3.47.63 \(\int \frac {32 x^3-16 x^5+2 x^7+(16 x^2-4 x^4) \log (4)+2 x \log ^2(4)+e^x (4 x^2+2 x^3-x^4+(1+x) \log (4))}{16 x^2-8 x^4+x^6+(8 x-2 x^3) \log (4)+\log ^2(4)} \, dx\)

Optimal. Leaf size=26 \[ -80+x^2+\frac {e^x x}{-x-x \left (-5+x^2\right )+\log (4)} \]

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Rubi [A]  time = 0.37, antiderivative size = 36, normalized size of antiderivative = 1.38, number of steps used = 3, number of rules used = 2, integrand size = 96, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6688, 2288} \begin {gather*} x^2+\frac {e^x \left (-x^4+4 x^2+x \log (4)\right )}{\left (-x^3+4 x+\log (4)\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(32*x^3 - 16*x^5 + 2*x^7 + (16*x^2 - 4*x^4)*Log[4] + 2*x*Log[4]^2 + E^x*(4*x^2 + 2*x^3 - x^4 + (1 + x)*Log
[4]))/(16*x^2 - 8*x^4 + x^6 + (8*x - 2*x^3)*Log[4] + Log[4]^2),x]

[Out]

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

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]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 21, normalized size = 0.81 \begin {gather*} x \left (x+\frac {e^x}{4 x-x^3+\log (4)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(32*x^3 - 16*x^5 + 2*x^7 + (16*x^2 - 4*x^4)*Log[4] + 2*x*Log[4]^2 + E^x*(4*x^2 + 2*x^3 - x^4 + (1 +
x)*Log[4]))/(16*x^2 - 8*x^4 + x^6 + (8*x - 2*x^3)*Log[4] + Log[4]^2),x]

[Out]

x*(x + E^x/(4*x - x^3 + Log[4]))

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fricas [A]  time = 0.54, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \relax (2) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/
(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="fricas")

[Out]

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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giac [A]  time = 0.21, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \relax (2) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/
(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="giac")

[Out]

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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maple [A]  time = 0.24, size = 41, normalized size = 1.58

