3.6.9 \(\int (a^{k x}-a^{l x})^3 \, dx\) [509]

Optimal. Leaf size=79 \[ \frac {a^{3 k x}}{3 k \log (a)}-\frac {a^{3 l x}}{3 l \log (a)}-\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)} \]

[Out]

1/3*a^(3*k*x)/k/ln(a)-1/3*a^(3*l*x)/l/ln(a)-3*a^((2*k+l)*x)/(2*k+l)/ln(a)+3*a^((k+2*l)*x)/(k+2*l)/ln(a)

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Rubi [A]
time = 0.06, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6874, 2225} \begin {gather*} -\frac {3 a^{x (2 k+l)}}{\log (a) (2 k+l)}+\frac {3 a^{x (k+2 l)}}{\log (a) (k+2 l)}+\frac {a^{3 k x}}{3 k \log (a)}-\frac {a^{3 l x}}{3 l \log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^(k*x) - a^(l*x))^3,x]

[Out]

a^(3*k*x)/(3*k*Log[a]) - a^(3*l*x)/(3*l*Log[a]) - (3*a^((2*k + l)*x))/((2*k + l)*Log[a]) + (3*a^((k + 2*l)*x))
/((k + 2*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 )^3 \, dx &=\frac {\text {Subst}\left (\int \left (e^{k x}-e^{l x}\right )^3 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int \left (e^{3 k x}-e^{3 l x}-3 e^{(2 k+l) x}+3 e^{(k+2 l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int e^{3 k x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac {\text {Subst}\left (\int e^{3 l x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac {3 \text {Subst}\left (\int e^{(2 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {3 \text {Subst}\left (\int e^{(k+2 l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {a^{3 k x}}{3 k \log (a)}-\frac {a^{3 l x}}{3 l \log (a)}-\frac {3 a^{(2 k+l) x}}{(2 k+l) \log (a)}+\frac {3 a^{(k+2 l) x}}{(k+2 l) \log (a)}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 66, normalized size = 0.84 \begin {gather*} \frac {\frac {a^{3 k x}}{k}-\frac {a^{3 l x}}{l}-\frac {9 a^{(2 k+l) x}}{2 k+l}+\frac {9 a^{(k+2 l) x}}{k+2 l}}{3 \log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^(k*x) - a^(l*x))^3,x]

[Out]

(a^(3*k*x)/k - a^(3*l*x)/l - (9*a^((2*k + l)*x))/(2*k + l) + (9*a^((k + 2*l)*x))/(k + 2*l))/(3*Log[a])

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 5.34, size = 633, normalized size = 8.01 \begin {gather*} \text {Piecewise}\left [\left \{\left \{0,a\text {==}1\text {\&\&}a\text {==}1\text {$\vert $$\vert $}k\text {==}0\text {\&\&}a\text {==}1\text {$\vert $$\vert $}l\text {==}0\right \},\left \{\frac {l x \text {Log}\left [a\right ]-3 a^{l x}-\frac {a^{3 l x}}{3}+\frac {3 a^{2 l x}}{2}}{l \text {Log}\left [a\right ]},k\text {==}0\right \},\left \{\frac {\left (-\frac {1}{6}+3 l x \text {Log}\left [a\right ] a^{6 l x}+a^{3 l x}-\frac {a^{9 l x}}{3}\right ) a^{-6 l x}}{l \text {Log}\left [a\right ]},k\text {==}-2 l\right \},\left \{\frac {-2 a^{\frac {-3 l x}{2}}}{3 l \text {Log}\left [a\right ]}-\frac {a^{3 l x}}{3 l \text {Log}\left [a\right ]}+\frac {2 a^{\frac {3 l x}{2}}}{l \text {Log}\left [a\right ]}-3 x,l\text {==}-2 k\right \},\left \{\frac {-k x \text {Log}\left [a\right ]-\frac {3 a^{2 k x}}{2}+\frac {a^{3 k x}}{3}+3 a^{k x}}{k \text {Log}\left [a\right ]},l\text {==}0\right \}\right \},\frac {-2 k^3 a^{3 l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}-\frac {9 k^2 l a^{2 k x} a^{l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}-\frac {5 k^2 l a^{3 l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}+\frac {2 k^2 l a^{3 k x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}+\frac {18 k^2 l a^{k x} a^{2 l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}-\frac {18 k l^2 a^{2 k x} a^{l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}-\frac {2 k l^2 a^{3 l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}+\frac {5 k l^2 a^{3 k x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}+\frac {9 k l^2 a^{k x} a^{2 l x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}+\frac {2 l^3 a^{3 k x}}{6 k^3 l \text {Log}\left [a\right ]+15 k^2 l^2 \text {Log}\left [a\right ]+6 k l^3 \text {Log}\left [a\right ]}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[(a^(k*x) - a^(l*x))^3,x]')

