Integrand size = 13, antiderivative size = 98 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {a^{4 k x}}{4 k \log (a)}+\frac {a^{4 l x}}{4 l \log (a)}+\frac {3 a^{2 (k+l) x}}{(k+l) \log (a)}+\frac {4 a^{(3 k+l) x}}{(3 k+l) \log (a)}+\frac {4 a^{(k+3 l) x}}{(k+3 l) \log (a)} \] Output:
1/4*a^(4*k*x)/k/ln(a)+1/4*a^(4*l*x)/l/ln(a)+3*a^(2*(k+l)*x)/(k+l)/ln(a)+4* a^((3*k+l)*x)/(3*k+l)/ln(a)+4*a^((k+3*l)*x)/(k+3*l)/ln(a)
Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {\frac {a^{4 k x}}{k}+\frac {a^{4 l x}}{l}+\frac {12 a^{2 (k+l) x}}{k+l}+\frac {16 a^{(3 k+l) x}}{3 k+l}+\frac {16 a^{(k+3 l) x}}{k+3 l}}{4 \log (a)} \] Input:
Integrate[(a^(k*x) + a^(l*x))^4,x]
Output:
(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])
Time = 0.34 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7281, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a^{k x}+a^{l x}\right )^4 \, dx\) |
| \(\Big \downarrow \) 7281 |
| \(\displaystyle \frac {\int \left (a^{k x}+a^{l x}\right )^4d(x \log (a))}{\log (a)}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {\int \left (a^{4 k x}+a^{4 l x}+6 a^{2 (k+l) x}+4 a^{(3 k+l) x}+4 a^{(k+3 l) x}\right )d(x \log (a))}{\log (a)}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3 a^{2 x (k+l)}}{k+l}+\frac {4 a^{x (3 k+l)}}{3 k+l}+\frac {4 a^{x (k+3 l)}}{k+3 l}+\frac {a^{4 k x}}{4 k}+\frac {a^{4 l x}}{4 l}}{\log (a)}\) |
Input:
Int[(a^(k*x) + a^(l*x))^4,x]
Output:
(a^(4*k*x)/(4*k) + a^(4*l*x)/(4*l) + (3*a^(2*(k + l)*x))/(k + l) + (4*a^(( 3*k + l)*x))/(3*k + l) + (4*a^((k + 3*l)*x))/(k + 3*l))/Log[a]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.37 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {a^{4 k x}}{4 k \ln \left (a \right )}+\frac {4 a^{3 k x} a^{l x}}{\ln \left (a \right ) \left (3 k +l \right )}+\frac {3 a^{2 k x} a^{2 l x}}{\ln \left (a \right ) \left (k +l \right )}+\frac {4 a^{k x} a^{3 l x}}{\ln \left (a \right ) \left (k +3 l \right )}+\frac {a^{4 l x}}{4 l \ln \left (a \right )}\) | \(109\) |
| meijerg | \(-\frac {1-{\mathrm e}^{4 k x \ln \left (a \right )}}{4 k \ln \left (a \right )}-\frac {4 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}-\frac {3 \left (1-{\mathrm e}^{2 x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}-\frac {4 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}-\frac {1-{\mathrm e}^{4 l x \ln \left (a \right )}}{4 l \ln \left (a \right )}\) | \(212\) |
| parallelrisch | \(\frac {3 a^{4 k x} k^{3} l +13 a^{4 k x} k^{2} l^{2}+13 a^{4 k x} k \,l^{3}+3 a^{4 k x} l^{4}+16 a^{3 k x} a^{l x} k^{3} l +64 a^{3 k x} a^{l x} k^{2} l^{2}+48 a^{3 k x} a^{l x} k \,l^{3}+36 a^{2 k x} a^{2 l x} k^{3} l +120 a^{2 k x} a^{2 l x} k^{2} l^{2}+36 a^{2 k x} a^{2 l x} k \,l^{3}+48 a^{k x} a^{3 l x} k^{3} l +64 a^{k x} a^{3 l x} k^{2} l^{2}+16 a^{k x} a^{3 l x} k \,l^{3}+3 a^{4 l x} k^{4}+13 a^{4 l x} k^{3} l +13 a^{4 l x} k^{2} l^{2}+3 a^{4 l x} k \,l^{3}}{4 \ln \left (a \right ) k \left (3 k +l \right ) \left (k +l \right ) \left (k +3 l \right ) l}\) | \(313\) |
