3.40.99 \(\int \frac {3+4 x+(1+x) \log (1+x)}{9 x^2+9 x^3+(6 x^2+6 x^3) \log (1+x)+(x^2+x^3) \log ^2(1+x)} \, dx\)

Optimal. Leaf size=14 \[ \frac {1}{x (-3-\log (1+x))} \]

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Rubi [A]  time = 0.17, antiderivative size = 13, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 57, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.035, Rules used = {6688, 6687} \begin {gather*} -\frac {1}{x (\log (x+1)+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 4*x + (1 + x)*Log[1 + x])/(9*x^2 + 9*x^3 + (6*x^2 + 6*x^3)*Log[1 + x] + (x^2 + x^3)*Log[1 + x]^2),x]

[Out]

-(1/(x*(3 + Log[1 + x])))

Rule 6687

Int[(u_)*(y_)^(m_.)*(z_)^(n_.), x_Symbol] :> With[{q = DerivativeDivides[y*z, u*z^(n - m), x]}, Simp[(q*y^(m +
1)*z^(m + 1))/(m + 1), x] /;  !FalseQ[q]] /; FreeQ[{m, n}, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3+4 x+(1+x) \log (1+x)}{x^2 (1+x) (3+\log (1+x))^2} \, dx\\ &=-\frac {1}{x (3+\log (1+x))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 13, normalized size = 0.93 \begin {gather*} -\frac {1}{x (3+\log (1+x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 4*x + (1 + x)*Log[1 + x])/(9*x^2 + 9*x^3 + (6*x^2 + 6*x^3)*Log[1 + x] + (x^2 + x^3)*Log[1 + x]^
2),x]

[Out]

-(1/(x*(3 + Log[1 + x])))

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fricas [A]  time = 0.70, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x \log \left (x + 1\right ) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(x+1)+3+4*x)/((x^3+x^2)*log(x+1)^2+(6*x^3+6*x^2)*log(x+1)+9*x^3+9*x^2),x, algorithm="frica
s")

[Out]

-1/(x*log(x + 1) + 3*x)

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giac [A]  time = 0.15, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x \log \left (x + 1\right ) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(x+1)+3+4*x)/((x^3+x^2)*log(x+1)^2+(6*x^3+6*x^2)*log(x+1)+9*x^3+9*x^2),x, algorithm="giac"
)

[Out]

-1/(x*log(x + 1) + 3*x)

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maple [A]  time = 0.04, size = 14, normalized size = 1.00

method	result	size
norman	\(-\frac {1}{x \left (3+\ln \left (x +1\right )\right )}\)	\(14\)
risch	\(-\frac {1}{x \left (3+\ln \left (x +1\right )\right )}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x+1)*ln(x+1)+3+4*x)/((x^3+x^2)*ln(x+1)^2+(6*x^3+6*x^2)*ln(x+1)+9*x^3+9*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/x/(3+ln(x+1))

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maxima [A]  time = 0.40, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x \log \left (x + 1\right ) + 3 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*log(x+1)+3+4*x)/((x^3+x^2)*log(x+1)^2+(6*x^3+6*x^2)*log(x+1)+9*x^3+9*x^2),x, algorithm="maxim
a")

[Out]

-1/(x*log(x + 1) + 3*x)

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mupad [B]  time = 2.88, size = 77, normalized size = 5.50 \begin {gather*} \frac {\frac {2\,x\,\ln \left (x+1\right )}{3}+\frac {4\,x^2\,\ln \left (x+1\right )}{3}+\frac {2\,x^3\,\ln \left (x+1\right )}{3}+3\,x^2+2\,x^3-1}{3\,x+x\,\ln \left (x+1\right )+2\,x^2\,\ln \left (x+1\right )+x^3\,\ln \left (x+1\right )+6\,x^2+3\,x^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + log(x + 1)*(x + 1) + 3)/(log(x + 1)^2*(x^2 + x^3) + log(x + 1)*(6*x^2 + 6*x^3) + 9*x^2 + 9*x^3),x)

[Out]

((2*x*log(x + 1))/3 + (4*x^2*log(x + 1))/3 + (2*x^3*log(x + 1))/3 + 3*x^2 + 2*x^3 - 1)/(3*x + x*log(x + 1) + 2
*x^2*log(x + 1) + x^3*log(x + 1) + 6*x^2 + 3*x^3)

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sympy [A]  time = 0.11, size = 12, normalized size = 0.86 \begin {gather*} - \frac {1}{x \log {\left (x + 1 \right )} + 3 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x+1)*ln(x+1)+3+4*x)/((x**3+x**2)*ln(x+1)**2+(6*x**3+6*x**2)*ln(x+1)+9*x**3+9*x**2),x)

[Out]

-1/(x*log(x + 1) + 3*x)

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