3.95.75 \(\int \frac {16+4 x^2+e^{2 x+2 x^3+2 x^2 \log (4+x^2)} (-8-26 x^2-4 x^3-6 x^4+(-16 x-4 x^3) \log (4+x^2))}{4+x^2} \, dx\)

Optimal. Leaf size=25 \[ -e^{2 x+2 x^2 \left (x+\log \left (4+x^2\right )\right )}+4 x \]

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Rubi [B]  time = 0.75, antiderivative size = 51, normalized size of antiderivative = 2.04, number of steps used = 3, number of rules used = 2, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {6725, 2288} \begin {gather*} 4 x-\frac {e^{2 x^3+2 x} \left (x^2+4\right )^{2 x^2-1} \left (3 x^4+13 x^2+4\right )}{3 x^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(16 + 4*x^2 + E^(2*x + 2*x^3 + 2*x^2*Log[4 + x^2])*(-8 - 26*x^2 - 4*x^3 - 6*x^4 + (-16*x - 4*x^3)*Log[4 +
x^2]))/(4 + x^2),x]

[Out]

4*x - (E^(2*x + 2*x^3)*(4 + x^2)^(-1 + 2*x^2)*(4 + 13*x^2 + 3*x^4))/(1 + 3*x^2)

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (4-2 e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+2 x^3+3 x^4+8 x \log \left (4+x^2\right )+2 x^3 \log \left (4+x^2\right )\right )\right ) \, dx\\ &=4 x-2 \int e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+2 x^3+3 x^4+8 x \log \left (4+x^2\right )+2 x^3 \log \left (4+x^2\right )\right ) \, dx\\ &=4 x-\frac {e^{2 x+2 x^3} \left (4+x^2\right )^{-1+2 x^2} \left (4+13 x^2+3 x^4\right )}{1+3 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.41, size = 26, normalized size = 1.04 \begin {gather*} 4 x-e^{2 \left (x+x^3\right )} \left (4+x^2\right )^{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16 + 4*x^2 + E^(2*x + 2*x^3 + 2*x^2*Log[4 + x^2])*(-8 - 26*x^2 - 4*x^3 - 6*x^4 + (-16*x - 4*x^3)*Lo
g[4 + x^2]))/(4 + x^2),x]

[Out]

4*x - E^(2*(x + x^3))*(4 + x^2)^(2*x^2)

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fricas [A]  time = 0.49, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3-16*x)*log(x^2+4)-6*x^4-4*x^3-26*x^2-8)*exp(x^2*log(x^2+4)+x^3+x)^2+4*x^2+16)/(x^2+4),x, al
gorithm="fricas")

[Out]

4*x - e^(2*x^3 + 2*x^2*log(x^2 + 4) + 2*x)

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giac [A]  time = 0.46, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3-16*x)*log(x^2+4)-6*x^4-4*x^3-26*x^2-8)*exp(x^2*log(x^2+4)+x^3+x)^2+4*x^2+16)/(x^2+4),x, al
gorithm="giac")

[Out]

4*x - e^(2*x^3 + 2*x^2*log(x^2 + 4) + 2*x)

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maple [A]  time = 0.59, size = 27, normalized size = 1.08

method	result	size
risch	\(-\left (x^{2}+4\right )^{2 x^{2}} {\mathrm e}^{2 x \left (x^{2}+1\right )}+4 x\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-4*x^3-16*x)*ln(x^2+4)-6*x^4-4*x^3-26*x^2-8)*exp(x^2*ln(x^2+4)+x^3+x)^2+4*x^2+16)/(x^2+4),x,method=_RET
URNVERBOSE)

[Out]

-((x^2+4)^(x^2))^2*exp(2*x*(x^2+1))+4*x

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maxima [A]  time = 0.51, size = 27, normalized size = 1.08 \begin {gather*} 4 \, x - e^{\left (2 \, x^{3} + 2 \, x^{2} \log \left (x^{2} + 4\right ) + 2 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x^3-16*x)*log(x^2+4)-6*x^4-4*x^3-26*x^2-8)*exp(x^2*log(x^2+4)+x^3+x)^2+4*x^2+16)/(x^2+4),x, al
gorithm="maxima")

[Out]

4*x - e^(2*x^3 + 2*x^2*log(x^2 + 4) + 2*x)

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mupad [B]  time = 8.41, size = 27, normalized size = 1.08 \begin {gather*} 4\,x-{\mathrm {e}}^{2\,x^3+2\,x}\,{\left (x^2+4\right )}^{2\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^2 - exp(2*x + 2*x^3 + 2*x^2*log(x^2 + 4))*(log(x^2 + 4)*(16*x + 4*x^3) + 26*x^2 + 4*x^3 + 6*x^4 + 8)
+ 16)/(x^2 + 4),x)

[Out]

4*x - exp(2*x + 2*x^3)*(x^2 + 4)^(2*x^2)

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sympy [A]  time = 0.39, size = 24, normalized size = 0.96 \begin {gather*} 4 x - e^{2 x^{3} + 2 x^{2} \log {\left (x^{2} + 4 \right )} + 2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-4*x**3-16*x)*ln(x**2+4)-6*x**4-4*x**3-26*x**2-8)*exp(x**2*ln(x**2+4)+x**3+x)**2+4*x**2+16)/(x**2
+4),x)

[Out]

4*x - exp(2*x**3 + 2*x**2*log(x**2 + 4) + 2*x)

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