3.82.5 \(\int \frac {4 x^4-x^6-90 x^7+(5 x^2-2 x^4-180 x^5) \log (4)+(3-x^2-90 x^3) \log ^2(4)}{x^6+2 x^4 \log (4)+x^2 \log ^2(4)} \, dx\)

Optimal. Leaf size=26 \[ 5-\frac {3}{x}-x-45 x^2-\frac {x}{x^2+\log (4)} \]

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Rubi [A]  time = 0.14, antiderivative size = 33, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, integrand size = 74, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.068, Rules used = {1594, 28, 1805, 1586, 14} \begin {gather*} -45 x^2-\frac {x}{x^2+\log (4)}-\frac {3}{x}-\frac {x \log (16)}{2 \log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*x^4 - x^6 - 90*x^7 + (5*x^2 - 2*x^4 - 180*x^5)*Log[4] + (3 - x^2 - 90*x^3)*Log[4]^2)/(x^6 + 2*x^4*Log[4
] + x^2*Log[4]^2),x]

[Out]

-3/x - 45*x^2 - x/(x^2 + Log[4]) - (x*Log[16])/(2*Log[4])

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

Rule 1805

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(c*x)^m*Pq,
a + b*x^2, x], f = Coeff[PolynomialRemainder[(c*x)^m*Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[
(c*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[((a*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*
(p + 1)), Int[(c*x)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Q)/(c*x)^m + (f*(2*p + 3))/(c*x)^m, x], x],
x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && LtQ[p, -1] && ILtQ[m, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 x^4-x^6-90 x^7+\left (5 x^2-2 x^4-180 x^5\right ) \log (4)+\left (3-x^2-90 x^3\right ) \log ^2(4)}{x^2 \left (x^4+2 x^2 \log (4)+\log ^2(4)\right )} \, dx\\ &=\int \frac {4 x^4-x^6-90 x^7+\left (5 x^2-2 x^4-180 x^5\right ) \log (4)+\left (3-x^2-90 x^3\right ) \log ^2(4)}{x^2 \left (x^2+\log (4)\right )^2} \, dx\\ &=-\frac {x}{x^2+\log (4)}-\frac {\int \frac {2 x^4 \log (4)+180 x^5 \log (4)-2 x^2 (3-\log (4)) \log (4)-6 \log ^2(4)+180 x^3 \log ^2(4)}{x^2 \left (x^2+\log (4)\right )} \, dx}{2 \log (4)}\\ &=-\frac {x}{x^2+\log (4)}-\frac {\int \frac {-6 \log (4)+2 x^2 \log (4)+180 x^3 \log (4)}{x^2} \, dx}{2 \log (4)}\\ &=-\frac {x}{x^2+\log (4)}-\frac {\int \left (-\frac {6 \log (4)}{x^2}+180 x \log (4)+\log (16)\right ) \, dx}{2 \log (4)}\\ &=-\frac {3}{x}-45 x^2-\frac {x}{x^2+\log (4)}-\frac {x \log (16)}{2 \log (4)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 25, normalized size = 0.96 \begin {gather*} -\frac {3}{x}-x-45 x^2-\frac {x}{x^2+\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*x^4 - x^6 - 90*x^7 + (5*x^2 - 2*x^4 - 180*x^5)*Log[4] + (3 - x^2 - 90*x^3)*Log[4]^2)/(x^6 + 2*x^4
*Log[4] + x^2*Log[4]^2),x]

[Out]

-3/x - x - 45*x^2 - x/(x^2 + Log[4])

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fricas [A]  time = 0.65, size = 41, normalized size = 1.58 \begin {gather*} -\frac {45 \, x^{5} + x^{4} + 4 \, x^{2} + 2 \, {\left (45 \, x^{3} + x^{2} + 3\right )} \log \relax (2)}{x^{3} + 2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-90*x^3-x^2+3)*log(2)^2+2*(-180*x^5-2*x^4+5*x^2)*log(2)-90*x^7-x^6+4*x^4)/(4*x^2*log(2)^2+4*x^4*
log(2)+x^6),x, algorithm="fricas")

[Out]