| orering | \(\frac {\left (3 k^{4}+38 k^{3} l +78 k^{2} l^{2}+38 k \,l^{3}+3 l^{4}\right ) \left (a^{k x}+a^{l x}\right )^{4}}{4 \ln \left (a \right ) l \left (3 k^{3}+13 l \,k^{2}+13 l^{2} k +3 l^{3}\right ) k}-\frac {5 \left (5 k^{2}+22 l k +5 l^{2}\right ) \left (a^{k x}+a^{l x}\right )^{3} \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )}{4 \ln \left (a \right )^{2} k \left (3 k^{2}+10 l k +3 l^{2}\right ) l}+\frac {5 \left (7 k^{2}+18 l k +7 l^{2}\right ) \left (12 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{2}+4 \left (a^{k x}+a^{l x}\right )^{3} \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )\right )}{32 \ln \left (a \right )^{3} l \left (3 k^{3}+13 l \,k^{2}+13 l^{2} k +3 l^{3}\right ) k}-\frac {5 \left (24 \left (a^{k x}+a^{l x}\right ) \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{3}+36 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right ) \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )+4 \left (a^{k x}+a^{l x}\right )^{3} \left (a^{k x} k^{3} \ln \left (a \right )^{3}+a^{l x} l^{3} \ln \left (a \right )^{3}\right )\right )}{16 \ln \left (a \right )^{4} k \left (3 k^{2}+10 l k +3 l^{2}\right ) l}+\frac {24 \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{4}+144 \left (a^{k x}+a^{l x}\right ) \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right )^{2} \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )+36 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k^{2} \ln \left (a \right )^{2}+a^{l x} l^{2} \ln \left (a \right )^{2}\right )^{2}+48 \left (a^{k x}+a^{l x}\right )^{2} \left (a^{k x} k \ln \left (a \right )+a^{l x} l \ln \left (a \right )\right ) \left (a^{k x} k^{3} \ln \left (a \right )^{3}+a^{l x} l^{3} \ln \left (a \right )^{3}\right )+4 \left (a^{k x}+a^{l x}\right )^{3} \left (a^{k x} k^{4} \ln \left (a \right )^{4}+a^{l x} l^{4} \ln \left (a \right )^{4}\right )}{32 \ln \left (a \right )^{5} l \left (3 k^{3}+13 l \,k^{2}+13 l^{2} k +3 l^{3}\right ) k}\) | \(726\) |
Input:
int((a^(k*x)+a^(l*x))^4,x,method=_RETURNVERBOSE)
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (94) = 188\).
Time = 0.07 (sec) , antiderivative size = 205, normalized size of antiderivative = 2.09 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {16 \, {\left (3 \, k^{3} l + 4 \, k^{2} l^{2} + k l^{3}\right )} a^{k x} a^{3 \, l x} + 12 \, {\left (3 \, k^{3} l + 10 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{2 \, k x} a^{2 \, l x} + 16 \, {\left (k^{3} l + 4 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{3 \, k x} a^{l x} + {\left (3 \, k^{3} l + 13 \, k^{2} l^{2} + 13 \, k l^{3} + 3 \, l^{4}\right )} a^{4 \, k x} + {\left (3 \, k^{4} + 13 \, k^{3} l + 13 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{4 \, l x}}{4 \, {\left (3 \, k^{4} l + 13 \, k^{3} l^{2} + 13 \, k^{2} l^{3} + 3 \, k l^{4}\right )} \log \left (a\right )} \] Input:
integrate((a^(k*x)+a^(l*x))^4,x, algorithm="fricas")
Output:
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))
Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (82) = 164\).
