3.6.3 \(\int (a^{k x}+a^{l x})^2 \, dx\) [503]

Optimal. Leaf size=53 \[ \frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)}+\frac {2 a^{(k+l) x}}{(k+l) \log (a)} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {2 a^{x (k+l)}}{\log (a) (k+l)}+\frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a^(k*x) + a^(l*x))^2,x]

[Out]

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

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Mathematica [A]
time = 0.05, size = 55, normalized size = 1.04 \begin {gather*} \frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)}+\frac {2 a^{k x+l x}}{(k+l) \log (a)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a^(k*x) + a^(l*x))^2,x]

[Out]

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

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Maple [A]
time = 0.04, size = 55, normalized size = 1.04

method	result	size
risch	\(\frac {a^{2 k x}}{2 k \ln \left (a \right )}+\frac {a^{2 l x}}{2 l \ln \left (a \right )}+\frac {2 a^{k x} a^{l x}}{\ln \left (a \right ) \left (k +l \right )}\)	\(55\)
norman	\(\frac {{\mathrm e}^{2 k x \ln \left (a \right )}}{2 k \ln \left (a \right )}+\frac {{\mathrm e}^{2 l x \ln \left (a \right )}}{2 l \ln \left (a \right )}+\frac {2 \,{\mathrm e}^{k x \ln \left (a \right )} {\mathrm e}^{l x \ln \left (a \right )}}{\ln \left (a \right ) \left (k +l \right )}\)	\(59\)
meijerg	\(-\frac {1-{\mathrm e}^{2 k x \ln \left (a \right )}}{2 k \ln \left (a \right )}-\frac {2 \left (1-{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}-\frac {1-{\mathrm e}^{2 l x \ln \left (a \right )}}{2 l \ln \left (a \right )}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x)+a^(l*x))^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 1.16, size = 51, normalized size = 0.96 \begin {gather*} \frac {2 \, a^{k x + l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac {a^{2 \, k x}}{2 \, k \log \left (a\right )} + \frac {a^{2 \, l x}}{2 \, 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))^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.53, size = 62, normalized size = 1.17 \begin {gather*} \frac {4 \, a^{k x} a^{l x} k l + {\left (k l + l^{2}\right )} a^{2 \, k x} + {\left (k^{2} + k l\right )} a^{2 \, l x}}{2 \, {\left (k^{2} l + k l^{2}\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))^2,x, algorithm="fricas")

[Out]

1/2*(4*a^(k*x)*a^(l*x)*k*l + (k*l + l^2)*a^(2*k*x) + (k^2 + k*l)*a^(2*l*x))/((k^2*l + k*l^2)*log(a))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (41) = 82\).
time = 0.47, size = 250, normalized size = 4.72 \begin {gather*} \begin {cases} 4 x & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{2 l x}}{2 l \log {\left (a \right )}} + \frac {2 a^{l x}}{l \log {\left (a \right )}} + x & \text {for}\: k = 0 \\\frac {a^{2 l x}}{2 l \log {\left (a \right )}} + 2 x - \frac {a^{- 2 l x}}{2 l \log {\left (a \right )}} & \text {for}\: k = - l \\\frac {a^{2 k x}}{2 k \log {\left (a \right )}} + \frac {2 a^{k x}}{k \log {\left (a \right )}} + x & \text {for}\: l = 0 \\\frac {a^{2 k x} k l}{2 k^{2} l \log {\left (a \right )} + 2 k l^{2} \log {\left (a \right )}} + \frac {a^{2 k x} l^{2}}{2 k^{2} l \log {\left (a \right )} + 2 k l^{2} \log {\left (a \right )}} + \frac {4 a^{k x} a^{l x} k l}{2 k^{2} l \log {\left (a \right )} + 2 k l^{2} \log {\left (a \right )}} + \frac {a^{2 l x} k^{2}}{2 k^{2} l \log {\left (a \right )} + 2 k l^{2} \log {\left (a \right )}} + \frac {a^{2 l x} k l}{2 k^{2} l \log {\left (a \right )} + 2 k l^{2} \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))**2,x)

[Out]

Piecewise((4*x, Eq(a, 1) & (Eq(a, 1) | Eq(k, 0)) & (Eq(a, 1) | Eq(l, 0))), (a**(2*l*x)/(2*l*log(a)) + 2*a**(l*
x)/(l*log(a)) + x, Eq(k, 0)), (a**(2*l*x)/(2*l*log(a)) + 2*x - 1/(2*a**(2*l*x)*l*log(a)), Eq(k, -l)), (a**(2*k
*x)/(2*k*log(a)) + 2*a**(k*x)/(k*log(a)) + x, Eq(l, 0)), (a**(2*k*x)*k*l/(2*k**2*l*log(a) + 2*k*l**2*log(a)) +
a**(2*k*x)*l**2/(2*k**2*l*log(a) + 2*k*l**2*log(a)) + 4*a**(k*x)*a**(l*x)*k*l/(2*k**2*l*log(a) + 2*k*l**2*log
(a)) + a**(2*l*x)*k**2/(2*k**2*l*log(a) + 2*k*l**2*log(a)) + a**(2*l*x)*k*l/(2*k**2*l*log(a) + 2*k*l**2*log(a)
), True))

