3.98.13 \(\int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} (\frac {1}{x^2})^{2 x/3} x^4+2^{2 x/3} (\frac {1}{x^2})^{x/3} (51 x^2+70 x^3+x^3 \log (\frac {4}{x^2}))}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} (\frac {1}{x^2})^{2 x/3} x^4+2^{2 x/3} (\frac {1}{x^2})^{x/3} (48 x^2+72 x^3)} \, dx\)

Optimal. Leaf size=33 \[ x-\frac {x}{4 \left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )} \]

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Rubi [F]  time = 1.64, 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 {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{48+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (48 x^2+72 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2))^(x/3)*(51*x^2 + 70
*x^3 + x^3*Log[4/x^2]))/(48 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2)
)^(x/3)*(48*x^2 + 72*x^3)),x]

[Out]

x - Defer[Int][(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^(-2), x] - (5*Defer[Int][x/(2 + 2^((2*x)/3)*(x^(-2)
)^(-1 + x/3) + 3*x)^2, x])/12 - (Log[4/x^2]*Defer[Int][x/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^2, x])/6
+ Defer[Int][x^2/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^2, x]/2 - (Log[4/x^2]*Defer[Int][x^2/(2 + 2^((2*x
)/3)*(x^(-2))^(-1 + x/3) + 3*x)^2, x])/4 + Defer[Int][(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^(-1), x]/4 -
Defer[Int][x/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x), x]/6 + (Log[4/x^2]*Defer[Int][x/(2 + 2^((2*x)/3)*(x
^(-2))^(-1 + x/3) + 3*x), x])/12 - Defer[Int][Defer[Int][x/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^2, x]/x
, x]/3 - Defer[Int][Defer[Int][x^2/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x)^2, x]/x, x]/2 + Defer[Int][Defe
r[Int][x/(2 + 2^((2*x)/3)*(x^(-2))^(-1 + x/3) + 3*x), x]/x, x]/6

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{12 \left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ &=\frac {1}{12} \int \frac {42+144 x+108 x^2+3\ 2^{2+\frac {4 x}{3}} \left (\frac {1}{x^2}\right )^{2 x/3} x^4+2^{2 x/3} \left (\frac {1}{x^2}\right )^{x/3} \left (51 x^2+70 x^3+x^3 \log \left (\frac {4}{x^2}\right )\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ &=\frac {1}{12} \int \left (12+\frac {3-2 x+x \log \left (\frac {4}{x^2}\right )}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}-\frac {12+5 x-6 x^2+2 x \log \left (\frac {4}{x^2}\right )+3 x^2 \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}\right ) \, dx\\ &=x+\frac {1}{12} \int \frac {3-2 x+x \log \left (\frac {4}{x^2}\right )}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{12} \int \frac {12+5 x-6 x^2+2 x \log \left (\frac {4}{x^2}\right )+3 x^2 \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ &=x-\frac {1}{12} \int \left (\frac {12}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}+\frac {5 x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}-\frac {6 x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}+\frac {2 x \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}+\frac {3 x^2 \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2}\right ) \, dx+\frac {1}{12} \int \left (\frac {3}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}-\frac {2 x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}+\frac {x \log \left (\frac {4}{x^2}\right )}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}\right ) \, dx\\ &=x+\frac {1}{12} \int \frac {x \log \left (\frac {4}{x^2}\right )}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{6} \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{6} \int \frac {x \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx+\frac {1}{4} \int \frac {1}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{4} \int \frac {x^2 \log \left (\frac {4}{x^2}\right )}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\frac {5}{12} \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx+\frac {1}{2} \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\int \frac {1}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ &=x-\frac {1}{12} \int -\frac {2 \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx}{x} \, dx-\frac {1}{6} \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx+\frac {1}{6} \int -\frac {2 \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx}{x} \, dx+\frac {1}{4} \int \frac {1}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx+\frac {1}{4} \int -\frac {2 \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx}{x} \, dx-\frac {5}{12} \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx+\frac {1}{2} \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx+\frac {1}{12} \log \left (\frac {4}{x^2}\right ) \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{6} \log \left (\frac {4}{x^2}\right ) \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\frac {1}{4} \log \left (\frac {4}{x^2}\right ) \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\int \frac {1}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ &=x-\frac {1}{6} \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx+\frac {1}{6} \int \frac {\int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx}{x} \, dx+\frac {1}{4} \int \frac {1}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{3} \int \frac {\int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx}{x} \, dx-\frac {5}{12} \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx+\frac {1}{2} \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\frac {1}{2} \int \frac {\int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx}{x} \, dx+\frac {1}{12} \log \left (\frac {4}{x^2}\right ) \int \frac {x}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x} \, dx-\frac {1}{6} \log \left (\frac {4}{x^2}\right ) \int \frac {x}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\frac {1}{4} \log \left (\frac {4}{x^2}\right ) \int \frac {x^2}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx-\int \frac {1}{\left (2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 35, normalized size = 1.06 \begin {gather*} \frac {1}{4} x \left (4-\frac {1}{2+2^{2 x/3} \left (\frac {1}{x^2}\right )^{-1+\frac {x}{3}}+3 x}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(42 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(x^(-2))^(x/3)*(51*x^
2 + 70*x^3 + x^3*Log[4/x^2]))/(48 + 144*x + 108*x^2 + 3*2^(2 + (4*x)/3)*(x^(-2))^((2*x)/3)*x^4 + 2^((2*x)/3)*(
x^(-2))^(x/3)*(48*x^2 + 72*x^3)),x]

