Integrand size = 26, antiderivative size = 117 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\frac {1}{b x}+\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt {b}}\right )}{2 \sqrt [4]{2} b^{3/2}}-\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt {b}}\right )}{2 \sqrt [4]{2} b^{3/2}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^2}{\sqrt {2} b}\right )}{2 \sqrt {2} b} \] Output:
1/b/x+1/4*a^(1/4)*arctan(1/2*a^(1/4)*x*2^(3/4)/b^(1/2))*2^(3/4)/b^(3/2)-1/ 4*a^(1/4)*arctanh(1/2*a^(1/4)*x*2^(3/4)/b^(1/2))*2^(3/4)/b^(3/2)-1/4*a^(1/ 2)*arctanh(1/2*a^(1/2)*x^2*2^(1/2)/b)*2^(1/2)/b
Time = 0.10 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.59 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\frac {\frac {16 b}{x}+4\ 2^{3/4} \sqrt [4]{a} \sqrt {b} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt {b}}\right )+2^{3/4} \sqrt [4]{a} \left (2 \sqrt {b}+2^{3/4} \sqrt [4]{a} b\right ) \log \left (2 \sqrt {b}-2^{3/4} \sqrt [4]{a} x\right )+2^{3/4} \sqrt [4]{a} \left (-2 \sqrt {b}+2^{3/4} \sqrt [4]{a} b\right ) \log \left (2 \sqrt {b}+2^{3/4} \sqrt [4]{a} x\right )-2 \sqrt {2} \sqrt {a} b \log \left (2 b+\sqrt {2} \sqrt {a} x^2\right )}{16 b^2} \] Input:
Integrate[(2*b + a*x^3)/(-2*b^2*x^2 + a*x^6),x]
Output:
((16*b)/x + 4*2^(3/4)*a^(1/4)*Sqrt[b]*ArcTan[(a^(1/4)*x)/(2^(1/4)*Sqrt[b]) ] + 2^(3/4)*a^(1/4)*(2*Sqrt[b] + 2^(3/4)*a^(1/4)*b)*Log[2*Sqrt[b] - 2^(3/4 )*a^(1/4)*x] + 2^(3/4)*a^(1/4)*(-2*Sqrt[b] + 2^(3/4)*a^(1/4)*b)*Log[2*Sqrt [b] + 2^(3/4)*a^(1/4)*x] - 2*Sqrt[2]*Sqrt[a]*b*Log[2*b + Sqrt[2]*Sqrt[a]*x ^2])/(16*b^2)
Time = 0.37 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2026, 2370, 2009}
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.
\(\displaystyle \int \frac {a x^3+2 b}{a x^6-2 b^2 x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {a x^3+2 b}{x^2 \left (a x^4-2 b^2\right )}dx\) |
\(\Big \downarrow \) 2370 |
\(\displaystyle \int \left (\frac {a x}{a x^4-2 b^2}+\frac {2 b}{x^2 \left (a x^4-2 b^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt {b}}\right )}{2 \sqrt [4]{2} b^{3/2}}-\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{2} \sqrt {b}}\right )}{2 \sqrt [4]{2} b^{3/2}}-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} x^2}{\sqrt {2} b}\right )}{2 \sqrt {2} b}+\frac {1}{b x}\) |
Input:
Int[(2*b + a*x^3)/(-2*b^2*x^2 + a*x^6),x]
Output:
1/(b*x) + (a^(1/4)*ArcTan[(a^(1/4)*x)/(2^(1/4)*Sqrt[b])])/(2*2^(1/4)*b^(3/ 2)) - (a^(1/4)*ArcTanh[(a^(1/4)*x)/(2^(1/4)*Sqrt[b])])/(2*2^(1/4)*b^(3/2)) - (Sqrt[a]*ArcTanh[(Sqrt[a]*x^2)/(Sqrt[2]*b)])/(2*Sqrt[2]*b)
