3.64.39 \(\int \frac {-11664 x^5-1296 x^6 \log ^2(2+x^4)+(324 x^2+162 x^6) \log ^3(2+x^4)+(90+36 x^3+45 x^4+18 x^7) \log ^5(2+x^4)}{(2 x^2+x^6) \log ^5(2+x^4)} \, dx\)

Optimal. Leaf size=23 \[ 9 \left (16-\frac {5}{x}+\left (x+\frac {9}{\log ^2\left (2+x^4\right )}\right )^2\right ) \]

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Rubi [F]  time = 0.63, 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 {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{\left (2 x^2+x^6\right ) \log ^5\left (2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-11664*x^5 - 1296*x^6*Log[2 + x^4]^2 + (324*x^2 + 162*x^6)*Log[2 + x^4]^3 + (90 + 36*x^3 + 45*x^4 + 18*x^
7)*Log[2 + x^4]^5)/((2*x^2 + x^6)*Log[2 + x^4]^5),x]

[Out]

-45/x + 9*x^2 + 729/Log[2 + x^4]^4 - 1296*Defer[Int][x^4/((2 + x^4)*Log[2 + x^4]^3), x] + 162*Defer[Int][Log[2
+ x^4]^(-2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-11664 x^5-1296 x^6 \log ^2\left (2+x^4\right )+\left (324 x^2+162 x^6\right ) \log ^3\left (2+x^4\right )+\left (90+36 x^3+45 x^4+18 x^7\right ) \log ^5\left (2+x^4\right )}{x^2 \left (2+x^4\right ) \log ^5\left (2+x^4\right )} \, dx\\ &=\int \left (\frac {9 \left (5+2 x^3\right )}{x^2}-\frac {11664 x^3}{\left (2+x^4\right ) \log ^5\left (2+x^4\right )}-\frac {1296 x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )}+\frac {162}{\log ^2\left (2+x^4\right )}\right ) \, dx\\ &=9 \int \frac {5+2 x^3}{x^2} \, dx+162 \int \frac {1}{\log ^2\left (2+x^4\right )} \, dx-1296 \int \frac {x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )} \, dx-11664 \int \frac {x^3}{\left (2+x^4\right ) \log ^5\left (2+x^4\right )} \, dx\\ &=9 \int \left (\frac {5}{x^2}+2 x\right ) \, dx+162 \int \frac {1}{\log ^2\left (2+x^4\right )} \, dx-1296 \int \frac {x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )} \, dx-2916 \operatorname {Subst}\left (\int \frac {1}{(2+x) \log ^5(2+x)} \, dx,x,x^4\right )\\ &=-\frac {45}{x}+9 x^2+162 \int \frac {1}{\log ^2\left (2+x^4\right )} \, dx-1296 \int \frac {x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )} \, dx-2916 \operatorname {Subst}\left (\int \frac {1}{x \log ^5(x)} \, dx,x,2+x^4\right )\\ &=-\frac {45}{x}+9 x^2+162 \int \frac {1}{\log ^2\left (2+x^4\right )} \, dx-1296 \int \frac {x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )} \, dx-2916 \operatorname {Subst}\left (\int \frac {1}{x^5} \, dx,x,\log \left (2+x^4\right )\right )\\ &=-\frac {45}{x}+9 x^2+\frac {729}{\log ^4\left (2+x^4\right )}+162 \int \frac {1}{\log ^2\left (2+x^4\right )} \, dx-1296 \int \frac {x^4}{\left (2+x^4\right ) \log ^3\left (2+x^4\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 32, normalized size = 1.39 \begin {gather*} 9 \left (-\frac {5}{x}+x^2+\frac {81}{\log ^4\left (2+x^4\right )}+\frac {18 x}{\log ^2\left (2+x^4\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-11664*x^5 - 1296*x^6*Log[2 + x^4]^2 + (324*x^2 + 162*x^6)*Log[2 + x^4]^3 + (90 + 36*x^3 + 45*x^4 +
18*x^7)*Log[2 + x^4]^5)/((2*x^2 + x^6)*Log[2 + x^4]^5),x]

[Out]

9*(-5/x + x^2 + 81/Log[2 + x^4]^4 + (18*x)/Log[2 + x^4]^2)

