3.510 \(\int (a^{k x}-a^{l x})^4 \, dx\)
Optimal. Leaf size=98 \[ \frac{3 a^{2 x (k+l)}}{\log (a) (k+l)}-\frac{4 a^{x (3 k+l)}}{\log (a) (3 k+l)}-\frac{4 a^{x (k+3 l)}}{\log (a) (k+3 l)}+\frac{a^{4 k x}}{4 k \log (a)}+\frac{a^{4 l x}}{4 l \log (a)} \]
[Out]
a^(4*k*x)/(4*k*Log[a]) + a^(4*l*x)/(4*l*Log[a]) + (3*a^(2*(k + l)*x))/((k + l)*Log[a]) - (4*a^((3*k + l)*x))/(
(3*k + l)*Log[a]) - (4*a^((k + 3*l)*x))/((k + 3*l)*Log[a])
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Rubi [A] time = 0.0984358, antiderivative size = 98, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used =
{6742, 2194} \[ \frac{3 a^{2 x (k+l)}}{\log (a) (k+l)}-\frac{4 a^{x (3 k+l)}}{\log (a) (3 k+l)}-\frac{4 a^{x (k+3 l)}}{\log (a) (k+3 l)}+\frac{a^{4 k x}}{4 k \log (a)}+\frac{a^{4 l x}}{4 l \log (a)} \]
Antiderivative was successfully verified.
[In]
Int[(a^(k*x) - a^(l*x))^4,x]
[Out]
a^(4*k*x)/(4*k*Log[a]) + a^(4*l*x)/(4*l*Log[a]) + (3*a^(2*(k + l)*x))/((k + l)*Log[a]) - (4*a^((3*k + l)*x))/(
(3*k + l)*Log[a]) - (4*a^((k + 3*l)*x))/((k + 3*l)*Log[a])
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rule 2194
Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]
Rubi steps
\begin{align*} \int \left (a^{k x}-a^{l x}\right )^4 \, dx &=\frac{\operatorname{Subst}\left (\int \left (e^{k x}-e^{l x}\right )^4 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{\operatorname{Subst}\left (\int \left (e^{4 k x}+e^{4 l x}+6 e^{2 (k+l) x}-4 e^{(3 k+l) x}-4 e^{(k+3 l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{\operatorname{Subst}\left (\int e^{4 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac{\operatorname{Subst}\left (\int e^{4 l x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac{4 \operatorname{Subst}\left (\int e^{(3 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac{4 \operatorname{Subst}\left (\int e^{(k+3 l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac{6 \operatorname{Subst}\left (\int e^{2 (k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac{a^{4 k x}}{4 k \log (a)}+\frac{a^{4 l x}}{4 l \log (a)}+\frac{3 a^{2 (k+l) x}}{(k+l) \log (a)}-\frac{4 a^{(3 k+l) x}}{(3 k+l) \log (a)}-\frac{4 a^{(k+3 l) x}}{(k+3 l) \log (a)}\\ \end{align*}
Mathematica [A] time = 0.0700836, size = 80, normalized size = 0.82 \[ \frac{\frac{12 a^{2 x (k+l)}}{k+l}-\frac{16 a^{x (3 k+l)}}{3 k+l}-\frac{16 a^{x (k+3 l)}}{k+3 l}+\frac{a^{4 k x}}{k}+\frac{a^{4 l x}}{l}}{4 \log (a)} \]
Antiderivative was successfully verified.
[In]
Integrate[(a^(k*x) - a^(l*x))^4,x]
[Out]
(a^(4*k*x)/k + a^(4*l*x)/l + (12*a^(2*(k + l)*x))/(k + l) - (16*a^((3*k + l)*x))/(3*k + l) - (16*a^((k + 3*l)*
x))/(k + 3*l))/(4*Log[a])
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Maple [A] time = 0.015, size = 109, normalized size = 1.1 \begin{align*}{\frac{ \left ({a}^{kx} \right ) ^{4}}{4\,k\ln \left ( a \right ) }}-4\,{\frac{ \left ({a}^{kx} \right ) ^{3}{a}^{lx}}{\ln \left ( a \right ) \left ( 3\,k+l \right ) }}+3\,{\frac{ \left ({a}^{kx} \right ) ^{2} \left ({a}^{lx} \right ) ^{2}}{\ln \left ( a \right ) \left ( k+l \right ) }}-4\,{\frac{{a}^{kx} \left ({a}^{lx} \right ) ^{3}}{\ln \left ( a \right ) \left ( k+3\,l \right ) }}+{\frac{ \left ({a}^{lx} \right ) ^{4}}{4\,l\ln \left ( a \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a^(k*x)-a^(l*x))^4,x)