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[ {v = Sum[(c*x)^(m + ii)*((Coeff[Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2) )/(c^ii*(a + b*x^n))), {ii, 0, n/2 - 1}]}, Int[v, x] /; SumQ[v]] /; FreeQ[{ a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91
method | result | size |
risch | \(\frac {1}{b x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (4 b^{6} \textit {\_Z}^{4}-4 a \,b^{4} \textit {\_Z}^{2}-8 a \,b^{2} \textit {\_Z} +a^{2} b^{2}-2 a \right )}{\sum }\textit {\_R} \ln \left (\left (-10 \textit {\_R}^{4} b^{6}+9 \textit {\_R}^{2} a \,b^{4}+17 \textit {\_R} a \,b^{2}-2 a^{2} b^{2}+4 a \right ) x +2 b^{5} \textit {\_R}^{3}+a \,b^{3} \textit {\_R} \right )\right )}{4}\) | \(106\) |
default | \(\frac {1}{b x}+\frac {a \left (\frac {b \sqrt {2}\, \ln \left (\frac {-2 b^{2}+x^{2} \sqrt {2}\, \sqrt {a \,b^{2}}}{-2 b^{2}-x^{2} \sqrt {2}\, \sqrt {a \,b^{2}}}\right )}{8 \sqrt {a \,b^{2}}}+\frac {2 \arctan \left (\frac {x}{\sqrt {\sqrt {2}\, \sqrt {\frac {b^{2}}{a}}}}\right )-\ln \left (\frac {x +\sqrt {\sqrt {2}\, \sqrt {\frac {b^{2}}{a}}}}{x -\sqrt {\sqrt {2}\, \sqrt {\frac {b^{2}}{a}}}}\right )}{4 a \sqrt {\sqrt {2}\, \sqrt {\frac {b^{2}}{a}}}}\right )}{b}\) | \(156\) |
Input:
int((a*x^3+2*b)/(a*x^6-2*b^2*x^2),x,method=_RETURNVERBOSE)
Output:
1/b/x+1/4*sum(_R*ln((-10*_R^4*b^6+9*_R^2*a*b^4+17*_R*a*b^2-2*a^2*b^2+4*a)* x+2*b^5*_R^3+a*b^3*_R),_R=RootOf(4*_Z^4*b^6-4*_Z^2*a*b^4-8*_Z*a*b^2+a^2*b^ 2-2*a))
Result contains complex when optimal does not.
Time = 0.82 (sec) , antiderivative size = 17307, normalized size of antiderivative = 147.92 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\text {Too large to display} \] Input:
integrate((a*x^3+2*b)/(a*x^6-2*b^2*x^2),x, algorithm="fricas")
Output:
Too large to include
Time = 0.38 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.81 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\operatorname {RootSum} {\left (1024 t^{4} b^{6} - 64 t^{2} a b^{4} - 32 t a b^{2} + a^{2} b^{2} - 2 a, \left ( t \mapsto t \log {\left (x + \frac {- 128 t^{3} b^{5} - 32 t^{2} a b^{5} + 12 t a b^{3} + a^{2} b^{3} + 3 a b}{2 a^{2} b^{2} + a} \right )} \right )\right )} + \frac {1}{b x} \] Input:
integrate((a*x**3+2*b)/(a*x**6-2*b**2*x**2),x)
Output:
RootSum(1024*_t**4*b**6 - 64*_t**2*a*b**4 - 32*_t*a*b**2 + a**2*b**2 - 2*a , Lambda(_t, _t*log(x + (-128*_t**3*b**5 - 32*_t**2*a*b**5 + 12*_t*a*b**3 + a**2*b**3 + 3*a*b)/(2*a**2*b**2 + a)))) + 1/(b*x)