[Out]

Piecewise[{{0, a == 1 && a == 1 || k == 0 && a == 1 || l == 0}, {(l x Log[a] - 3 a ^ (l x) - a ^ (3 l x) / 3 +
3 a ^ (2 l x) / 2) / (l Log[a]), k == 0}, {(-1 / 6 + 3 l x Log[a] a ^ (6 l x) + a ^ (3 l x) - a ^ (9 l x) / 3
) a ^ (-6 l x) / (l Log[a]), k == -2 l}, {-2 a ^ (-3 l x / 2) / (3 l Log[a]) - a ^ (3 l x) / (3 l Log[a]) + 2
a ^ (3 l x / 2) / (l Log[a]) - 3 x, l == -2 k}, {(-k x Log[a] - 3 a ^ (2 k x) / 2 + a ^ (3 k x) / 3 + 3 a ^ (k
x)) / (k Log[a]), l == 0}}, -2 k ^ 3 a ^ (3 l x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[
a]) - 9 k ^ 2 l a ^ (2 k x) a ^ (l x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) - 5 k ^
2 l a ^ (3 l x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) + 2 k ^ 2 l a ^ (3 k x) / (6 k
^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) + 18 k ^ 2 l a ^ (k x) a ^ (2 l x) / (6 k ^ 3 l Log[
a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) - 18 k l ^ 2 a ^ (2 k x) a ^ (l x) / (6 k ^ 3 l Log[a] + 15 k ^
2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) - 2 k l ^ 2 a ^ (3 l x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k
l ^ 3 Log[a]) + 5 k l ^ 2 a ^ (3 k x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) + 9 k l
^ 2 a ^ (k x) a ^ (2 l x) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a]) + 2 l ^ 3 a ^ (3 k x
) / (6 k ^ 3 l Log[a] + 15 k ^ 2 l ^ 2 Log[a] + 6 k l ^ 3 Log[a])]

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Maple [A]
time = 0.04, size = 84, normalized size = 1.06

method	result	size
risch	\(\frac {a^{3 k x}}{3 k \ln \left (a \right )}-\frac {a^{3 l x}}{3 l \ln \left (a \right )}+\frac {3 a^{k x} a^{2 l x}}{\ln \left (a \right ) \left (k +2 l \right )}-\frac {3 a^{2 k x} a^{l x}}{\ln \left (a \right ) \left (2 k +l \right )}\)	\(84\)
norman	\(\frac {{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}-\frac {{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}+\frac {3 \,{\mathrm e}^{k x \ln \left (a \right )} {\mathrm e}^{2 l x \ln \left (a \right )}}{\ln \left (a \right ) \left (k +2 l \right )}-\frac {3 \,{\mathrm e}^{2 k x \ln \left (a \right )} {\mathrm e}^{l x \ln \left (a \right )}}{\ln \left (a \right ) \left (2 k +l \right )}\)	\(90\)
meijerg	\(-\frac {1-{\mathrm e}^{3 k x \ln \left (a \right )}}{3 k \ln \left (a \right )}+\frac {3-3 \,{\mathrm e}^{x k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}}{k \ln \left (a \right ) \left (2+\frac {l}{k}\right )}-\frac {3 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {1}{1+\frac {k}{l}}\right )}+\frac {1-{\mathrm e}^{3 l x \ln \left (a \right )}}{3 l \ln \left (a \right )}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x)-a^(l*x))^3,x,method=_RETURNVERBOSE)

