3.33.26 $\int \frac{-20+22{\log }^2(4)+(4-2{\log }^2(4))\log (4+\log (5))}{-2+{\log }^2(4)}dx$

Optimal. Leaf size=30 $$x(4+2(3(3-\frac{4}{2-{\log }^2(4)})-\log (4+\log (5))))$$

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Rubi [A]  time = 0.02, antiderivative size = 36, normalized size of antiderivative = 1.20, number of steps used = 1, number of rules used = 1, integrand size = 31, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.032, Rules used = {8} $$\begin{array}{c}\frac{2x(10-11{\log }^2(4)-(2-{\log }^2(4))\log (4+\log (5)))}{2-{\log }^2(4)}\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-20 + 22*Log[4]^2 + (4 - 2*Log[4]^2)*Log[4 + Log[5]])/(-2 + Log[4]^2),x]

[Out]

(2*x*(10 - 11*Log[4]^2 - (2 - Log[4]^2)*Log[4 + Log[5]]))/(2 - Log[4]^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\frac{2x(10-11{\log }^2(4)-(2-{\log }^2(4))\log (4+\log (5)))}{2-{\log }^2(4)}\end{array}\end{array}$$

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Mathematica [A]  time = 0.00, size = 50, normalized size = 1.67 $$\begin{array}{c}-\frac{20x}{-2+{\log }^2(4)}+\frac{22x{\log }^2(4)}{-2+{\log }^2(4)}+\frac{x(4-2{\log }^2(4))\log (4+\log (5))}{-2+{\log }^2(4)}\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 22*Log[4]^2 + (4 - 2*Log[4]^2)*Log[4 + Log[5]])/(-2 + Log[4]^2),x]

[Out]

(-20*x)/(-2 + Log[4]^2) + (22*x*Log[4]^2)/(-2 + Log[4]^2) + (x*(4 - 2*Log[4]^2)*Log[4 + Log[5]])/(-2 + Log[4]^
2)

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fricas [A]  time = 0.70, size = 41, normalized size = 1.37 $$\begin{array}{c}\frac{2(22x\log (2{)}^2-(2x\log (2{)}^2-x)\log (\log (5)+4)-5x)}{2\log (2{)}^2-1}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*log(2)^2+4)*log(log(5)+4)+88*log(2)^2-20)/(4*log(2)^2-2),x, algorithm="fricas")

[Out]

2*(22*x*log(2)^2 - (2*x*log(2)^2 - x)*log(log(5) + 4) - 5*x)/(2*log(2)^2 - 1)

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giac [A]  time = 0.16, size = 36, normalized size = 1.20 $$\begin{array}{c}\frac{2(22\log (2{)}^2-(2\log (2{)}^2-1)\log (\log (5)+4)-5)x}{2\log (2{)}^2-1}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*log(2)^2+4)*log(log(5)+4)+88*log(2)^2-20)/(4*log(2)^2-2),x, algorithm="giac")

[Out]

2*(22*log(2)^2 - (2*log(2)^2 - 1)*log(log(5) + 4) - 5)*x/(2*log(2)^2 - 1)

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maple [A]  time = 0.02, size = 35, normalized size = 1.17

method	result	size
default	$\frac{((-8\ln (2{)}^2+4)\ln (\ln (5)+4)+88\ln (2{)}^2-20)x}{4\ln (2{)}^2-2}$	$35$
norman	$-\frac{2(2\ln (2{)}^2\ln (\ln (5)+4)-22\ln (2{)}^2-\ln (\ln (5)+4)+5)x}{2\ln (2{)}^2-1}$	$40$
risch	$-\frac{8x\ln (2{)}^2\ln (\ln (5)+4)}{4\ln (2{)}^2-2}+\frac{88x\ln (2{)}^2}{4\ln (2{)}^2-2}+\frac{4x\ln (\ln (5)+4)}{4\ln (2{)}^2-2}-\frac{20x}{4\ln (2{)}^2-2}$	$72$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-8*ln(2)^2+4)*ln(ln(5)+4)+88*ln(2)^2-20)/(4*ln(2)^2-2),x,method=_RETURNVERBOSE)

[Out]

((-8*ln(2)^2+4)*ln(ln(5)+4)+88*ln(2)^2-20)/(4*ln(2)^2-2)*x

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maxima [A]  time = 0.46, size = 36, normalized size = 1.20 $$\begin{array}{c}\frac{2(22\log (2{)}^2-(2\log (2{)}^2-1)\log (\log (5)+4)-5)x}{2\log (2{)}^2-1}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*log(2)^2+4)*log(log(5)+4)+88*log(2)^2-20)/(4*log(2)^2-2),x, algorithm="maxima")

[Out]

2*(22*log(2)^2 - (2*log(2)^2 - 1)*log(log(5) + 4) - 5)*x/(2*log(2)^2 - 1)

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mupad [B]  time = 0.00, size = 35, normalized size = 1.17 $$\begin{array}{c}-\frac{x(\ln (\ln (5)+4)(8{\ln (2)}^2-4)-88{\ln (2)}^2+20)}{4{\ln (2)}^2-2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(log(5) + 4)*(8*log(2)^2 - 4) - 88*log(2)^2 + 20)/(4*log(2)^2 - 2),x)

[Out]

-(x*(log(log(5) + 4)*(8*log(2)^2 - 4) - 88*log(2)^2 + 20))/(4*log(2)^2 - 2)

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sympy [A]  time = 0.06, size = 32, normalized size = 1.07 $$\begin{array}{c}\frac{x(-20+(4-8\log {(2)}^2)\log (\log (5)+4)+88\log {(2)}^2)}{-2+4\log {(2)}^2}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-8*ln(2)**2+4)*ln(ln(5)+4)+88*ln(2)**2-20)/(4*ln(2)**2-2),x)

[Out]

x*(-20 + (4 - 8*log(2)**2)*log(log(5) + 4) + 88*log(2)**2)/(-2 + 4*log(2)**2)

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