3.34.48 \(\int \frac {(-x^3-25 x^5) \log (4)+(3 x^3+15 x^5) \log (4) \log (x+10 x^3+25 x^5)+(-1-5 x^2) \log ^2(x+10 x^3+25 x^5)+(1+5 x^2) \log (4^{-\frac {x^3}{\log (x+10 x^3+25 x^5)}} x) \log ^2(x+10 x^3+25 x^5)}{(1+5 x^2) \log ^2(4^{-\frac {x^3}{\log (x+10 x^3+25 x^5)}} x) \log ^2(x+10 x^3+25 x^5)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \]

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Rubi [F]  time = 2.55, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-x^3-25 x^5\right ) \log (4)+\left (3 x^3+15 x^5\right ) \log (4) \log \left (x+10 x^3+25 x^5\right )+\left (-1-5 x^2\right ) \log ^2\left (x+10 x^3+25 x^5\right )+\left (1+5 x^2\right ) \log \left (4^{-\frac {x^3}{\log \left (x+10 x^3+25 x^5\right )}} x\right ) \log ^2\left (x+10 x^3+25 x^5\right )}{\left (1+5 x^2\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x+10 x^3+25 x^5\right )}} x\right ) \log ^2\left (x+10 x^3+25 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-x^3 - 25*x^5)*Log[4] + (3*x^3 + 15*x^5)*Log[4]*Log[x + 10*x^3 + 25*x^5] + (-1 - 5*x^2)*Log[x + 10*x^3 +
25*x^5]^2 + (1 + 5*x^2)*Log[x/4^(x^3/Log[x + 10*x^3 + 25*x^5])]*Log[x + 10*x^3 + 25*x^5]^2)/((1 + 5*x^2)*Log[
x/4^(x^3/Log[x + 10*x^3 + 25*x^5])]^2*Log[x + 10*x^3 + 25*x^5]^2),x]

[Out]

-Defer[Int][Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^(-2), x] + Defer[Int][Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^(-1)
, x] + (4*Log[4]*Defer[Int][x/(Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^2*Log[x*(1 + 5*x^2)^2]^2), x])/5 - 5*Log[4]
*Defer[Int][x^3/(Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^2*Log[x*(1 + 5*x^2)^2]^2), x] + (2*Log[4]*Defer[Int][1/((
I - Sqrt[5]*x)*Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^2*Log[x*(1 + 5*x^2)^2]^2), x])/(5*Sqrt[5]) - (2*Log[4]*Defe
r[Int][1/((I + Sqrt[5]*x)*Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^2*Log[x*(1 + 5*x^2)^2]^2), x])/(5*Sqrt[5]) + Log
[64]*Defer[Int][x^3/(Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]^2*Log[x*(1 + 5*x^2)^2]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-1+\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )}-\frac {x^3 \left (1+25 x^2\right ) \log (4)}{\left (1+5 x^2\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}+\frac {x^3 \log (64)}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )}\right ) \, dx\\ &=-\left (\log (4) \int \frac {x^3 \left (1+25 x^2\right )}{\left (1+5 x^2\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx\right )+\log (64) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )} \, dx+\int \frac {-1+\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx\\ &=-\left (\log (4) \int \left (-\frac {4 x}{5 \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}+\frac {5 x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}+\frac {4 x}{5 \left (1+5 x^2\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}\right ) \, dx\right )+\log (64) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )} \, dx+\int \left (-\frac {1}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )}+\frac {1}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )}\right ) \, dx\\ &=\frac {1}{5} (4 \log (4)) \int \frac {x}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx-\frac {1}{5} (4 \log (4)) \int \frac {x}{\left (1+5 x^2\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx-(5 \log (4)) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx+\log (64) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )} \, dx-\int \frac {1}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx+\int \frac {1}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx\\ &=-\left (\frac {1}{5} (4 \log (4)) \int \left (-\frac {1}{2 \sqrt {5} \left (i-\sqrt {5} x\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}+\frac {1}{2 \sqrt {5} \left (i+\sqrt {5} x\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )}\right ) \, dx\right )+\frac {1}{5} (4 \log (4)) \int \frac {x}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx-(5 \log (4)) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx+\log (64) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )} \, dx-\int \frac {1}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx+\int \frac {1}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx\\ &=\frac {1}{5} (4 \log (4)) \int \frac {x}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx-(5 \log (4)) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx+\frac {(2 \log (4)) \int \frac {1}{\left (i-\sqrt {5} x\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx}{5 \sqrt {5}}-\frac {(2 \log (4)) \int \frac {1}{\left (i+\sqrt {5} x\right ) \log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log ^2\left (x \left (1+5 x^2\right )^2\right )} \, dx}{5 \sqrt {5}}+\log (64) \int \frac {x^3}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right ) \log \left (x \left (1+5 x^2\right )^2\right )} \, dx-\int \frac {1}{\log ^2\left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx+\int \frac {1}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.40, size = 28, normalized size = 1.00 \begin {gather*} \frac {x}{\log \left (4^{-\frac {x^3}{\log \left (x \left (1+5 x^2\right )^2\right )}} x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-x^3 - 25*x^5)*Log[4] + (3*x^3 + 15*x^5)*Log[4]*Log[x + 10*x^3 + 25*x^5] + (-1 - 5*x^2)*Log[x + 10
*x^3 + 25*x^5]^2 + (1 + 5*x^2)*Log[x/4^(x^3/Log[x + 10*x^3 + 25*x^5])]*Log[x + 10*x^3 + 25*x^5]^2)/((1 + 5*x^2
)*Log[x/4^(x^3/Log[x + 10*x^3 + 25*x^5])]^2*Log[x + 10*x^3 + 25*x^5]^2),x]