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fricas [A]  time = 0.78, size = 44, normalized size = 1.91 \begin {gather*} \frac {9 \, {\left ({\left (x^{3} - 5\right )} \log \left (x^{4} + 2\right )^{4} + 18 \, x^{2} \log \left (x^{4} + 2\right )^{2} + 81 \, x\right )}}{x \log \left (x^{4} + 2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^4+2)^3-1296*x^6*log(x^4+2)^2-11664*x
^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algorithm="fricas")

[Out]

9*((x^3 - 5)*log(x^4 + 2)^4 + 18*x^2*log(x^4 + 2)^2 + 81*x)/(x*log(x^4 + 2)^4)

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giac [B]  time = 0.81, size = 78, normalized size = 3.39 \begin {gather*} 9 \, x^{2} + \frac {81 \, {\left (2 \, x^{11} \log \left (x^{4} + 2\right )^{2} + 9 \, x^{10} + 4 \, x^{7} \log \left (x^{4} + 2\right )^{2} + 18 \, x^{6}\right )}}{x^{10} \log \left (x^{4} + 2\right )^{4} + 2 \, x^{6} \log \left (x^{4} + 2\right )^{4}} - \frac {45}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^4+2)^3-1296*x^6*log(x^4+2)^2-11664*x
^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algorithm="giac")

[Out]

9*x^2 + 81*(2*x^11*log(x^4 + 2)^2 + 9*x^10 + 4*x^7*log(x^4 + 2)^2 + 18*x^6)/(x^10*log(x^4 + 2)^4 + 2*x^6*log(x
^4 + 2)^4) - 45/x

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maple [A]  time = 0.36, size = 35, normalized size = 1.52

method	result	size
risch	\(\frac {9 x^{3}-45}{x}+\frac {162 x \ln \left (x^{4}+2\right )^{2}+729}{\ln \left (x^{4}+2\right )^{4}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((18*x^7+45*x^4+36*x^3+90)*ln(x^4+2)^5+(162*x^6+324*x^2)*ln(x^4+2)^3-1296*x^6*ln(x^4+2)^2-11664*x^5)/(x^6+
2*x^2)/ln(x^4+2)^5,x,method=_RETURNVERBOSE)

[Out]

9*(x^3-5)/x+81*(2*x*ln(x^4+2)^2+9)/ln(x^4+2)^4

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maxima [A]  time = 0.50, size = 44, normalized size = 1.91 \begin {gather*} \frac {9 \, {\left ({\left (x^{3} - 5\right )} \log \left (x^{4} + 2\right )^{2} + 18 \, x^{2}\right )}}{x \log \left (x^{4} + 2\right )^{2}} + \frac {729}{\log \left (x^{4} + 2\right )^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x^7+45*x^4+36*x^3+90)*log(x^4+2)^5+(162*x^6+324*x^2)*log(x^4+2)^3-1296*x^6*log(x^4+2)^2-11664*x
^5)/(x^6+2*x^2)/log(x^4+2)^5,x, algorithm="maxima")

[Out]

9*((x^3 - 5)*log(x^4 + 2)^2 + 18*x^2)/(x*log(x^4 + 2)^2) + 729/log(x^4 + 2)^4

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mupad [B]  time = 0.30, size = 32, normalized size = 1.39 \begin {gather*} \frac {729}{{\ln \left (x^4+2\right )}^4}-\frac {45}{x}+9\,x^2+\frac {162\,x}{{\ln \left (x^4+2\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x^4 + 2)^3*(324*x^2 + 162*x^6) - 1296*x^6*log(x^4 + 2)^2 + log(x^4 + 2)^5*(36*x^3 + 45*x^4 + 18*x^7 +
90) - 11664*x^5)/(log(x^4 + 2)^5*(2*x^2 + x^6)),x)

[Out]

729/log(x^4 + 2)^4 - 45/x + 9*x^2 + (162*x)/log(x^4 + 2)^2

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sympy [A]  time = 0.16, size = 29, normalized size = 1.26 \begin {gather*} 9 x^{2} + \frac {162 x \log {\left (x^{4} + 2 \right )}^{2} + 729}{\log {\left (x^{4} + 2 \right )}^{4}} - \frac {45}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((18*x**7+45*x**4+36*x**3+90)*ln(x**4+2)**5+(162*x**6+324*x**2)*ln(x**4+2)**3-1296*x**6*ln(x**4+2)**
2-11664*x**5)/(x**6+2*x**2)/ln(x**4+2)**5,x)

[Out]

9*x**2 + (162*x*log(x**4 + 2)**2 + 729)/log(x**4 + 2)**4 - 45/x

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