[Out]

1/3/k/ln(a)*(a^(k*x))^3-1/3/l/ln(a)*(a^(l*x))^3+3/ln(a)/(k+2*l)*a^(k*x)*(a^(l*x))^2-3/ln(a)/(2*k+l)*(a^(k*x))^
2*a^(l*x)

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Maxima [A]
time = 0.25, size = 77, normalized size = 0.97 \begin {gather*} -\frac {3 \, a^{2 \, k x + l x}}{{\left (2 \, k + l\right )} \log \left (a\right )} + \frac {3 \, a^{k x + 2 \, l x}}{{\left (k + 2 \, l\right )} \log \left (a\right )} + \frac {a^{3 \, k x}}{3 \, k \log \left (a\right )} - \frac {a^{3 \, l x}}{3 \, 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))^3,x, algorithm="maxima")

[Out]

-3*a^(2*k*x + l*x)/((2*k + l)*log(a)) + 3*a^(k*x + 2*l*x)/((k + 2*l)*log(a)) + 1/3*a^(3*k*x)/(k*log(a)) - 1/3*
a^(3*l*x)/(l*log(a))

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Fricas [A]
time = 0.31, size = 131, normalized size = 1.66 \begin {gather*} \frac {9 \, {\left (2 \, k^{2} l + k l^{2}\right )} a^{k x} a^{2 \, l x} - 9 \, {\left (k^{2} l + 2 \, k l^{2}\right )} a^{2 \, k x} a^{l x} + {\left (2 \, k^{2} l + 5 \, k l^{2} + 2 \, l^{3}\right )} a^{3 \, k x} - {\left (2 \, k^{3} + 5 \, k^{2} l + 2 \, k l^{2}\right )} a^{3 \, l x}}{3 \, {\left (2 \, k^{3} l + 5 \, k^{2} l^{2} + 2 \, k l^{3}\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))^3,x, algorithm="fricas")

[Out]

1/3*(9*(2*k^2*l + k*l^2)*a^(k*x)*a^(2*l*x) - 9*(k^2*l + 2*k*l^2)*a^(2*k*x)*a^(l*x) + (2*k^2*l + 5*k*l^2 + 2*l^
3)*a^(3*k*x) - (2*k^3 + 5*k^2*l + 2*k*l^2)*a^(3*l*x))/((2*k^3*l + 5*k^2*l^2 + 2*k*l^3)*log(a))

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Sympy [A]
time = 2.71, size = 663, normalized size = 8.39 \begin {gather*} \begin {cases} 0 & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\- \frac {a^{3 l x}}{3 l \log {\left (a \right )}} + \frac {3 a^{2 l x}}{2 l \log {\left (a \right )}} - \frac {3 a^{l x}}{l \log {\left (a \right )}} + x & \text {for}\: k = 0 \\- \frac {a^{3 l x}}{3 l \log {\left (a \right )}} + 3 x + \frac {a^{- 3 l x}}{l \log {\left (a \right )}} - \frac {a^{- 6 l x}}{6 l \log {\left (a \right )}} & \text {for}\: k = - 2 l \\\frac {2 a^{\frac {3 l x}{2}}}{l \log {\left (a \right )}} - \frac {a^{3 l x}}{3 l \log {\left (a \right )}} - 3 x - \frac {2 a^{- \frac {3 l x}{2}}}{3 l \log {\left (a \right )}} & \text {for}\: k = - \frac {l}{2} \\\frac {a^{3 k x}}{3 k \log {\left (a \right )}} - \frac {3 a^{2 k x}}{2 k \log {\left (a \right )}} + \frac {3 a^{k x}}{k \log {\left (a \right )}} - x & \text {for}\: l = 0 \\\frac {2 a^{3 k x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {5 a^{3 k x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {2 a^{3 k x} l^{3}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} - \frac {9 a^{2 k x} a^{l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} - \frac {18 a^{2 k x} a^{l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {18 a^{k x} a^{2 l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} + \frac {9 a^{k x} a^{2 l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} - \frac {2 a^{3 l x} k^{3}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} - \frac {5 a^{3 l x} k^{2} l}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \log {\left (a \right )}} - \frac {2 a^{3 l x} k l^{2}}{6 k^{3} l \log {\left (a \right )} + 15 k^{2} l^{2} \log {\left (a \right )} + 6 k l^{3} \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))**3,x)