[Out]

x/Log[x/4^(x^3/Log[x*(1 + 5*x^2)^2])]

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fricas [A]  time = 0.52, size = 31, normalized size = 1.11 \begin {gather*} \frac {x}{\log \left (\frac {x}{2^{\frac {2 \, x^{3}}{\log \left (25 \, x^{5} + 10 \, x^{3} + x\right )}}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2+1)*log(25*x^5+10*x^3+x)^2*log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))+(-5*x^2-1)*log(25*x^
5+10*x^3+x)^2+2*(15*x^5+3*x^3)*log(2)*log(25*x^5+10*x^3+x)+2*(-25*x^5-x^3)*log(2))/(5*x^2+1)/log(25*x^5+10*x^3
+x)^2/log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))^2,x, algorithm="fricas")

[Out]

x/log(x/2^(2*x^3/log(25*x^5 + 10*x^3 + x)))

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giac [A]  time = 0.39, size = 46, normalized size = 1.64 \begin {gather*} -\frac {2 \, x^{4} \log \relax (2)}{2 \, x^{3} \log \relax (2) \log \relax (x) - 2 \, \log \left (5 \, x^{2} + 1\right ) \log \relax (x)^{2} - \log \relax (x)^{3}} + \frac {x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2+1)*log(25*x^5+10*x^3+x)^2*log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))+(-5*x^2-1)*log(25*x^
5+10*x^3+x)^2+2*(15*x^5+3*x^3)*log(2)*log(25*x^5+10*x^3+x)+2*(-25*x^5-x^3)*log(2))/(5*x^2+1)/log(25*x^5+10*x^3
+x)^2/log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))^2,x, algorithm="giac")

[Out]

-2*x^4*log(2)/(2*x^3*log(2)*log(x) - 2*log(5*x^2 + 1)*log(x)^2 - log(x)^3) + x/log(x)

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maple [F]  time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (5 x^{2}+1\right ) \ln \left (25 x^{5}+10 x^{3}+x \right )^{2} \ln \left (x \,{\mathrm e}^{-\frac {2 x^{3} \ln \relax (2)}{\ln \left (25 x^{5}+10 x^{3}+x \right )}}\right )+\left (-5 x^{2}-1\right ) \ln \left (25 x^{5}+10 x^{3}+x \right )^{2}+2 \left (15 x^{5}+3 x^{3}\right ) \ln \relax (2) \ln \left (25 x^{5}+10 x^{3}+x \right )+2 \left (-25 x^{5}-x^{3}\right ) \ln \relax (2)}{\left (5 x^{2}+1\right ) \ln \left (25 x^{5}+10 x^{3}+x \right )^{2} \ln \left (x \,{\mathrm e}^{-\frac {2 x^{3} \ln \relax (2)}{\ln \left (25 x^{5}+10 x^{3}+x \right )}}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^2+1)*ln(25*x^5+10*x^3+x)^2*ln(x/exp(2*x^3*ln(2)/ln(25*x^5+10*x^3+x)))+(-5*x^2-1)*ln(25*x^5+10*x^3+x)
^2+2*(15*x^5+3*x^3)*ln(2)*ln(25*x^5+10*x^3+x)+2*(-25*x^5-x^3)*ln(2))/(5*x^2+1)/ln(25*x^5+10*x^3+x)^2/ln(x/exp(
2*x^3*ln(2)/ln(25*x^5+10*x^3+x)))^2,x)