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Giac [C] Result contains complex when optimal does not.
time = 0.84, size = 691, normalized size = 13.04 \begin {gather*} {\left (\frac {2 \, k \cos \left (-\pi k x \mathrm {sgn}\left (a\right ) + \pi k x\right ) \log \left ({\left | a \right |}\right )}{4 \, k^{2} \log \left ({\left | a \right |}\right )^{2} + {\left (\pi k \mathrm {sgn}\left (a\right ) - \pi k\right )}^{2}} - \frac {{\left (\pi k \mathrm {sgn}\left (a\right ) - \pi k\right )} \sin \left (-\pi k x \mathrm {sgn}\left (a\right ) + \pi k x\right )}{4 \, k^{2} \log \left ({\left | a \right |}\right )^{2} + {\left (\pi k \mathrm {sgn}\left (a\right ) - \pi k\right )}^{2}}\right )} {\left | a \right |}^{2 \, k x} + {\left (\frac {2 \, l \cos \left (-\pi l x \mathrm {sgn}\left (a\right ) + \pi l x\right ) \log \left ({\left | a \right |}\right )}{4 \, l^{2} \log \left ({\left | a \right |}\right )^{2} + {\left (\pi l \mathrm {sgn}\left (a\right ) - \pi l\right )}^{2}} - \frac {{\left (\pi l \mathrm {sgn}\left (a\right ) - \pi l\right )} \sin \left (-\pi l x \mathrm {sgn}\left (a\right ) + \pi l x\right )}{4 \, l^{2} \log \left ({\left | a \right |}\right )^{2} + {\left (\pi l \mathrm {sgn}\left (a\right ) - \pi l\right )}^{2}}\right )} {\left | a \right |}^{2 \, l x} - \frac {1}{2} i \, {\left | a \right |}^{2 \, k x} {\left (-\frac {i \, e^{\left (i \, \pi k x \mathrm {sgn}\left (a\right ) - i \, \pi k x\right )}}{i \, \pi k \mathrm {sgn}\left (a\right ) - i \, \pi k + 2 \, k \log \left ({\left | a \right |}\right )} + \frac {i \, e^{\left (-i \, \pi k x \mathrm {sgn}\left (a\right ) + i \, \pi k x\right )}}{-i \, \pi k \mathrm {sgn}\left (a\right ) + i \, \pi k + 2 \, k \log \left ({\left | a \right |}\right )}\right )} - \frac {1}{2} i \, {\left | a \right |}^{2 \, l x} {\left (-\frac {i \, e^{\left (i \, \pi l x \mathrm {sgn}\left (a\right ) - i \, \pi l x\right )}}{i \, \pi l \mathrm {sgn}\left (a\right ) - i \, \pi l + 2 \, l \log \left ({\left | a \right |}\right )} + \frac {i \, e^{\left (-i \, \pi l x \mathrm {sgn}\left (a\right ) + i \, \pi l x\right )}}{-i \, \pi l \mathrm {sgn}\left (a\right ) + i \, \pi l + 2 \, l \log \left ({\left | a \right |}\right )}\right )} + 4 \, {\left (\frac {2 \, {\left (k \log \left ({\left | a \right |}\right ) + l \log \left ({\left | a \right |}\right )\right )} \cos \left (-\frac {1}{2} \, \pi k x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi l x \mathrm {sgn}\left (a\right ) + \frac {1}{2} \, \pi k x + \frac {1}{2} \, \pi l x\right )}{{\left (\pi k \mathrm {sgn}\left (a\right ) + \pi l \mathrm {sgn}\left (a\right ) - \pi k - \pi l\right )}^{2} + 4 \, {\left (k \log \left ({\left | a \right |}\right ) + l \log \left ({\left | a \right |}\right )\right )}^{2}} - \frac {{\left (\pi k \mathrm {sgn}\left (a\right ) + \pi l \mathrm {sgn}\left (a\right ) - \pi k - \pi l\right )} \sin \left (-\frac {1}{2} \, \pi k x \mathrm {sgn}\left (a\right ) - \frac {1}{2} \, \pi l x \mathrm {sgn}\left (a\right ) + \frac {1}{2} \, \pi k x + \frac {1}{2} \, \pi l x\right )}{{\left (\pi k \mathrm {sgn}\left (a\right ) + \pi l \mathrm {sgn}\left (a\right ) - \pi k - \pi l\right )}^{2} + 4 \, {\left (k \log \left ({\left | a \right |}\right ) + l \log \left ({\left | a \right |}\right )\right )}^{2}}\right )} e^{\left ({\left (k \log \left ({\left | a \right |}\right ) + l \log \left ({\left | a \right |}\right )\right )} x\right )} + 2 i \, {\left (\frac {i \, e^{\left (\frac {1}{2} i \, \pi k x \mathrm {sgn}\left (a\right ) + \frac {1}{2} i \, \pi l x \mathrm {sgn}\left (a\right ) - \frac {1}{2} i \, \pi k x - \frac {1}{2} i \, \pi l x\right )}}{i \, \pi k \mathrm {sgn}\left (a\right ) + i \, \pi l \mathrm {sgn}\left (a\right ) - i \, \pi k - i \, \pi l + 2 \, k \log \left ({\left | a \right |}\right ) + 2 \, l \log \left ({\left | a \right |}\right )} - \frac {i \, e^{\left (-\frac {1}{2} i \, \pi k x \mathrm {sgn}\left (a\right ) - \frac {1}{2} i \, \pi l x \mathrm {sgn}\left (a\right ) + \frac {1}{2} i \, \pi k x + \frac {1}{2} i \, \pi l x\right )}}{-i \, \pi k \mathrm {sgn}\left (a\right ) - i \, \pi l \mathrm {sgn}\left (a\right ) + i \, \pi k + i \, \pi l + 2 \, k \log \left ({\left | a \right |}\right ) + 2 \, l \log \left ({\left | a \right |}\right )}\right )} e^{\left ({\left (k \log \left ({\left | a \right |}\right ) + l \log \left ({\left | a \right |}\right )\right )} x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^(k*x)+a^(l*x))^2,x, algorithm="giac")