[Out]

Piecewise((0, Eq(a, 1) & (Eq(a, 1) | Eq(k, 0)) & (Eq(a, 1) | Eq(l, 0))), (-a**(3*l*x)/(3*l*log(a)) + 3*a**(2*l
*x)/(2*l*log(a)) - 3*a**(l*x)/(l*log(a)) + x, Eq(k, 0)), (-a**(3*l*x)/(3*l*log(a)) + 3*x + 1/(a**(3*l*x)*l*log
(a)) - 1/(6*a**(6*l*x)*l*log(a)), Eq(k, -2*l)), (2*a**(3*l*x/2)/(l*log(a)) - a**(3*l*x)/(3*l*log(a)) - 3*x - 2
/(3*a**(3*l*x/2)*l*log(a)), Eq(k, -l/2)), (a**(3*k*x)/(3*k*log(a)) - 3*a**(2*k*x)/(2*k*log(a)) + 3*a**(k*x)/(k
*log(a)) - x, Eq(l, 0)), (2*a**(3*k*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 5*a*
*(3*k*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 2*a**(3*k*x)*l**3/(6*k**3*l*log(a)
+ 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) - 9*a**(2*k*x)*a**(l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a
) + 6*k*l**3*log(a)) - 18*a**(2*k*x)*a**(l*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a))
+ 18*a**(k*x)*a**(2*l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) + 9*a**(k*x)*a**(2*
l*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)) - 2*a**(3*l*x)*k**3/(6*k**3*l*log(a) + 1
5*k**2*l**2*log(a) + 6*k*l**3*log(a)) - 5*a**(3*l*x)*k**2*l/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*
log(a)) - 2*a**(3*l*x)*k*l**2/(6*k**3*l*log(a) + 15*k**2*l**2*log(a) + 6*k*l**3*log(a)), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.02, size = 1035, normalized size = 13.10

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)-a^(l*x))^3,x)

[Out]