[Out]

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

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fricas [A]  time = 0.87, size = 45, normalized size = 1.36 \begin {gather*} \frac {4 \, x^{3} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 12 \, x^{2} + 7 \, x}{4 \, {\left (x^{2} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+4
2)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="frica
s")

[Out]

1/4*(4*x^3*(4/x^2)^(1/3*x) + 12*x^2 + 7*x)/(x^2*(4/x^2)^(1/3*x) + 3*x + 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {12 \, x^{4} \left (\frac {4}{x^{2}}\right )^{\frac {2}{3} \, x} + 108 \, x^{2} + {\left (x^{3} \log \left (\frac {4}{x^{2}}\right ) + 70 \, x^{3} + 51 \, x^{2}\right )} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 144 \, x + 42}{12 \, {\left (x^{4} \left (\frac {4}{x^{2}}\right )^{\frac {2}{3} \, x} + 9 \, x^{2} + 2 \, {\left (3 \, x^{3} + 2 \, x^{2}\right )} \left (\frac {4}{x^{2}}\right )^{\frac {1}{3} \, x} + 12 \, x + 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+4
2)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="giac"
)

[Out]

integrate(1/12*(12*x^4*(4/x^2)^(2/3*x) + 108*x^2 + (x^3*log(4/x^2) + 70*x^3 + 51*x^2)*(4/x^2)^(1/3*x) + 144*x
+ 42)/(x^4*(4/x^2)^(2/3*x) + 9*x^2 + 2*(3*x^3 + 2*x^2)*(4/x^2)^(1/3*x) + 12*x + 4), x)

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maple [A]  time = 0.19, size = 26, normalized size = 0.79

method	result	size
risch	\(x -\frac {x}{4 \left (3 x +\left (\frac {4}{x^{2}}\right )^{\frac {x}{3}} x^{2}+2\right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((12*x^4*exp(1/3*x*ln(4/x^2))^2+(x^3*ln(4/x^2)+70*x^3+51*x^2)*exp(1/3*x*ln(4/x^2))+108*x^2+144*x+42)/(12*x^
4*exp(1/3*x*ln(4/x^2))^2+(72*x^3+48*x^2)*exp(1/3*x*ln(4/x^2))+108*x^2+144*x+48),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.49, size = 58, normalized size = 1.76 \begin {gather*} \frac {{\left (12 \, x^{3} + x^{2}\right )} 2^{\frac {2}{3} \, x} + 2 \, {\left (18 \, x^{2} + 12 \, x + 1\right )} x^{\frac {2}{3} \, x}}{12 \, {\left (2^{\frac {2}{3} \, x} x^{2} + {\left (3 \, x + 2\right )} x^{\frac {2}{3} \, x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x^4*exp(1/3*x*log(4/x^2))^2+(x^3*log(4/x^2)+70*x^3+51*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+4
2)/(12*x^4*exp(1/3*x*log(4/x^2))^2+(72*x^3+48*x^2)*exp(1/3*x*log(4/x^2))+108*x^2+144*x+48),x, algorithm="maxim
a")

[Out]

1/12*((12*x^3 + x^2)*2^(2/3*x) + 2*(18*x^2 + 12*x + 1)*x^(2/3*x))/(2^(2/3*x)*x^2 + (3*x + 2)*x^(2/3*x))

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mupad [B]  time = 6.08, size = 49, normalized size = 1.48 \begin {gather*} \frac {x\,\left (12\,x+2^{\frac {2\,x}{3}+2}\,x^2\,{\left (\frac {1}{x^2}\right )}^{x/3}+7\right )}{4\,\left (3\,x+2^{\frac {2\,x}{3}}\,x^2\,{\left (\frac {1}{x^2}\right )}^{x/3}+2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*x + 12*x^4*exp((2*x*log(4/x^2))/3) + 108*x^2 + exp((x*log(4/x^2))/3)*(51*x^2 + 70*x^3 + x^3*log(4/x^2
)) + 42)/(144*x + exp((x*log(4/x^2))/3)*(48*x^2 + 72*x^3) + 12*x^4*exp((2*x*log(4/x^2))/3) + 108*x^2 + 48),x)

[Out]

(x*(12*x + 2^((2*x)/3 + 2)*x^2*(1/x^2)^(x/3) + 7))/(4*(3*x + 2^((2*x)/3)*x^2*(1/x^2)^(x/3) + 2))

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sympy [A]  time = 0.34, size = 24, normalized size = 0.73 \begin {gather*} x - \frac {x}{4 x^{2} e^{\frac {x \log {\left (\frac {4}{x^{2}} \right )}}{3}} + 12 x + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((12*x**4*exp(1/3*x*ln(4/x**2))**2+(x**3*ln(4/x**2)+70*x**3+51*x**2)*exp(1/3*x*ln(4/x**2))+108*x**2+1
44*x+42)/(12*x**4*exp(1/3*x*ln(4/x**2))**2+(72*x**3+48*x**2)*exp(1/3*x*ln(4/x**2))+108*x**2+144*x+48),x)

[Out]

x - x/(4*x**2*exp(x*log(4/x**2)/3) + 12*x + 8)

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