3.2.65 \(\int k^{x^2} x \, dx\) [165]

Optimal. Leaf size=13 \[ \frac {k^{x^2}}{2 \log (k)} \]

[Out]

1/2*k^(x^2)/ln(k)

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Rubi [A]
time = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2240} \begin {gather*} \frac {k^{x^2}}{2 \log (k)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[k^x^2*x,x]

[Out]

k^x^2/(2*Log[k])

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin {align*} \int k^{x^2} x \, dx &=\frac {k^{x^2}}{2 \log (k)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {k^{x^2}}{2 \log (k)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[k^x^2*x,x]

[Out]

k^x^2/(2*Log[k])

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Maple [A]
time = 0.02, size = 12, normalized size = 0.92

method	result	size
gosper	\(\frac {k^{x^{2}}}{2 \ln \left (k \right )}\)	\(12\)
derivativedivides	\(\frac {k^{x^{2}}}{2 \ln \left (k \right )}\)	\(12\)
default	\(\frac {k^{x^{2}}}{2 \ln \left (k \right )}\)	\(12\)
risch	\(\frac {k^{x^{2}}}{2 \ln \left (k \right )}\)	\(12\)
norman	\(\frac {{\mathrm e}^{\ln \left (k \right ) x^{2}}}{2 \ln \left (k \right )}\)	\(14\)
meijerg	\(-\frac {1-{\mathrm e}^{\ln \left (k \right ) x^{2}}}{2 \ln \left (k \right )}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*k^(x^2)/ln(k)

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Maxima [A]
time = 1.84, size = 11, normalized size = 0.85 \begin {gather*} \frac {k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k^(x^2)*x,x, algorithm="maxima")

[Out]

1/2*k^(x^2)/log(k)

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Fricas [A]
time = 0.59, size = 11, normalized size = 0.85 \begin {gather*} \frac {k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k^(x^2)*x,x, algorithm="fricas")

[Out]

1/2*k^(x^2)/log(k)

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Sympy [A]
time = 0.03, size = 15, normalized size = 1.15 \begin {gather*} \begin {cases} \frac {k^{x^{2}}}{2 \log {\left (k \right )}} & \text {for}\: \log {\left (k \right )} \neq 0 \\\frac {x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(k**(x**2)*x,x)

[Out]

Piecewise((k**(x**2)/(2*log(k)), Ne(log(k), 0)), (x**2/2, True))

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Giac [A]
time = 0.75, size = 11, normalized size = 0.85 \begin {gather*} \frac {k^{\left (x^{2}\right )}}{2 \, \log \left (k\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*k^(x^2)/log(k)

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Mupad [B]
time = 0.03, size = 11, normalized size = 0.85 \begin {gather*} \frac {k^{x^2}}{2\,\ln \left (k\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(k^(x^2)*x,x)

[Out]

k^(x^2)/(2*log(k))

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