2/3*(2*k*cos(-3/2*pi*k*x*sgn(a) + 3/2*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(-3/2*pi*k*x*sgn(a) + 3/2*pi*k*x)/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2))*abs(a)^
(3*k*x) - 2/3*(2*l*cos(-3/2*pi*l*x*sgn(a) + 3/2*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(-3/2*pi*l*x*sgn(a) + 3/2*pi*l*x)/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l)^2
))*abs(a)^(3*l*x) + 6*e^((k*log(abs(a)) + 2*l*log(abs(a)))*x)*(2*(k*log(abs(a)) + 2*l*log(abs(a)))*cos(-1/2*pi
*k*x*sgn(a) - pi*l*x*sgn(a) + 1/2*pi*k*x + pi*l*x)/((pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)^2 + 4*(k*log
(abs(a)) + 2*l*log(abs(a)))^2) - (pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)*sin(-1/2*pi*k*x*sgn(a) - pi*l*x
*sgn(a) + 1/2*pi*k*x + pi*l*x)/((pi*k*sgn(a) + 2*pi*l*sgn(a) - pi*k - 2*pi*l)^2 + 4*(k*log(abs(a)) + 2*l*log(a
bs(a)))^2)) - 6*e^((2*k*log(abs(a)) + l*log(abs(a)))*x)*(2*(2*k*log(abs(a)) + l*log(abs(a)))*cos(-pi*k*x*sgn(a
) - 1/2*pi*l*x*sgn(a) + pi*k*x + 1/2*pi*l*x)/((2*pi*k*sgn(a) + pi*l*sgn(a) - 2*pi*k - pi*l)^2 + 4*(2*k*log(abs
(a)) + l*log(abs(a)))^2) - (2*pi*k*sgn(a) + pi*l*sgn(a) - 2*pi*k - pi*l)*sin(-pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a
) + pi*k*x + 1/2*pi*l*x)/((2*pi*k*sgn(a) + pi*l*sgn(a) - 2*pi*k - pi*l)^2 + 4*(2*k*log(abs(a)) + l*log(abs(a))
)^2)) + I*abs(a)^(3*k*x)*(I*e^(3/2*I*pi*k*x*sgn(a) - 3/2*I*pi*k*x)/(3*I*pi*k*sgn(a) - 3*I*pi*k + 6*k*log(abs(a
))) - I*e^(-3/2*I*pi*k*x*sgn(a) + 3/2*I*pi*k*x)/(-3*I*pi*k*sgn(a) + 3*I*pi*k + 6*k*log(abs(a)))) + 3*e^((2*k*l
og(abs(a)) + l*log(abs(a)))*x)*(e^(I*pi*k*x*sgn(a) + 1/2*I*pi*l*x*sgn(a) - I*pi*k*x - 1/2*I*pi*l*x)/(2*I*pi*k*
sgn(a) + I*pi*l*sgn(a) - 2*I*pi*k - I*pi*l + 4*k*log(abs(a)) + 2*l*log(abs(a))) - e^(-I*pi*k*x*sgn(a) - 1/2*I*
pi*l*x*sgn(a) + I*pi*k*x + 1/2*I*pi*l*x)/(-2*I*pi*k*sgn(a) - I*pi*l*sgn(a) + 2*I*pi*k + I*pi*l + 4*k*log(abs(a
)) + 2*l*log(abs(a)))) - 3*e^((k*log(abs(a)) + 2*l*log(abs(a)))*x)*(e^(1/2*I*pi*k*x*sgn(a) + I*pi*l*x*sgn(a) -
1/2*I*pi*k*x - I*pi*l*x)/(I*pi*k*sgn(a) + 2*I*pi*l*sgn(a) - I*pi*k - 2*I*pi*l + 2*k*log(abs(a)) + 4*l*log(abs
(a))) - e^(-1/2*I*pi*k*x*sgn(a) - I*pi*l*x*sgn(a) + 1/2*I*pi*k*x + I*pi*l*x)/(-I*pi*k*sgn(a) - 2*I*pi*l*sgn(a)
+ I*pi*k + 2*I*pi*l + 2*k*log(abs(a)) + 4*l*log(abs(a)))) + I*abs(a)^(3*l*x)*(-I*e^(3/2*I*pi*l*x*sgn(a) - 3/2
*I*pi*l*x)/(3*I*pi*l*sgn(a) - 3*I*pi*l + 6*l*log(abs(a))) + I*e^(-3/2*I*pi*l*x*sgn(a) + 3/2*I*pi*l*x)/(-3*I*pi
*l*sgn(a) + 3*I*pi*l + 6*l*log(abs(a))))

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Mupad [B]
time = 0.36, size = 81, normalized size = 1.03 \begin {gather*} \frac {3\,a^{k\,x}\,a^{2\,l\,x}}{k\,\ln \left (a\right )+2\,l\,\ln \left (a\right )}-\frac {3\,a^{2\,k\,x}\,a^{l\,x}}{2\,k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {a^{3\,k\,x}}{3\,k\,\ln \left (a\right )}-\frac {a^{3\,l\,x}}{3\,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))^3,x)

[Out]

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

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