Time = 0.10 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.23 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=-\frac {a {\left (\frac {\sqrt {2} \log \left (\sqrt {a} x^{2} + \sqrt {2} b\right )}{\sqrt {a}} - \frac {\sqrt {2} \log \left (\sqrt {a} x^{2} - \sqrt {2} b\right )}{\sqrt {a}} - \frac {2 \cdot 2^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {a} x}{2 \, \sqrt {\sqrt {a} b}}\right )}{\sqrt {\sqrt {a} b} \sqrt {a}} - \frac {2^{\frac {3}{4}} \log \left (\frac {\sqrt {a} x - \sqrt {\sqrt {2} \sqrt {a} b}}{\sqrt {a} x + \sqrt {\sqrt {2} \sqrt {a} b}}\right )}{\sqrt {\sqrt {a} b} \sqrt {a}}\right )}}{8 \, b} + \frac {1}{b x} \] Input:
integrate((a*x^3+2*b)/(a*x^6-2*b^2*x^2),x, algorithm="maxima")
Output:
-1/8*a*(sqrt(2)*log(sqrt(a)*x^2 + sqrt(2)*b)/sqrt(a) - sqrt(2)*log(sqrt(a) *x^2 - sqrt(2)*b)/sqrt(a) - 2*2^(3/4)*arctan(1/2*2^(3/4)*sqrt(a)*x/sqrt(sq rt(a)*b))/(sqrt(sqrt(a)*b)*sqrt(a)) - 2^(3/4)*log((sqrt(a)*x - sqrt(sqrt(2 )*sqrt(a)*b))/(sqrt(a)*x + sqrt(sqrt(2)*sqrt(a)*b)))/(sqrt(sqrt(a)*b)*sqrt (a)))/b + 1/(b*x)
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (82) = 164\).
Time = 0.13 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.29 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\frac {1}{b x} + \frac {\sqrt {2} {\left (2 \, \sqrt {-a^{3} b^{2}} a b - 2^{\frac {3}{4}} \left (-a^{3} b^{2}\right )^{\frac {3}{4}}\right )} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}} + 2 \, x\right )}}{2 \, \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3}} - \frac {\sqrt {2} {\left (2 \, \sqrt {-a^{3} b^{2}} a b + 2^{\frac {3}{4}} \left (-a^{3} b^{2}\right )^{\frac {3}{4}}\right )} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}} - 2 \, x\right )}}{2 \, \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}}}\right )}{8 \, a^{2} b^{3}} + \frac {2^{\frac {1}{4}} \left (-a^{3} b^{2}\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}} x + x^{2} + \sqrt {2} \sqrt {-\frac {b^{2}}{a}}\right )}{8 \, a^{2} b^{3}} - \frac {2^{\frac {1}{4}} \left (-a^{3} b^{2}\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-\frac {b^{2}}{a}\right )^{\frac {1}{4}} x + x^{2} + \sqrt {2} \sqrt {-\frac {b^{2}}{a}}\right )}{8 \, a^{2} b^{3}} \] Input:
integrate((a*x^3+2*b)/(a*x^6-2*b^2*x^2),x, algorithm="giac")
Output:
1/(b*x) + 1/8*sqrt(2)*(2*sqrt(-a^3*b^2)*a*b - 2^(3/4)*(-a^3*b^2)^(3/4))*ar ctan(1/2*2^(1/4)*(2^(3/4)*(-b^2/a)^(1/4) + 2*x)/(-b^2/a)^(1/4))/(a^2*b^3) - 1/8*sqrt(2)*(2*sqrt(-a^3*b^2)*a*b + 2^(3/4)*(-a^3*b^2)^(3/4))*arctan(-1/ 2*2^(1/4)*(2^(3/4)*(-b^2/a)^(1/4) - 2*x)/(-b^2/a)^(1/4))/(a^2*b^3) + 1/8*2 ^(1/4)*(-a^3*b^2)^(3/4)*log(2^(3/4)*(-b^2/a)^(1/4)*x + x^2 + sqrt(2)*sqrt( -b^2/a))/(a^2*b^3) - 1/8*2^(1/4)*(-a^3*b^2)^(3/4)*log(-2^(3/4)*(-b^2/a)^(1 /4)*x + x^2 + sqrt(2)*sqrt(-b^2/a))/(a^2*b^3)
