Integrand size = 22, antiderivative size = 56 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {\log \left (a-b+2 a x^2+a x^4\right )}{4 a} \] Output:
1/2*arctanh(a^(1/2)*(x^2+1)/b^(1/2))/a^(1/2)/b^(1/2)+1/4*ln(a*x^4+2*a*x^2+ a-b)/a
Time = 0.01 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.91 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\frac {\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \left (1+x^2\right )}{\sqrt {b}}\right )}{\sqrt {b}}+\log \left (-b+a \left (1+x^2\right )^2\right )}{4 a} \] Input:
Integrate[x^3/(a - b + 2*a*x^2 + a*x^4),x]
Output:
((2*Sqrt[a]*ArcTanh[(Sqrt[a]*(1 + x^2))/Sqrt[b]])/Sqrt[b] + Log[-b + a*(1 + x^2)^2])/(4*a)
Time = 0.38 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.16, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {1434, 1142, 27, 1083, 219, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^3}{a x^4+2 a x^2+a-b} \, dx\) |
\(\Big \downarrow \) 1434 |
\(\displaystyle \frac {1}{2} \int \frac {x^2}{a x^4+2 a x^2+a-b}dx^2\) |
\(\Big \downarrow \) 1142 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {2 a \left (x^2+1\right )}{a x^4+2 a x^2+a-b}dx^2}{2 a}-\int \frac {1}{a x^4+2 a x^2+a-b}dx^2\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2-\int \frac {1}{a x^4+2 a x^2+a-b}dx^2\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{2} \left (2 \int \frac {1}{4 a b-x^4}d\left (2 a x^2+2 a\right )+\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\int \frac {x^2+1}{a x^4+2 a x^2+a-b}dx^2+\frac {\text {arctanh}\left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {2 a x^2+2 a}{2 \sqrt {a} \sqrt {b}}\right )}{\sqrt {a} \sqrt {b}}+\frac {\log \left (a x^4+2 a x^2+a-b\right )}{2 a}\right )\) |
Input:
Int[x^3/(a - b + 2*a*x^2 + a*x^4),x]
Output:
(ArcTanh[(2*a + 2*a*x^2)/(2*Sqrt[a]*Sqrt[b])]/(Sqrt[a]*Sqrt[b]) + Log[a - b + 2*a*x^2 + a*x^4]/(2*a))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp [1/2 Subst[Int[x^((m - 1)/2)*(a + b*x + c*x^2)^p, x], x, x^2], x] /; Free Q[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {\ln \left (a \,x^{4}+2 a \,x^{2}+a -b \right )}{4 a}+\frac {\operatorname {arctanh}\left (\frac {2 a \,x^{2}+2 a}{2 \sqrt {a b}}\right )}{2 \sqrt {a b}}\) | \(49\) |
risch | \(\frac {\ln \left (\left (\sqrt {a b}\, a -a b \right ) x^{2}+\sqrt {a b}\, a -\sqrt {a b}\, b \right )}{4 a}+\frac {\ln \left (\left (\sqrt {a b}\, a -a b \right ) x^{2}+\sqrt {a b}\, a -\sqrt {a b}\, b \right ) \sqrt {a b}}{4 b a}+\frac {\ln \left (\left (-\sqrt {a b}\, a -a b \right ) x^{2}-\sqrt {a b}\, a +\sqrt {a b}\, b \right )}{4 a}-\frac {\ln \left (\left (-\sqrt {a b}\, a -a b \right ) x^{2}-\sqrt {a b}\, a +\sqrt {a b}\, b \right ) \sqrt {a b}}{4 b a}\) | \(172\) |
Input:
int(x^3/(a*x^4+2*a*x^2+a-b),x,method=_RETURNVERBOSE)
Output:
1/4*ln(a*x^4+2*a*x^2+a-b)/a+1/2/(a*b)^(1/2)*arctanh(1/2*(2*a*x^2+2*a)/(a*b )^(1/2))
Time = 0.08 (sec) , antiderivative size = 134, normalized size of antiderivative = 2.39 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\left [\frac {b \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) + \sqrt {a b} \log \left (\frac {a x^{4} + 2 \, a x^{2} + 2 \, \sqrt {a b} {\left (x^{2} + 1\right )} + a + b}{a x^{4} + 2 \, a x^{2} + a - b}\right )}{4 \, a b}, \frac {b \log \left (a x^{4} + 2 \, a x^{2} + a - b\right ) - 2 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{a x^{2} + a}\right )}{4 \, a b}\right ] \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="fricas")
Output:
[1/4*(b*log(a*x^4 + 2*a*x^2 + a - b) + sqrt(a*b)*log((a*x^4 + 2*a*x^2 + 2* sqrt(a*b)*(x^2 + 1) + a + b)/(a*x^4 + 2*a*x^2 + a - b)))/(a*b), 1/4*(b*log (a*x^4 + 2*a*x^2 + a - b) - 2*sqrt(-a*b)*arctan(sqrt(-a*b)/(a*x^2 + a)))/( a*b)]
Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (48) = 96\).
Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.96 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\left (\frac {1}{4 a} - \frac {\sqrt {a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (\frac {1}{4 a} - \frac {\sqrt {a^{3} b}}{4 a^{2} b}\right ) + a - b}{a} \right )} + \left (\frac {1}{4 a} + \frac {\sqrt {a^{3} b}}{4 a^{2} b}\right ) \log {\left (x^{2} + \frac {4 a b \left (\frac {1}{4 a} + \frac {\sqrt {a^{3} b}}{4 a^{2} b}\right ) + a - b}{a} \right )} \] Input:
integrate(x**3/(a*x**4+2*a*x**2+a-b),x)
Output:
(1/(4*a) - sqrt(a**3*b)/(4*a**2*b))*log(x**2 + (4*a*b*(1/(4*a) - sqrt(a**3 *b)/(4*a**2*b)) + a - b)/a) + (1/(4*a) + sqrt(a**3*b)/(4*a**2*b))*log(x**2 + (4*a*b*(1/(4*a) + sqrt(a**3*b)/(4*a**2*b)) + a - b)/a)
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=-\frac {\log \left (\frac {a x^{2} + a - \sqrt {a b}}{a x^{2} + a + \sqrt {a b}}\right )}{4 \, \sqrt {a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, a} \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="maxima")
Output:
-1/4*log((a*x^2 + a - sqrt(a*b))/(a*x^2 + a + sqrt(a*b)))/sqrt(a*b) + 1/4* log(a*x^4 + 2*a*x^2 + a - b)/a
Time = 0.16 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=-\frac {\arctan \left (\frac {a x^{2} + a}{\sqrt {-a b}}\right )}{2 \, \sqrt {-a b}} + \frac {\log \left (a x^{4} + 2 \, a x^{2} + a - b\right )}{4 \, a} \] Input:
integrate(x^3/(a*x^4+2*a*x^2+a-b),x, algorithm="giac")
Output:
-1/2*arctan((a*x^2 + a)/sqrt(-a*b))/sqrt(-a*b) + 1/4*log(a*x^4 + 2*a*x^2 + a - b)/a
Time = 0.18 (sec) , antiderivative size = 153, normalized size of antiderivative = 2.73 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\frac {\ln \left (x^2\,\sqrt {a^3\,b}+a\,b-a^2-a^2\,x^2\right )}{4\,a}+\frac {\ln \left (x^2\,\sqrt {a^3\,b}-a\,b+a^2+a^2\,x^2\right )}{4\,a}-\frac {\ln \left (x^2\,\sqrt {a^3\,b}-a\,b+a^2+a^2\,x^2\right )\,\sqrt {a^3\,b}}{4\,a^2\,b}+\frac {\ln \left (x^2\,\sqrt {a^3\,b}+a\,b-a^2-a^2\,x^2\right )\,\sqrt {a^3\,b}}{4\,a^2\,b} \] Input:
int(x^3/(a - b + 2*a*x^2 + a*x^4),x)
Output:
log(x^2*(a^3*b)^(1/2) + a*b - a^2 - a^2*x^2)/(4*a) + log(x^2*(a^3*b)^(1/2) - a*b + a^2 + a^2*x^2)/(4*a) - (log(x^2*(a^3*b)^(1/2) - a*b + a^2 + a^2*x ^2)*(a^3*b)^(1/2))/(4*a^2*b) + (log(x^2*(a^3*b)^(1/2) + a*b - a^2 - a^2*x^ 2)*(a^3*b)^(1/2))/(4*a^2*b)
Time = 0.16 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.25 \[ \int \frac {x^3}{a-b+2 a x^2+a x^4} \, dx=\frac {-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )-\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right )+\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right )+\mathrm {log}\left (-\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b +\mathrm {log}\left (\sqrt {\sqrt {b}\, \sqrt {a}-a}+\sqrt {a}\, x \right ) b +\mathrm {log}\left (\sqrt {b}\, \sqrt {a}+a \,x^{2}+a \right ) b}{4 a b} \] Input:
int(x^3/(a*x^4+2*a*x^2+a-b),x)
Output:
( - sqrt(b)*sqrt(a)*log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) - sqrt(b )*sqrt(a)*log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x) + sqrt(b)*sqrt(a)*log (sqrt(b)*sqrt(a) + a*x**2 + a) + log( - sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a )*x)*b + log(sqrt(sqrt(b)*sqrt(a) - a) + sqrt(a)*x)*b + log(sqrt(b)*sqrt(a ) + a*x**2 + a)*b)/(4*a*b)