x \]

[Out]

x - (4*a^(m*x))/(m*Log[a]) + (3*a^(2*m*x))/(m*Log[a]) - (4*a^(3*m*x))/(3*m*Log[a]) + a^(4*m*x)/(4*m*Log[a])

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Rubi [A]  time = 0.0191933, antiderivative size = 65, normalized size of antiderivative = 1., 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.

[In]

Int[(1 - a^(m*x))^4,x]

[Out]

x - (4*a^(m*x))/(m*Log[a]) + (3*a^(2*m*x))/(m*Log[a]) - (4*a^(3*m*x))/(3*m*Log[a]) + a^(4*m*x)/(4*m*Log[a])

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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*}

Mathematica [A]  time = 0.0161545, 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.

[In]

Integrate[(1 - a^(m*x))^4,x]

[Out]

(-4*a^(m*x) + 3*a^(2*m*x) - (4*a^(3*m*x))/3 + a^(4*m*x)/4 + m*x*Log[a])/(m*Log[a])

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Maple [A]  time = 0.002, size = 78, normalized size = 1.2 \begin{align*}{\frac{ \left ({a}^{mx} \right ) ^{4}}{4\,m\ln \left ( a \right ) }}-{\frac{4\, \left ({a}^{mx} \right ) ^{3}}{3\,m\ln \left ( a \right ) }}+3\,{\frac{ \left ({a}^{mx} \right ) ^{2}}{m\ln \left ( a \right ) }}-4\,{\frac{{a}^{mx}}{m\ln \left ( a \right ) }}+{\frac{\ln \left ({a}^{mx} \right ) }{m\ln \left ( a \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-a^(m*x))^4,x)

[Out]

1/4/m/ln(a)*(a^(m*x))^4-4/3/m/ln(a)*(a^(m*x))^3+3/m/ln(a)*(a^(m*x))^2-4*a^(m*x)/m/ln(a)+1/m/ln(a)*ln(a^(m*x))

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Maxima [A]  time = 0.929881, size = 82, normalized size = 1.26 \begin{align*} x + \frac{a^{4 \, m x}}{4 \, m \log \left (a\right )} - \frac{4 \, a^{3 \, m x}}{3 \, m \log \left (a\right )} + \frac{3 \, a^{2 \, m x}}{m \log \left (a\right )} - \frac{4 \, a^{m x}}{m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a^(m*x))^4,x, algorithm="maxima")

[Out]

x + 1/4*a^(4*m*x)/(m*log(a)) - 4/3*a^(3*m*x)/(m*log(a)) + 3*a^(2*m*x)/(m*log(a)) - 4*a^(m*x)/(m*log(a))

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Fricas [A]  time = 1.84396, size = 122, normalized size = 1.88 \begin{align*} \frac{12 \, m x \log \left (a\right ) + 3 \, a^{4 \, m x} - 16 \, a^{3 \, m x} + 36 \, a^{2 \, m x} - 48 \, a^{m x}}{12 \, m \log \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a^(m*x))^4,x, algorithm="fricas")

[Out]

1/12*(12*m*x*log(a) + 3*a^(4*m*x) - 16*a^(3*m*x) + 36*a^(2*m*x) - 48*a^(m*x))/(m*log(a))

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Sympy [A]  time = 0.155208, size = 88, normalized size = 1.35 \begin{align*} x + \begin{cases} \frac{3 a^{4 m x} m^{3} \log{\left (a \right )}^{3} - 16 a^{3 m x} m^{3} \log{\left (a \right )}^{3} + 36 a^{2 m x} m^{3} \log{\left (a \right )}^{3} - 48 a^{m x} m^{3} \log{\left (a \right )}^{3}}{12 m^{4} \log{\left (a \right )}^{4}} & \text{for}\: 12 m^{4} \log{\left (a \right )}^{4} \neq 0 \\- x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a**(m*x))**4,x)

[Out]

x + Piecewise(((3*a**(4*m*x)*m**3*log(a)**3 - 16*a**(3*m*x)*m**3*log(a)**3 + 36*a**(2*m*x)*m**3*log(a)**3 - 48
*a**(m*x)*m**3*log(a)**3)/(12*m**4*log(a)**4), Ne(12*m**4*log(a)**4, 0)), (-x, True))

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Giac [A]  time = 1.10778, size = 65, normalized size = 1. \begin{align*} \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 \left (a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-a^(m*x))^4,x, algorithm="giac")

[Out]

1/12*(12*m*x*log(abs(a)) + 3*a^(4*m*x) - 16*a^(3*m*x) + 36*a^(2*m*x) - 48*a^(m*x))/(m*log(a))