\(\int (40+32 x+(-60 x^2-64 x^3) \log ^2(4)+24 x^5 \log ^4(4)) \, dx\) [6715]

Optimal result

Integrand size = 30, antiderivative size = 19 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=\left (-3+2 \left (-1-2 x+x^3 \log ^2(4)\right )\right )^2 \]

[Out]

(8*x^3*ln(2)^2-4*x-5)^2

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=4 x^6 \log ^4(4)-16 x^4 \log ^2(4)-20 x^3 \log ^2(4)+16 x^2+40 x \]

[In]

Int[40 + 32*x + (-60*x^2 - 64*x^3)*Log[4]^2 + 24*x^5*Log[4]^4,x]

[Out]

40*x + 16*x^2 - 20*x^3*Log[4]^2 - 16*x^4*Log[4]^2 + 4*x^6*Log[4]^4

Rubi steps \begin{align*} \text {integral}& = 40 x+16 x^2+4 x^6 \log ^4(4)+\log ^2(4) \int \left (-60 x^2-64 x^3\right ) \, dx \\ & = 40 x+16 x^2-20 x^3 \log ^2(4)-16 x^4 \log ^2(4)+4 x^6 \log ^4(4) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=40 x+16 x^2-20 x^3 \log ^2(4)-16 x^4 \log ^2(4)+4 x^6 \log ^4(4) \]

[In]

Integrate[40 + 32*x + (-60*x^2 - 64*x^3)*Log[4]^2 + 24*x^5*Log[4]^4,x]

[Out]

40*x + 16*x^2 - 20*x^3*Log[4]^2 - 16*x^4*Log[4]^2 + 4*x^6*Log[4]^4

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89

method	result	size
default	\(\left (8 x^{3} \ln \left (2\right )^{2}-4 x -5\right )^{2}\)	\(17\)
gosper	\(8 x \left (8 x^{5} \ln \left (2\right )^{4}-8 x^{3} \ln \left (2\right )^{2}-10 x^{2} \ln \left (2\right )^{2}+2 x +5\right )\)	\(36\)
norman	\(40 x +16 x^{2}-80 x^{3} \ln \left (2\right )^{2}-64 x^{4} \ln \left (2\right )^{2}+64 x^{6} \ln \left (2\right )^{4}\)	\(37\)
risch	\(40 x +16 x^{2}-80 x^{3} \ln \left (2\right )^{2}-64 x^{4} \ln \left (2\right )^{2}+64 x^{6} \ln \left (2\right )^{4}\)	\(37\)
parallelrisch	\(40 x +16 x^{2}-80 x^{3} \ln \left (2\right )^{2}-64 x^{4} \ln \left (2\right )^{2}+64 x^{6} \ln \left (2\right )^{4}\)	\(37\)
parts	\(40 x +16 x^{2}-80 x^{3} \ln \left (2\right )^{2}-64 x^{4} \ln \left (2\right )^{2}+64 x^{6} \ln \left (2\right )^{4}\)	\(37\)

[In]

int(384*x^5*ln(2)^4+4*(-64*x^3-60*x^2)*ln(2)^2+32*x+40,x,method=_RETURNVERBOSE)

[Out]

(8*x^3*ln(2)^2-4*x-5)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=64 \, x^{6} \log \left (2\right )^{4} - 16 \, {\left (4 \, x^{4} + 5 \, x^{3}\right )} \log \left (2\right )^{2} + 16 \, x^{2} + 40 \, x \]

[In]

integrate(384*x^5*log(2)^4+4*(-64*x^3-60*x^2)*log(2)^2+32*x+40,x, algorithm="fricas")

[Out]

64*x^6*log(2)^4 - 16*(4*x^4 + 5*x^3)*log(2)^2 + 16*x^2 + 40*x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (15) = 30\).

Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.95 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=64 x^{6} \log {\left (2 \right )}^{4} - 64 x^{4} \log {\left (2 \right )}^{2} - 80 x^{3} \log {\left (2 \right )}^{2} + 16 x^{2} + 40 x \]

[In]

integrate(384*x**5*ln(2)**4+4*(-64*x**3-60*x**2)*ln(2)**2+32*x+40,x)

[Out]

64*x**6*log(2)**4 - 64*x**4*log(2)**2 - 80*x**3*log(2)**2 + 16*x**2 + 40*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=64 \, x^{6} \log \left (2\right )^{4} - 16 \, {\left (4 \, x^{4} + 5 \, x^{3}\right )} \log \left (2\right )^{2} + 16 \, x^{2} + 40 \, x \]

[In]

integrate(384*x^5*log(2)^4+4*(-64*x^3-60*x^2)*log(2)^2+32*x+40,x, algorithm="maxima")

[Out]

64*x^6*log(2)^4 - 16*(4*x^4 + 5*x^3)*log(2)^2 + 16*x^2 + 40*x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 35 vs. \(2 (16) = 32\).

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=64 \, x^{6} \log \left (2\right )^{4} - 16 \, {\left (4 \, x^{4} + 5 \, x^{3}\right )} \log \left (2\right )^{2} + 16 \, x^{2} + 40 \, x \]

[In]

integrate(384*x^5*log(2)^4+4*(-64*x^3-60*x^2)*log(2)^2+32*x+40,x, algorithm="giac")

[Out]

64*x^6*log(2)^4 - 16*(4*x^4 + 5*x^3)*log(2)^2 + 16*x^2 + 40*x

Mupad [B] (verification not implemented)

Time = 11.78 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.89 \[ \int \left (40+32 x+\left (-60 x^2-64 x^3\right ) \log ^2(4)+24 x^5 \log ^4(4)\right ) \, dx=64\,{\ln \left (2\right )}^4\,x^6-64\,{\ln \left (2\right )}^2\,x^4-80\,{\ln \left (2\right )}^2\,x^3+16\,x^2+40\,x \]

[In]

int(32*x + 384*x^5*log(2)^4 - 4*log(2)^2*(60*x^2 + 64*x^3) + 40,x)

[Out]

40*x - 80*x^3*log(2)^2 - 64*x^4*log(2)^2 + 64*x^6*log(2)^4 + 16*x^2