3.18.17 \(\int (-1+8 x+2^{2 \log ^2(x)} (3 x^2+4 x^2 \log (2) \log (x))) \, dx\)

Optimal. Leaf size=22 \[ -x+x^2 \left (4+2^{2 \log ^2(x)} x\right )+\log (3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 21, normalized size of antiderivative = 0.95, number of steps used = 2, number of rules used = 1, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2288} \begin {gather*} x^3 2^{2 \log ^2(x)}+4 x^2-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-1 + 8*x + 2^(2*Log[x]^2)*(3*x^2 + 4*x^2*Log[2]*Log[x]),x]

[Out]

-x + 4*x^2 + 2^(2*Log[x]^2)*x^3

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x+4 x^2+\int 2^{2 \log ^2(x)} \left (3 x^2+4 x^2 \log (2) \log (x)\right ) \, dx\\ &=-x+4 x^2+2^{2 \log ^2(x)} x^3\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.02, size = 29, normalized size = 1.32 \begin {gather*} -x+4 x^2+\frac {2^{-1+2 \log ^2(x)} x^3 \log (16)}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-1 + 8*x + 2^(2*Log[x]^2)*(3*x^2 + 4*x^2*Log[2]*Log[x]),x]

[Out]

-x + 4*x^2 + (2^(-1 + 2*Log[x]^2)*x^3*Log[16])/Log[4]

________________________________________________________________________________________

fricas [A]  time = 0.75, size = 21, normalized size = 0.95 \begin {gather*} 2^{2 \, \log \relax (x)^{2}} x^{3} + 4 \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(2)*log(x)+3*x^2)*exp(log(2)*log(x)^2)^2+8*x-1,x, algorithm="fricas")

[Out]

2^(2*log(x)^2)*x^3 + 4*x^2 - x

________________________________________________________________________________________

giac [A]  time = 0.19, size = 21, normalized size = 0.95 \begin {gather*} 2^{2 \, \log \relax (x)^{2}} x^{3} + 4 \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(2)*log(x)+3*x^2)*exp(log(2)*log(x)^2)^2+8*x-1,x, algorithm="giac")

[Out]

2^(2*log(x)^2)*x^3 + 4*x^2 - x

________________________________________________________________________________________

maple [A]  time = 0.15, size = 22, normalized size = 1.00

method	result	size
risch	\(-x +x^{3} 2^{2 \ln \relax (x )^{2}}+4 x^{2}\)	\(22\)
default	\(-x +x^{3} {\mathrm e}^{2 \ln \relax (2) \ln \relax (x )^{2}}+4 x^{2}\)	\(24\)
norman	\(-x +x^{3} {\mathrm e}^{2 \ln \relax (2) \ln \relax (x )^{2}}+4 x^{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2*ln(2)*ln(x)+3*x^2)*exp(ln(2)*ln(x)^2)^2+8*x-1,x,method=_RETURNVERBOSE)

[Out]

-x+x^3*(2^(ln(x)^2))^2+4*x^2

________________________________________________________________________________________

maxima [A]  time = 0.64, size = 21, normalized size = 0.95 \begin {gather*} 2^{2 \, \log \relax (x)^{2}} x^{3} + 4 \, x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^2*log(2)*log(x)+3*x^2)*exp(log(2)*log(x)^2)^2+8*x-1,x, algorithm="maxima")

[Out]

2^(2*log(x)^2)*x^3 + 4*x^2 - x

________________________________________________________________________________________

mupad [B]  time = 1.11, size = 21, normalized size = 0.95 \begin {gather*} 2^{2\,{\ln \relax (x)}^2}\,x^3-x+4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x + exp(2*log(2)*log(x)^2)*(3*x^2 + 4*x^2*log(2)*log(x)) - 1,x)

[Out]

2^(2*log(x)^2)*x^3 - x + 4*x^2

________________________________________________________________________________________

sympy [A]  time = 0.30, size = 20, normalized size = 0.91 \begin {gather*} x^{3} e^{2 \log {\relax (2 )} \log {\relax (x )}^{2}} + 4 x^{2} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**2*ln(2)*ln(x)+3*x**2)*exp(ln(2)*ln(x)**2)**2+8*x-1,x)

[Out]

x**3*exp(2*log(2)*log(x)**2) + 4*x**2 - x

________________________________________________________________________________________