[Out]
1/4/ln(a)/k*(a^(k*x))^4-4*(a^(k*x))^3/ln(a)/(3*k+l)*a^(l*x)+3*(a^(k*x))^2/ln(a)/(k+l)*(a^(l*x))^2-4*a^(k*x)/ln
(a)/(k+3*l)*(a^(l*x))^3+1/4/ln(a)/l*(a^(l*x))^4
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Maxima [A] time = 0.931511, size = 134, normalized size = 1.37 \begin{align*} -\frac{4 \, a^{3 \, k x + l x}}{{\left (3 \, k + l\right )} \log \left (a\right )} - \frac{4 \, a^{k x + 3 \, l x}}{{\left (k + 3 \, l\right )} \log \left (a\right )} + \frac{3 \, a^{2 \, k x + 2 \, l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac{a^{4 \, k x}}{4 \, k \log \left (a\right )} + \frac{a^{4 \, l x}}{4 \, l \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a^(k*x)-a^(l*x))^4,x, algorithm="maxima")
[Out]
-4*a^(3*k*x + l*x)/((3*k + l)*log(a)) - 4*a^(k*x + 3*l*x)/((k + 3*l)*log(a)) + 3*a^(2*k*x + 2*l*x)/((k + l)*lo
g(a)) + 1/4*a^(4*k*x)/(k*log(a)) + 1/4*a^(4*l*x)/(l*log(a))
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Fricas [B] time = 1.94092, size = 450, normalized size = 4.59 \begin{align*} -\frac{16 \,{\left (3 \, k^{3} l + 4 \, k^{2} l^{2} + k l^{3}\right )} a^{k x} a^{3 \, l x} - 12 \,{\left (3 \, k^{3} l + 10 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{2 \, k x} a^{2 \, l x} + 16 \,{\left (k^{3} l + 4 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{3 \, k x} a^{l x} -{\left (3 \, k^{3} l + 13 \, k^{2} l^{2} + 13 \, k l^{3} + 3 \, l^{4}\right )} a^{4 \, k x} -{\left (3 \, k^{4} + 13 \, k^{3} l + 13 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{4 \, l x}}{4 \,{\left (3 \, k^{4} l + 13 \, k^{3} l^{2} + 13 \, k^{2} l^{3} + 3 \, k l^{4}\right )} \log \left (a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a^(k*x)-a^(l*x))^4,x, algorithm="fricas")
[Out]
-1/4*(16*(3*k^3*l + 4*k^2*l^2 + k*l^3)*a^(k*x)*a^(3*l*x) - 12*(3*k^3*l + 10*k^2*l^2 + 3*k*l^3)*a^(2*k*x)*a^(2*
l*x) + 16*(k^3*l + 4*k^2*l^2 + 3*k*l^3)*a^(3*k*x)*a^(l*x) - (3*k^3*l + 13*k^2*l^2 + 13*k*l^3 + 3*l^4)*a^(4*k*x
) - (3*k^4 + 13*k^3*l + 13*k^2*l^2 + 3*k*l^3)*a^(4*l*x))/((3*k^4*l + 13*k^3*l^2 + 13*k^2*l^3 + 3*k*l^4)*log(a)
)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a**(k*x)-a**(l*x))**4,x)
[Out]
Timed out
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Giac [C] time = 1.35651, size = 1835, normalized size = 18.72 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a^(k*x)-a^(l*x))^4,x, algorithm="giac")
[Out]
1/2*(2*k*cos(-2*pi*k*x*sgn(a) + 2*pi*k*x)*log(abs(a))/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2) - (pi*k*s
gn(a) - pi*k)*sin(-2*pi*k*x*sgn(a) + 2*pi*k*x)/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2))*abs(a)^(4*k*x)
+ 1/2*(2*l*cos(-2*pi*l*x*sgn(a) + 2*pi*l*x)*log(abs(a))/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l)^2) - (pi*l