method	result	size
norman	\(\frac {16 x +{\mathrm e}^{x} x -x^{5}+2 x^{2} \ln \relax (2)+8 \ln \relax (2)}{-x^{3}+2 \ln \relax (2)+4 x}\)	\(41\)
default	\(x^{2}-\ln \relax (2) \left (\munderset {\textit {\_R2} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \textit {\_R2}^{2} \ln \relax (2)-9 \ln \relax (2) \textit {\_R2} -24 \ln \relax (2)-16 \textit {\_R2} +32\right ) {\mathrm e}^{\textit {\_R2}} \expIntegralEi \left (1, -x +\textit {\_R2} \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (3 \textit {\_R2}^{2}-4\right )}\right )-\ln \relax (2) \left (\munderset {\textit {\_R2} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \ln \relax (2) \textit {\_R2} -12 \textit {\_R2}^{2}-18 \ln \relax (2)+12 \textit {\_R2} +32\right ) {\mathrm e}^{\textit {\_R2}} \expIntegralEi \left (1, -x +\textit {\_R2} \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (3 \textit {\_R2}^{2}-4\right )}\right )+\frac {8 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (45 \ln \relax (2)^{3} \textit {\_R} +108 \textit {\_R}^{2} \ln \relax (2)^{2}+8 \ln \relax (2)^{2}-112 \ln \relax (2) \textit {\_R} -256 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}+\frac {32 \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (-27 \textit {\_R}^{2} \ln \relax (2)^{2}-12 \ln \relax (2)^{2}+8 \ln \relax (2) \textit {\_R} +64 \textit {\_R}^{2}\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}-\frac {8 \left (-\frac {\ln \relax (2) \left (9 \ln \relax (2)^{2}-16\right ) x^{2}}{4 \left (27 \ln \relax (2)^{2}-64\right )}-\frac {2 \left (13 \ln \relax (2)^{2}-32\right ) x}{27 \ln \relax (2)^{2}-64}-\frac {4 \ln \relax (2) \left (3 \ln \relax (2)^{2}-8\right )}{27 \ln \relax (2)^{2}-64}\right )}{-\frac {x^{3}}{2}+\ln \relax (2)+2 x}-\frac {16 \left (\frac {4 \ln \relax (2) x^{2}}{27 \ln \relax (2)^{2}-64}+\frac {8 \left (3 \ln \relax (2)^{2}-8\right ) x}{27 \ln \relax (2)^{2}-64}+\frac {\ln \relax (2) \left (9 \ln \relax (2)^{2}-32\right )}{27 \ln \relax (2)^{2}-64}\right )}{-\frac {x^{3}}{2}+\ln \relax (2)+2 x}-\frac {24 \ln \relax (2)^{2} {\mathrm e}^{x}}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}+\frac {32 \ln \relax (2) {\mathrm e}^{x}}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}-\frac {{\mathrm e}^{x} \left (9 x^{2} \ln \relax (2)^{2}+12 \ln \relax (2)^{2}+8 x \ln \relax (2)-32 x^{2}\right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}+\frac {2 \,{\mathrm e}^{x} \left (9 x \ln \relax (2)^{2}+6 x^{2} \ln \relax (2)-16 \ln \relax (2)-32 x \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}+\frac {4 \,{\mathrm e}^{x} \left (9 \ln \relax (2)^{2}+6 x \ln \relax (2)-8 x^{2}\right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}-\frac {18 \ln \relax (2)^{3} x^{2}}{\left (-\frac {x^{3}}{2}+\ln \relax (2)+2 x \right ) \left (27 \ln \relax (2)^{2}-64\right )}+\frac {32 \ln \relax (2)^{2} x}{\left (-\frac {x^{3}}{2}+\ln \relax (2)+2 x \right ) \left (27 \ln \relax (2)^{2}-64\right )}+\frac {48 \ln \relax (2)^{3}}{\left (-\frac {x^{3}}{2}+\ln \relax (2)+2 x \right ) \left (27 \ln \relax (2)^{2}-64\right )}+\frac {9 \ln \relax (2)^{2} {\mathrm e}^{x} x^{2}}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}-\frac {16 \,{\mathrm e}^{x} \ln \relax (2) x}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}+\left (\munderset {\textit {\_R2} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \textit {\_R2}^{2} \ln \relax (2)^{2}+18 \ln \relax (2)^{2} \textit {\_R2} +12 \ln \relax (2)^{2}+8 \ln \relax (2) \textit {\_R2} -32 \textit {\_R2}^{2}-16 \ln \relax (2)-32 \textit {\_R2} \right ) {\mathrm e}^{\textit {\_R2}} \expIntegralEi \left (1, -x +\textit {\_R2} \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (3 \textit {\_R2}^{2}-4\right )}\right )-4 \left (\munderset {\textit {\_R2} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \ln \relax (2)^{2}+6 \ln \relax (2) \textit {\_R2} -8 \textit {\_R2}^{2}-12 \ln \relax (2)+8 \textit {\_R2} \right ) {\mathrm e}^{\textit {\_R2}} \expIntegralEi \left (1, -x +\textit {\_R2} \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (3 \textit {\_R2}^{2}-4\right )}\right )-2 \left (\munderset {\textit {\_R1} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \ln \relax (2)^{2} \textit {\_R1} +6 \textit {\_R1}^{2} \ln \relax (2)+9 \ln \relax (2)^{2}-6 \ln \relax (2) \textit {\_R1} -16 \ln \relax (2)-32 \textit {\_R1} \right ) {\mathrm e}^{\textit {\_R1}} \expIntegralEi \left (1, -x +\textit {\_R1} \right )}{\left (27 \ln \relax (2)^{2}-64\right ) \left (3 \textit {\_R1}^{2}-4\right )}\right )-\frac {8 \ln \relax (2)^{2} \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (9 \ln \relax (2) \textit {\_R} -32\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}+\frac {32 \ln \relax (2) \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (-9 \ln \relax (2)^{2} \textit {\_R} +8 \ln \relax (2)+16 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}-\frac {192 \ln \relax (2) \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (-3 \ln \relax (2)+2 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}-\frac {256 \ln \relax (2) \left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{3}-2 \ln \relax (2)-4 \textit {\_Z} \right )}{\sum }\frac {\left (3 \ln \relax (2)-2 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-4}\right )}{54 \ln \relax (2)^{2}-128}+\frac {\frac {96 \ln \relax (2) x^{2}}{27 \ln \relax (2)^{2}-64}+\frac {32 \left (9 \ln \relax (2)^{2}-32\right ) x}{54 \ln \relax (2)^{2}-128}-\frac {256 \ln \relax (2)}{27 \ln \relax (2)^{2}-64}}{-\frac {x^{3}}{2}+\ln \relax (2)+2 x}+\frac {9 \,{\mathrm e}^{x} \ln \relax (2)^{2} x}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}-\frac {12 \ln \relax (2) {\mathrm e}^{x} x^{2}}{\left (27 \ln \relax (2)^{2}-64\right ) \left (-x^{3}+2 \ln \relax (2)+4 x \right )}\)	\(1409\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(2)*(x+1)-x^4+2*x^3+4*x^2)*exp(x)+8*x*ln(2)^2+2*(-4*x^4+16*x^2)*ln(2)+2*x^7-16*x^5+32*x^3)/(4*ln(2)^
2+2*(-2*x^3+8*x)*ln(2)+x^6-8*x^4+16*x^2),x,method=_RETURNVERBOSE)