[Out]

(2*k*cos(-pi*k*x*sgn(a) + 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(-pi*k*x*sgn(a) + pi*k*x)/(4*k^2*log(abs(a))^2 + (pi*k*sgn(a) - pi*k)^2))*abs(a)^(2*k*x) + (2*l*cos(-
pi*l*x*sgn(a) + 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(
-pi*l*x*sgn(a) + pi*l*x)/(4*l^2*log(abs(a))^2 + (pi*l*sgn(a) - pi*l)^2))*abs(a)^(2*l*x) - 1/2*I*abs(a)^(2*k*x)
*(-I*e^(I*pi*k*x*sgn(a) - I*pi*k*x)/(I*pi*k*sgn(a) - I*pi*k + 2*k*log(abs(a))) + I*e^(-I*pi*k*x*sgn(a) + I*pi*
k*x)/(-I*pi*k*sgn(a) + I*pi*k + 2*k*log(abs(a)))) - 1/2*I*abs(a)^(2*l*x)*(-I*e^(I*pi*l*x*sgn(a) - I*pi*l*x)/(I
*pi*l*sgn(a) - I*pi*l + 2*l*log(abs(a))) + I*e^(-I*pi*l*x*sgn(a) + I*pi*l*x)/(-I*pi*l*sgn(a) + I*pi*l + 2*l*lo
g(abs(a)))) + 4*(2*(k*log(abs(a)) + l*log(abs(a)))*cos(-1/2*pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a) + 1/2*pi*k*x + 1
/2*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(-1/2*pi*k*x*sgn(a) - 1/2*pi*l*x*sgn(a) + 1/2*pi*k*x + 1/2*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^((k*log(abs(a)) + l*log(abs(a)))*x
) + 2*I*(I*e^(1/2*I*pi*k*x*sgn(a) + 1/2*I*pi*l*x*sgn(a) - 1/2*I*pi*k*x - 1/2*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^(-1/2*I*pi*k*x*sgn(a) - 1/2*I*pi*l*x*sgn(
a) + 1/2*I*pi*k*x + 1/2*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*lo
g(abs(a))))*e^((k*log(abs(a)) + l*log(abs(a)))*x)

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Mupad [B]
time = 0.38, size = 68, normalized size = 1.28 \begin {gather*} \frac {a^{2\,k\,x}}{2\,k\,\ln \left (a\right )}+\frac {\frac {a^{2\,l\,x}\,k^2}{2}+l\,\left (2\,a^{k\,x+l\,x}\,k+\frac {a^{2\,l\,x}\,k}{2}\right )}{k\,l\,\ln \left (a\right )\,\left (k+l\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^(k*x) + a^(l*x))^2,x)

[Out]

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

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