\(\int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx\) [6837]

Optimal result

Integrand size = 27, antiderivative size = 22 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=\log \left (e^{\frac {16 (7+x) \left (x+x^2\right )}{x^2}} \left (1+x^2\right )\right ) \]

[Out]

ln(exp(16*(x+7)*(x^2+x)/x^2)*(x^2+1))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1607, 1816, 266} \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=\log \left (x^2+1\right )+16 x+\frac {112}{x} \]

[In]

Int[(-112 - 96*x^2 + 2*x^3 + 16*x^4)/(x^2 + x^4),x]

[Out]

112/x + 16*x + Log[1 + x^2]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 1607

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

Rule 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2 \left (1+x^2\right )} \, dx \\ & = \int \left (16-\frac {112}{x^2}+\frac {2 x}{1+x^2}\right ) \, dx \\ & = \frac {112}{x}+16 x+2 \int \frac {x}{1+x^2} \, dx \\ & = \frac {112}{x}+16 x+\log \left (1+x^2\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.95 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=2 \left (\frac {56}{x}+8 x+\frac {1}{2} \log \left (1+x^2\right )\right ) \]

[In]

Integrate[(-112 - 96*x^2 + 2*x^3 + 16*x^4)/(x^2 + x^4),x]

[Out]

2*(56/x + 8*x + Log[1 + x^2]/2)

Maple [A] (verified)

Time = 0.48 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73

method	result	size
default	\(16 x +\frac {112}{x}+\ln \left (x^{2}+1\right )\)	\(16\)
meijerg	\(16 x +\frac {112}{x}+\ln \left (x^{2}+1\right )\)	\(16\)
risch	\(16 x +\frac {112}{x}+\ln \left (x^{2}+1\right )\)	\(16\)
norman	\(\frac {16 x^{2}+112}{x}+\ln \left (x^{2}+1\right )\)	\(19\)
parallelrisch	\(\frac {\ln \left (x^{2}+1\right ) x +16 x^{2}+112}{x}\)	\(20\)

[In]

int((16*x^4+2*x^3-96*x^2-112)/(x^4+x^2),x,method=_RETURNVERBOSE)

[Out]

16*x+112/x+ln(x^2+1)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=\frac {16 \, x^{2} + x \log \left (x^{2} + 1\right ) + 112}{x} \]

[In]

integrate((16*x^4+2*x^3-96*x^2-112)/(x^4+x^2),x, algorithm="fricas")

[Out]

(16*x^2 + x*log(x^2 + 1) + 112)/x

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.55 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=16 x + \log {\left (x^{2} + 1 \right )} + \frac {112}{x} \]

[In]

integrate((16*x**4+2*x**3-96*x**2-112)/(x**4+x**2),x)

[Out]

16*x + log(x**2 + 1) + 112/x

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=16 \, x + \frac {112}{x} + \log \left (x^{2} + 1\right ) \]

[In]

integrate((16*x^4+2*x^3-96*x^2-112)/(x^4+x^2),x, algorithm="maxima")

[Out]

16*x + 112/x + log(x^2 + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=16 \, x + \frac {112}{x} + \log \left (x^{2} + 1\right ) \]

[In]

integrate((16*x^4+2*x^3-96*x^2-112)/(x^4+x^2),x, algorithm="giac")

[Out]

16*x + 112/x + log(x^2 + 1)

Mupad [B] (verification not implemented)

Time = 12.68 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \frac {-112-96 x^2+2 x^3+16 x^4}{x^2+x^4} \, dx=16\,x+\ln \left (x^2+1\right )+\frac {112}{x} \]

[In]

int(-(96*x^2 - 2*x^3 - 16*x^4 + 112)/(x^2 + x^4),x)

[Out]

16*x + log(x^2 + 1) + 112/x