Time = 15.02 (sec) , antiderivative size = 1350, normalized size of antiderivative = 13.78 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\text {Too large to display} \] Input:
integrate((a**(k*x)+a**(l*x))**4,x)
Output:
Piecewise((16*x, Eq(a, 1) & (Eq(a, 1) | Eq(k, 0)) & (Eq(a, 1) | Eq(l, 0))) , (a**(4*l*x)/(4*l*log(a)) + 4*a**(3*l*x)/(3*l*log(a)) + 3*a**(2*l*x)/(l*l og(a)) + 4*a**(l*x)/(l*log(a)) + x, Eq(k, 0)), (a**(4*l*x)/(4*l*log(a)) + 4*x - 3/(2*a**(4*l*x)*l*log(a)) - 1/(2*a**(8*l*x)*l*log(a)) - 1/(12*a**(12 *l*x)*l*log(a)), Eq(k, -3*l)), (a**(4*l*x)/(4*l*log(a)) + 2*a**(2*l*x)/(l* log(a)) + 6*x - 2/(a**(2*l*x)*l*log(a)) - 1/(4*a**(4*l*x)*l*log(a)), Eq(k, -l)), (3*a**(8*l*x/3)/(2*l*log(a)) + 9*a**(4*l*x/3)/(2*l*log(a)) + a**(4* l*x)/(4*l*log(a)) + 4*x - 3/(4*a**(4*l*x/3)*l*log(a)), Eq(k, -l/3)), (a**( 4*k*x)/(4*k*log(a)) + 4*a**(3*k*x)/(3*k*log(a)) + 3*a**(2*k*x)/(k*log(a)) + 4*a**(k*x)/(k*log(a)) + x, Eq(l, 0)), (3*a**(4*k*x)*k**3*l/(12*k**4*l*lo g(a) + 52*k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 13* a**(4*k*x)*k**2*l**2/(12*k**4*l*log(a) + 52*k**3*l**2*log(a) + 52*k**2*l** 3*log(a) + 12*k*l**4*log(a)) + 13*a**(4*k*x)*k*l**3/(12*k**4*l*log(a) + 52 *k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 3*a**(4*k*x) *l**4/(12*k**4*l*log(a) + 52*k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k *l**4*log(a)) + 16*a**(3*k*x)*a**(l*x)*k**3*l/(12*k**4*l*log(a) + 52*k**3* l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 64*a**(3*k*x)*a**( l*x)*k**2*l**2/(12*k**4*l*log(a) + 52*k**3*l**2*log(a) + 52*k**2*l**3*log( a) + 12*k*l**4*log(a)) + 48*a**(3*k*x)*a**(l*x)*k*l**3/(12*k**4*l*log(a) + 52*k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 36*a**...
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.01 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {4 \, a^{3 \, k x + l x}}{{\left (3 \, k + l\right )} \log \left (a\right )} + \frac {4 \, a^{k x + 3 \, l x}}{{\left (k + 3 \, l\right )} \log \left (a\right )} + \frac {3 \, a^{2 \, k x + 2 \, l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac {a^{4 \, k x}}{4 \, k \log \left (a\right )} + \frac {a^{4 \, l x}}{4 \, l \log \left (a\right )} \] Input:
integrate((a^(k*x)+a^(l*x))^4,x, algorithm="maxima")
Output:
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)*log(a)) + 1/4*a^(4*k*x)/(k*log(a)) + 1/4* a^(4*l*x)/(l*log(a))
Result contains complex when optimal does not.