-(45*x^5 + x^4 + 4*x^2 + 2*(45*x^3 + x^2 + 3)*log(2))/(x^3 + 2*x*log(2))

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giac [A]  time = 0.17, size = 32, normalized size = 1.23 \begin {gather*} -45 \, x^{2} - x - \frac {2 \, {\left (2 \, x^{2} + 3 \, \log \relax (2)\right )}}{x^{3} + 2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-90*x^3-x^2+3)*log(2)^2+2*(-180*x^5-2*x^4+5*x^2)*log(2)-90*x^7-x^6+4*x^4)/(4*x^2*log(2)^2+4*x^4*
log(2)+x^6),x, algorithm="giac")

[Out]

-45*x^2 - x - 2*(2*x^2 + 3*log(2))/(x^3 + 2*x*log(2))

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maple [A]  time = 0.08, size = 28, normalized size = 1.08

method	result	size
default	\(-x -45 x^{2}-\frac {3}{x}-\frac {x}{2 \left (\frac {x^{2}}{2}+\ln \relax (2)\right )}\)	\(28\)
risch	\(-45 x^{2}-x +\frac {-4 x^{2}-6 \ln \relax (2)}{x \left (x^{2}+2 \ln \relax (2)\right )}\)	\(35\)
norman	\(\frac {180 x \ln \relax (2)^{2}+\left (-2 \ln \relax (2)-4\right ) x^{2}-x^{4}-45 x^{5}-6 \ln \relax (2)}{x \left (x^{2}+2 \ln \relax (2)\right )}\)	\(47\)
gosper	\(\frac {-45 x^{5}-x^{4}+180 x \ln \relax (2)^{2}-2 x^{2} \ln \relax (2)-4 x^{2}-6 \ln \relax (2)}{x \left (x^{2}+2 \ln \relax (2)\right )}\)	\(49\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(-90*x^3-x^2+3)*ln(2)^2+2*(-180*x^5-2*x^4+5*x^2)*ln(2)-90*x^7-x^6+4*x^4)/(4*x^2*ln(2)^2+4*x^4*ln(2)+x^6
),x,method=_RETURNVERBOSE)

[Out]

-x-45*x^2-3/x-1/2*x/(1/2*x^2+ln(2))

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maxima [A]  time = 0.36, size = 32, normalized size = 1.23 \begin {gather*} -45 \, x^{2} - x - \frac {2 \, {\left (2 \, x^{2} + 3 \, \log \relax (2)\right )}}{x^{3} + 2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-90*x^3-x^2+3)*log(2)^2+2*(-180*x^5-2*x^4+5*x^2)*log(2)-90*x^7-x^6+4*x^4)/(4*x^2*log(2)^2+4*x^4*
log(2)+x^6),x, algorithm="maxima")

[Out]

-45*x^2 - x - 2*(2*x^2 + 3*log(2))/(x^3 + 2*x*log(2))

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mupad [B]  time = 0.15, size = 32, normalized size = 1.23 \begin {gather*} -x-\frac {4\,x^2+6\,\ln \relax (2)}{x^3+2\,\ln \relax (2)\,x}-45\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*log(2)^2*(x^2 + 90*x^3 - 3) + 2*log(2)*(2*x^4 - 5*x^2 + 180*x^5) - 4*x^4 + x^6 + 90*x^7)/(4*x^2*log(2)
^2 + 4*x^4*log(2) + x^6),x)

[Out]

- x - (6*log(2) + 4*x^2)/(2*x*log(2) + x^3) - 45*x^2

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sympy [A]  time = 0.27, size = 27, normalized size = 1.04 \begin {gather*} - 45 x^{2} - x - \frac {4 x^{2} + 6 \log {\relax (2 )}}{x^{3} + 2 x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(-90*x**3-x**2+3)*ln(2)**2+2*(-180*x**5-2*x**4+5*x**2)*ln(2)-90*x**7-x**6+4*x**4)/(4*x**2*ln(2)**
2+4*x**4*ln(2)+x**6),x)

[Out]

-45*x**2 - x - (4*x**2 + 6*log(2))/(x**3 + 2*x*log(2))

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