3.6.5 \(\int (a^{k x}+a^{l x})^4 \, dx\) [505]

Optimal. Leaf size=98 \[ \frac {a^{4 k x}}{4 k \log (a)}+\frac {a^{4 l x}}{4 l \log (a)}+\frac {3 a^{2 (k+l) x}}{(k+l) \log (a)}+\frac {4 a^{(3 k+l) x}}{(3 k+l) \log (a)}+\frac {4 a^{(k+3 l) x}}{(k+3 l) \log (a)} \]

[Out]

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)

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Rubi [A]
time = 0.09, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6874, 2225} \begin {gather*} \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)} \end {gather*}

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 2225

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]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \left (a^{k x}+a^{l x}\right )^4 \, dx &=\frac {\text {Subst}\left (\int \left (e^{k x}+e^{l x}\right )^4 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {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 {\text {Subst}\left (\int e^{4 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\text {Subst}\left (\int e^{4 l x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {4 \text {Subst}\left (\int e^{(3 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {4 \text {Subst}\left (\int e^{(k+3 l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {6 \text {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*}

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Mathematica [A]
time = 0.09, size = 80, normalized size = 0.82 \begin {gather*} \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)} \end {gather*}

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 = 0.04, size = 109, normalized size = 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	\(-\frac {1-{\mathrm e}^{4 k x \ln \left (a \right )}}{4 k \ln \left (a \right )}-\frac {4 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}-\frac {3 \left (1-{\mathrm e}^{2 x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}-\frac {4 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}-\frac {1-{\mathrm e}^{4 l x \ln \left (a \right )}}{4 l \ln \left (a \right )}\)	\(212\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x)+a^(l*x))^4,x,method=_RETURNVERBOSE)

[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 = 1.08, size = 99, normalized size = 1.01 \begin {gather*} \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 {gather*}

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)*log
(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] Leaf count of result is larger than twice the leaf count of optimal. 205 vs. \(2 (94) = 188\).
time = 0.60, size = 205, normalized size = 2.09 \begin {gather*} \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 {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 1350 vs. \(2 (82) = 164\).
time = 20.57, size = 1350, normalized size = 13.78 \begin {gather*} \begin {cases} 16 x & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} + \frac {4 a^{3 l x}}{3 l \log {\left (a \right )}} + \frac {3 a^{2 l x}}{l \log {\left (a \right )}} + \frac {4 a^{l x}}{l \log {\left (a \right )}} + x & \text {for}\: k = 0 \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} + 4 x - \frac {3 a^{- 4 l x}}{2 l \log {\left (a \right )}} - \frac {a^{- 8 l x}}{2 l \log {\left (a \right )}} - \frac {a^{- 12 l x}}{12 l \log {\left (a \right )}} & \text {for}\: k = - 3 l \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} + \frac {2 a^{2 l x}}{l \log {\left (a \right )}} + 6 x - \frac {2 a^{- 2 l x}}{l \log {\left (a \right )}} - \frac {a^{- 4 l x}}{4 l \log {\left (a \right )}} & \text {for}\: k = - l \\\frac {3 a^{\frac {8 l x}{3}}}{2 l \log {\left (a \right )}} + \frac {9 a^{\frac {4 l x}{3}}}{2 l \log {\left (a \right )}} + \frac {a^{4 l x}}{4 l \log {\left (a \right )}} + 4 x - \frac {3 a^{- \frac {4 l x}{3}}}{4 l \log {\left (a \right )}} & \text {for}\: k = - \frac {l}{3} \\\frac {a^{4 k x}}{4 k \log {\left (a \right )}} + \frac {4 a^{3 k x}}{3 k \log {\left (a \right )}} + \frac {3 a^{2 k x}}{k \log {\left (a \right )}} + \frac {4 a^{k x}}{k \log {\left (a \right )}} + x & \text {for}\: l = 0 \\\frac {3 a^{4 k x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 k x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 k x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 k x} l^{4}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {16 a^{3 k x} a^{l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {64 a^{3 k x} a^{l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {48 a^{3 k x} a^{l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {36 a^{2 k x} a^{2 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {120 a^{2 k x} a^{2 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {36 a^{2 k x} a^{2 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {48 a^{k x} a^{3 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {64 a^{k x} a^{3 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {16 a^{k x} a^{3 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 l x} k^{4}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**(k*x)+a**(l*x))**4,x)

[Out]

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*log(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)), E
q(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*lo
g(a)) + 4*a**(k*x)/(k*log(a)) + x, Eq(l, 0)), (3*a**(4*k*x)*k**3*l/(12*k**4*l*log(a) + 52*k**3*l**2*log(a) + 5
2*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) + 5
2*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 36*a**(2*k*x)*a**(2*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)) + 120*a**(2*k*x)*a**(2*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)) + 36*a**(2*k*x)*a**(2*l*x)*k*l**3/(12*k**4*l*log(a) + 5
2*k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 48*a**(k*x)*a**(3*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**(k*x)*a**(3*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)) + 16*a**(k*x)*a**(3*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)) + 3*a**(4*l*x)*k**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)) + 13*a**(4*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)) + 13*a**(4*l*x)*k**2*l**2/(12*k**4*l*log(a) + 5
2*k**3*l**2*log(a) + 52*k**2*l**3*log(a) + 12*k*l**4*log(a)) + 3*a**(4*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)), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.75, size = 1359, normalized size = 13.87 \begin {gather*} \text {Too large to display} \end {gather*}

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) + 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*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(ab
s(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*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) - (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) + 4*I*(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))) - 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(ab
s(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) + 3*I*(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))) - I*e^(-I*p
i*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))))*e^(2*(k*log(abs(a)) + l*log(abs(a)))*x)

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Mupad [B]
time = 0.42, size = 106, normalized size = 1.08 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x) + a^(l*x))^4,x)

[Out]

(3*a^(2*k*x)*a^(2*l*x))/(k*log(a) + l*log(a)) + (4*a^(k*x)*a^(3*l*x))/(k*log(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))

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