[Out]

int(((5*x^2+1)*ln(25*x^5+10*x^3+x)^2*ln(x/exp(2*x^3*ln(2)/ln(25*x^5+10*x^3+x)))+(-5*x^2-1)*ln(25*x^5+10*x^3+x)
^2+2*(15*x^5+3*x^3)*ln(2)*ln(25*x^5+10*x^3+x)+2*(-25*x^5-x^3)*ln(2))/(5*x^2+1)/ln(25*x^5+10*x^3+x)^2/ln(x/exp(
2*x^3*ln(2)/ln(25*x^5+10*x^3+x)))^2,x)

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maxima [A]  time = 0.93, size = 34, normalized size = 1.21 \begin {gather*} -\frac {x}{2 \, \log \left (2^{\frac {x^{3}}{2 \, \log \left (5 \, x^{2} + 1\right ) + \log \relax (x)}}\right ) - \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^2+1)*log(25*x^5+10*x^3+x)^2*log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))+(-5*x^2-1)*log(25*x^
5+10*x^3+x)^2+2*(15*x^5+3*x^3)*log(2)*log(25*x^5+10*x^3+x)+2*(-25*x^5-x^3)*log(2))/(5*x^2+1)/log(25*x^5+10*x^3
+x)^2/log(x/exp(2*x^3*log(2)/log(25*x^5+10*x^3+x)))^2,x, algorithm="maxima")

[Out]

-x/(2*log(2^(x^3/(2*log(5*x^2 + 1) + log(x)))) - log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\ln \left (25\,x^5+10\,x^3+x\right )}^2\,\left (5\,x^2+1\right )+2\,\ln \relax (2)\,\left (25\,x^5+x^3\right )-2\,\ln \relax (2)\,\ln \left (25\,x^5+10\,x^3+x\right )\,\left (15\,x^5+3\,x^3\right )-\ln \left (x\,{\mathrm {e}}^{-\frac {2\,x^3\,\ln \relax (2)}{\ln \left (25\,x^5+10\,x^3+x\right )}}\right )\,{\ln \left (25\,x^5+10\,x^3+x\right )}^2\,\left (5\,x^2+1\right )}{{\ln \left (x\,{\mathrm {e}}^{-\frac {2\,x^3\,\ln \relax (2)}{\ln \left (25\,x^5+10\,x^3+x\right )}}\right )}^2\,{\ln \left (25\,x^5+10\,x^3+x\right )}^2\,\left (5\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1) + 2*log(2)*(x^3 + 25*x^5) - 2*log(2)*log(x + 10*x^3 + 25*x^5)*(3*
x^3 + 15*x^5) - log(x*exp(-(2*x^3*log(2))/log(x + 10*x^3 + 25*x^5)))*log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1))/(
log(x*exp(-(2*x^3*log(2))/log(x + 10*x^3 + 25*x^5)))^2*log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1)),x)

[Out]

int(-(log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1) + 2*log(2)*(x^3 + 25*x^5) - 2*log(2)*log(x + 10*x^3 + 25*x^5)*(3*
x^3 + 15*x^5) - log(x*exp(-(2*x^3*log(2))/log(x + 10*x^3 + 25*x^5)))*log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1))/(
log(x*exp(-(2*x^3*log(2))/log(x + 10*x^3 + 25*x^5)))^2*log(x + 10*x^3 + 25*x^5)^2*(5*x^2 + 1)), x)

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sympy [A]  time = 2.39, size = 27, normalized size = 0.96 \begin {gather*} \frac {x}{\log {\left (x e^{- \frac {2 x^{3} \log {\relax (2 )}}{\log {\left (25 x^{5} + 10 x^{3} + x \right )}}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**2+1)*ln(25*x**5+10*x**3+x)**2*ln(x/exp(2*x**3*ln(2)/ln(25*x**5+10*x**3+x)))+(-5*x**2-1)*ln(25
*x**5+10*x**3+x)**2+2*(15*x**5+3*x**3)*ln(2)*ln(25*x**5+10*x**3+x)+2*(-25*x**5-x**3)*ln(2))/(5*x**2+1)/ln(25*x
**5+10*x**3+x)**2/ln(x/exp(2*x**3*ln(2)/ln(25*x**5+10*x**3+x)))**2,x)

[Out]

x/log(x*exp(-2*x**3*log(2)/log(25*x**5 + 10*x**3 + x)))

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