Time = 0.26 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.68 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\left (\sum _{k=1}^4\ln \left (a^4\,b^4\,\left (\mathrm {root}\left (1024\,b^6\,z^4-64\,a\,b^4\,z^2-32\,a\,b^2\,z+a^2\,b^2-2\,a,z,k\right )\,b^2\,16-{\mathrm {root}\left (1024\,b^6\,z^4-64\,a\,b^4\,z^2-32\,a\,b^2\,z+a^2\,b^2-2\,a,z,k\right )}^2\,b^3\,x\,32-\mathrm {root}\left (1024\,b^6\,z^4-64\,a\,b^4\,z^2-32\,a\,b^2\,z+a^2\,b^2-2\,a,z,k\right )\,b\,x\,8+a\,b\,x+2\right )\right )\,\mathrm {root}\left (1024\,b^6\,z^4-64\,a\,b^4\,z^2-32\,a\,b^2\,z+a^2\,b^2-2\,a,z,k\right )\right )+\frac {1}{b\,x} \] Input:
int((2*b + a*x^3)/(a*x^6 - 2*b^2*x^2),x)
Output:
symsum(log(a^4*b^4*(16*root(1024*b^6*z^4 - 64*a*b^4*z^2 - 32*a*b^2*z + a^2 *b^2 - 2*a, z, k)*b^2 - 32*root(1024*b^6*z^4 - 64*a*b^4*z^2 - 32*a*b^2*z + a^2*b^2 - 2*a, z, k)^2*b^3*x - 8*root(1024*b^6*z^4 - 64*a*b^4*z^2 - 32*a* b^2*z + a^2*b^2 - 2*a, z, k)*b*x + a*b*x + 2))*root(1024*b^6*z^4 - 64*a*b^ 4*z^2 - 32*a*b^2*z + a^2*b^2 - 2*a, z, k), k, 1, 4) + 1/(b*x)
Time = 0.17 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.32 \[ \int \frac {2 b+a x^3}{-2 b^2 x^2+a x^6} \, dx=\frac {2 \sqrt {b}\, a^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {a}\, x 2^{\frac {3}{4}}}{2 \sqrt {b}\, a^{\frac {1}{4}}}\right ) x +\sqrt {b}\, a^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}} x -\sqrt {b}\, 2^{\frac {1}{4}}\right ) x -\sqrt {b}\, a^{\frac {1}{4}} \sqrt {2}\, 2^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}} x +\sqrt {b}\, 2^{\frac {1}{4}}\right ) x +\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (a^{\frac {1}{4}} x -\sqrt {b}\, 2^{\frac {1}{4}}\right ) b x +\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (a^{\frac {1}{4}} x +\sqrt {b}\, 2^{\frac {1}{4}}\right ) b x -\sqrt {a}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {a}\, x^{2}+\sqrt {2}\, b \right ) b x +8 b}{8 b^{2} x} \] Input:
int((a*x^3+2*b)/(a*x^6-2*b^2*x^2),x)
Output:
(2*sqrt(b)*a**(1/4)*sqrt(2)*2**(1/4)*atan((sqrt(a)*x)/(sqrt(b)*a**(1/4)*2* *(1/4)))*x + sqrt(b)*a**(1/4)*sqrt(2)*2**(1/4)*log(a**(1/4)*x - sqrt(b)*2* *(1/4))*x - sqrt(b)*a**(1/4)*sqrt(2)*2**(1/4)*log(a**(1/4)*x + sqrt(b)*2** (1/4))*x + sqrt(a)*sqrt(2)*log(a**(1/4)*x - sqrt(b)*2**(1/4))*b*x + sqrt(a )*sqrt(2)*log(a**(1/4)*x + sqrt(b)*2**(1/4))*b*x - sqrt(a)*sqrt(2)*log(sqr t(a)*x**2 + sqrt(2)*b)*b*x + 8*b)/(8*b**2*x)