Time = 0.18 (sec) , antiderivative size = 1359, normalized size of antiderivative = 13.87 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\text {Too large to display} \] Input:
integrate((a^(k*x)+a^(l*x))^4,x, algorithm="giac")
Output:
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*sgn(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*sgn(a) + 2*I*pi*l*x)/(-2*I*pi*l*sgn(a) + 2*I*p i*l + 4*l*log(abs(a)))) + 8*(2*(3*k*log(abs(a)) + l*log(abs(a)))*cos(-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*lo g(abs(a)) + l*log(abs(a)))*x) + 4*I*(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))) - I*e^(-3/2*I*...
Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.08 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {3\,a^{2\,k\,x}\,a^{2\,l\,x}}{k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {4\,a^{k\,x}\,a^{3\,l\,x}}{k\,\ln \left (a\right )+3\,l\,\ln \left (a\right )}+\frac {4\,a^{3\,k\,x}\,a^{l\,x}}{3\,k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {a^{4\,k\,x}}{4\,k\,\ln \left (a\right )}+\frac {a^{4\,l\,x}}{4\,l\,\ln \left (a\right )} \] Input:
int((a^(k*x) + a^(l*x))^4,x)
Output:
(3*a^(2*k*x)*a^(2*l*x))/(k*log(a) + l*log(a)) + (4*a^(k*x)*a^(3*l*x))/(k*l og(a) + 3*l*log(a)) + (4*a^(3*k*x)*a^(l*x))/(3*k*log(a) + l*log(a)) + a^(4 *k*x)/(4*k*log(a)) + a^(4*l*x)/(4*l*log(a))
Time = 0.16 (sec) , antiderivative size = 289, normalized size of antiderivative = 2.95 \[ \int \left (a^{k x}+a^{l x}\right )^4 \, dx=\frac {3 a^{4 k x} k^{3} l +13 a^{4 k x} k^{2} l^{2}+13 a^{4 k x} k \,l^{3}+3 a^{4 k x} l^{4}+16 a^{3 k x +l x} k^{3} l +64 a^{3 k x +l x} k^{2} l^{2}+48 a^{3 k x +l x} k \,l^{3}+36 a^{2 k x +2 l x} k^{3} l +120 a^{2 k x +2 l x} k^{2} l^{2}+36 a^{2 k x +2 l x} k \,l^{3}+48 a^{k x +3 l x} k^{3} l +64 a^{k x +3 l x} k^{2} l^{2}+16 a^{k x +3 l x} k \,l^{3}+3 a^{4 l x} k^{4}+13 a^{4 l x} k^{3} l +13 a^{4 l x} k^{2} l^{2}+3 a^{4 l x} k \,l^{3}}{4 \,\mathrm {log}\left (a \right ) k l \left (3 k^{3}+13 k^{2} l +13 k \,l^{2}+3 l^{3}\right )} \] Input:
int((a^(k*x)+a^(l*x))^4,x)
Output:
(3*a**(4*k*x)*k**3*l + 13*a**(4*k*x)*k**2*l**2 + 13*a**(4*k*x)*k*l**3 + 3* a**(4*k*x)*l**4 + 16*a**(3*k*x + l*x)*k**3*l + 64*a**(3*k*x + l*x)*k**2*l* *2 + 48*a**(3*k*x + l*x)*k*l**3 + 36*a**(2*k*x + 2*l*x)*k**3*l + 120*a**(2 *k*x + 2*l*x)*k**2*l**2 + 36*a**(2*k*x + 2*l*x)*k*l**3 + 48*a**(k*x + 3*l* x)*k**3*l + 64*a**(k*x + 3*l*x)*k**2*l**2 + 16*a**(k*x + 3*l*x)*k*l**3 + 3 *a**(4*l*x)*k**4 + 13*a**(4*l*x)*k**3*l + 13*a**(4*l*x)*k**2*l**2 + 3*a**( 4*l*x)*k*l**3)/(4*log(a)*k*l*(3*k**3 + 13*k**2*l + 13*k*l**2 + 3*l**3))