Optimal. Leaf size=53 \[ \frac {a^{2 k x}}{2 k \log (a)}+\frac {a^{2 l x}}{2 l \log (a)}-\frac {2 a^{(k+l) x}}{(k+l) \log (a)} \]
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Rubi [A]
time = 0.05, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps
used = 6, 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 {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.
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Rule 2225
Rule 6874
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.
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Maple [A]
time = 0.03, 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 | \(-\frac {1-{\mathrm e}^{2 k x \ln \left (a \right )}}{2 k \ln \left (a \right )}+\frac {2-2 \,{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}-\frac {1-{\mathrm e}^{2 l x \ln \left (a \right )}}{2 l \ln \left (a \right )}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.09, 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.
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Fricas [A]
time = 0.71, size = 64, normalized size = 1.21 \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.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs.
\(2 (41) = 82\).
time = 0.47, size = 248, normalized size = 4.68 \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^{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.
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Giac [C] Result contains complex when optimal does not.
time = 1.07, 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.
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Mupad [B]
time = 0.36, size = 69, normalized size = 1.30 \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.
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