[Out]

(16*x+exp(x)*x-x^5+2*x^2*ln(2)+8*ln(2))/(-x^3+2*ln(2)+4*x)

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maxima [A]  time = 0.45, size = 35, normalized size = 1.35 \begin {gather*} \frac {x^{5} - 4 \, x^{3} - 2 \, x^{2} \log \relax (2) - x e^{x}}{x^{3} - 4 \, x - 2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)*(x+1)-x^4+2*x^3+4*x^2)*exp(x)+8*x*log(2)^2+2*(-4*x^4+16*x^2)*log(2)+2*x^7-16*x^5+32*x^3)/
(4*log(2)^2+2*(-2*x^3+8*x)*log(2)+x^6-8*x^4+16*x^2),x, algorithm="maxima")

[Out]

(x^5 - 4*x^3 - 2*x^2*log(2) - x*e^x)/(x^3 - 4*x - 2*log(2))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,\ln \relax (2)\,\left (16\,x^2-4\,x^4\right )+8\,x\,{\ln \relax (2)}^2+32\,x^3-16\,x^5+2\,x^7+{\mathrm {e}}^x\,\left (2\,\ln \relax (2)\,\left (x+1\right )+4\,x^2+2\,x^3-x^4\right )}{2\,\ln \relax (2)\,\left (8\,x-2\,x^3\right )+4\,{\ln \relax (2)}^2+16\,x^2-8\,x^4+x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(16*x^2 - 4*x^4) + 8*x*log(2)^2 + 32*x^3 - 16*x^5 + 2*x^7 + exp(x)*(2*log(2)*(x + 1) + 4*x^2 + 2
*x^3 - x^4))/(2*log(2)*(8*x - 2*x^3) + 4*log(2)^2 + 16*x^2 - 8*x^4 + x^6),x)

[Out]

int((2*log(2)*(16*x^2 - 4*x^4) + 8*x*log(2)^2 + 32*x^3 - 16*x^5 + 2*x^7 + exp(x)*(2*log(2)*(x + 1) + 4*x^2 + 2
*x^3 - x^4))/(2*log(2)*(8*x - 2*x^3) + 4*log(2)^2 + 16*x^2 - 8*x^4 + x^6), x)

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sympy [A]  time = 0.18, size = 19, normalized size = 0.73 \begin {gather*} x^{2} - \frac {x e^{x}}{x^{3} - 4 x - 2 \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(2)*(x+1)-x**4+2*x**3+4*x**2)*exp(x)+8*x*ln(2)**2+2*(-4*x**4+16*x**2)*ln(2)+2*x**7-16*x**5+32*
x**3)/(4*ln(2)**2+2*(-2*x**3+8*x)*ln(2)+x**6-8*x**4+16*x**2),x)

[Out]

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

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