*sgn(a) - pi*l)*sin(-2*pi*l*x*sgn(a) + 2*pi*l*x)/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l)^2))*abs(a)^(4*l*x
) - 1/2*I*abs(a)^(4*k*x)*(-I*e^(2*I*pi*k*x*sgn(a) - 2*I*pi*k*x)/(2*I*pi*k*sgn(a) - 2*I*pi*k + 4*k*log(abs(a)))
+ I*e^(-2*I*pi*k*x*sgn(a) + 2*I*pi*k*x)/(-2*I*pi*k*sgn(a) + 2*I*pi*k + 4*k*log(abs(a)))) - 1/2*I*abs(a)^(4*l*
x)*(-I*e^(2*I*pi*l*x*sgn(a) - 2*I*pi*l*x)/(2*I*pi*l*sgn(a) - 2*I*pi*l + 4*l*log(abs(a))) + I*e^(-2*I*pi*l*x*sg
n(a) + 2*I*pi*l*x)/(-2*I*pi*l*sgn(a) + 2*I*pi*l + 4*l*log(abs(a)))) - 8*(2*(3*k*log(abs(a)) + l*log(abs(a)))*c
os(-3/2*pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a) + 3/2*pi*k*x + 1/2*pi*l*x)/((3*pi*k*sgn(a) + pi*l*sgn(a) - 3*pi*k -
pi*l)^2 + 4*(3*k*log(abs(a)) + l*log(abs(a)))^2) - (3*pi*k*sgn(a) + pi*l*sgn(a) - 3*pi*k - pi*l)*sin(-3/2*pi*k
*x*sgn(a) - 1/2*pi*l*x*sgn(a) + 3/2*pi*k*x + 1/2*pi*l*x)/((3*pi*k*sgn(a) + pi*l*sgn(a) - 3*pi*k - pi*l)^2 + 4*
(3*k*log(abs(a)) + l*log(abs(a)))^2))*e^((3*k*log(abs(a)) + l*log(abs(a)))*x) - 1/2*I*(8*I*e^(3/2*I*pi*k*x*sgn
(a) + 1/2*I*pi*l*x*sgn(a) - 3/2*I*pi*k*x - 1/2*I*pi*l*x)/(3*I*pi*k*sgn(a) + I*pi*l*sgn(a) - 3*I*pi*k - I*pi*l
+ 6*k*log(abs(a)) + 2*l*log(abs(a))) - 8*I*e^(-3/2*I*pi*k*x*sgn(a) - 1/2*I*pi*l*x*sgn(a) + 3/2*I*pi*k*x + 1/2*
I*pi*l*x)/(-3*I*pi*k*sgn(a) - I*pi*l*sgn(a) + 3*I*pi*k + I*pi*l + 6*k*log(abs(a)) + 2*l*log(abs(a))))*e^((3*k*
log(abs(a)) + l*log(abs(a)))*x) - 8*(2*(k*log(abs(a)) + 3*l*log(abs(a)))*cos(-1/2*pi*k*x*sgn(a) - 3/2*pi*l*x*s
gn(a) + 1/2*pi*k*x + 3/2*pi*l*x)/((pi*k*sgn(a) + 3*pi*l*sgn(a) - pi*k - 3*pi*l)^2 + 4*(k*log(abs(a)) + 3*l*log
(abs(a)))^2) - (pi*k*sgn(a) + 3*pi*l*sgn(a) - pi*k - 3*pi*l)*sin(-1/2*pi*k*x*sgn(a) - 3/2*pi*l*x*sgn(a) + 1/2*
pi*k*x + 3/2*pi*l*x)/((pi*k*sgn(a) + 3*pi*l*sgn(a) - pi*k - 3*pi*l)^2 + 4*(k*log(abs(a)) + 3*l*log(abs(a)))^2)
)*e^((k*log(abs(a)) + 3*l*log(abs(a)))*x) - 1/2*I*(8*I*e^(1/2*I*pi*k*x*sgn(a) + 3/2*I*pi*l*x*sgn(a) - 1/2*I*pi
*k*x - 3/2*I*pi*l*x)/(I*pi*k*sgn(a) + 3*I*pi*l*sgn(a) - I*pi*k - 3*I*pi*l + 2*k*log(abs(a)) + 6*l*log(abs(a)))
- 8*I*e^(-1/2*I*pi*k*x*sgn(a) - 3/2*I*pi*l*x*sgn(a) + 1/2*I*pi*k*x + 3/2*I*pi*l*x)/(-I*pi*k*sgn(a) - 3*I*pi*l
*sgn(a) + I*pi*k + 3*I*pi*l + 2*k*log(abs(a)) + 6*l*log(abs(a))))*e^((k*log(abs(a)) + 3*l*log(abs(a)))*x) + 6*
(2*(k*log(abs(a)) + l*log(abs(a)))*cos(-pi*k*x*sgn(a) - pi*l*x*sgn(a) + pi*k*x + pi*l*x)/((pi*k*sgn(a) + pi*l*
sgn(a) - pi*k - pi*l)^2 + 4*(k*log(abs(a)) + l*log(abs(a)))^2) - (pi*k*sgn(a) + pi*l*sgn(a) - pi*k - pi*l)*sin
(-pi*k*x*sgn(a) - pi*l*x*sgn(a) + pi*k*x + pi*l*x)/((pi*k*sgn(a) + pi*l*sgn(a) - pi*k - pi*l)^2 + 4*(k*log(abs
(a)) + l*log(abs(a)))^2))*e^(2*(k*log(abs(a)) + l*log(abs(a)))*x) - 1/2*I*(-6*I*e^(I*pi*k*x*sgn(a) + I*pi*l*x*
sgn(a) - I*pi*k*x - I*pi*l*x)/(I*pi*k*sgn(a) + I*pi*l*sgn(a) - I*pi*k - I*pi*l + 2*k*log(abs(a)) + 2*l*log(abs
(a))) + 6*I*e^(-I*pi*k*x*sgn(a) - I*pi*l*x*sgn(a) + I*pi*k*x + I*pi*l*x)/(-I*pi*k*sgn(a) - I*pi*l*sgn(a) + I*p
i*k + I*pi*l + 2*k*log(abs(a)) + 2*l*log(abs(a))))*e^(2*(k*log(abs(a)) + l*log(abs(a)))*x)