Integrand size = 22, antiderivative size = 85 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=-\frac {1}{x}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {a \log (x)}{b}+\frac {a \log \left (b-a x^4\right )}{4 b} \] Output:
-1/x-1/2*a^(1/4)*arctan(a^(1/4)*x/b^(1/4))/b^(1/4)+1/2*a^(1/4)*arctanh(a^( 1/4)*x/b^(1/4))/b^(1/4)-a*ln(x)/b+1/4*a*ln(-a*x^4+b)/b
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.35 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=-\frac {1}{x}-\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}-\frac {a \log (x)}{b}-\frac {\sqrt [4]{a} \log \left (\sqrt [4]{b}-\sqrt [4]{a} x\right )}{4 \sqrt [4]{b}}+\frac {\sqrt [4]{a} \log \left (\sqrt [4]{b}+\sqrt [4]{a} x\right )}{4 \sqrt [4]{b}}+\frac {a \log \left (b-a x^4\right )}{4 b} \] Input:
Integrate[(-b + a*x)/(-(b*x^2) + a*x^6),x]
Output:
-x^(-1) - (a^(1/4)*ArcTan[(a^(1/4)*x)/b^(1/4)])/(2*b^(1/4)) - (a*Log[x])/b - (a^(1/4)*Log[b^(1/4) - a^(1/4)*x])/(4*b^(1/4)) + (a^(1/4)*Log[b^(1/4) + a^(1/4)*x])/(4*b^(1/4)) + (a*Log[b - a*x^4])/(4*b)
Time = 0.31 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, 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.
\(\displaystyle \int \frac {a x-b}{a x^6-b x^2} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {a x-b}{x^2 \left (a x^4-b\right )}dx\) |
\(\Big \downarrow \) 2370 |
\(\displaystyle \int \left (\frac {a}{x \left (a x^4-b\right )}-\frac {b}{x^2 \left (a x^4-b\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\sqrt [4]{a} \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}+\frac {\sqrt [4]{a} \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt [4]{b}}+\frac {a \log \left (b-a x^4\right )}{4 b}-\frac {a \log (x)}{b}-\frac {1}{x}\) |
Input:
Int[(-b + a*x)/(-(b*x^2) + a*x^6),x]
Output:
-x^(-1) - (a^(1/4)*ArcTan[(a^(1/4)*x)/b^(1/4)])/(2*b^(1/4)) + (a^(1/4)*Arc Tanh[(a^(1/4)*x)/b^(1/4)])/(2*b^(1/4)) - (a*Log[x])/b + (a*Log[b - a*x^4]) /(4*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
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {1}{x}-\frac {a \ln \left (x \right )}{b}+\frac {a \left (-\frac {b \left (2 \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {b}{a}\right )^{\frac {1}{4}}}{x -\left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )\right )}{4 a \left (\frac {b}{a}\right )^{\frac {1}{4}}}+\frac {\ln \left (a \,x^{4}-b \right )}{4}\right )}{b}\) | \(85\) |
risch | \(-\frac {1}{x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} \textit {\_Z}^{4}-4 a \,b^{3} \textit {\_Z}^{3}+6 a^{2} b^{2} \textit {\_Z}^{2}-4 a^{3} b \textit {\_Z} +a^{4}-a \,b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4} b^{3}+15 \textit {\_R}^{3} a \,b^{2}-15 \textit {\_R}^{2} a^{2} b +5 \textit {\_R} \,a^{3}+4 a \,b^{2}\right ) x -b^{3} \textit {\_R}^{3}-2 a \,b^{2} \textit {\_R}^{2}+7 a^{2} b \textit {\_R} -4 a^{3}\right )\right )}{4}-\frac {a \ln \left (-x \right )}{b}\) | \(141\) |
Input:
int((a*x-b)/(a*x^6-b*x^2),x,method=_RETURNVERBOSE)
Output:
-1/x-a*ln(x)/b+a/b*(-1/4*b/a/(b/a)^(1/4)*(2*arctan(x/(b/a)^(1/4))-ln((x+(b /a)^(1/4))/(x-(b/a)^(1/4))))+1/4*ln(a*x^4-b))
Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (63) = 126\).
Time = 0.08 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.00 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=-\frac {4 \, a x \log \left (x\right ) + {\left (b x \sqrt {-\sqrt {\frac {a}{b}}} - a x\right )} \log \left (a x + b \sqrt {-\sqrt {\frac {a}{b}}} \sqrt {\frac {a}{b}}\right ) - {\left (b x \sqrt {-\sqrt {\frac {a}{b}}} + a x\right )} \log \left (a x - b \sqrt {-\sqrt {\frac {a}{b}}} \sqrt {\frac {a}{b}}\right ) - {\left (b x \left (\frac {a}{b}\right )^{\frac {1}{4}} + a x\right )} \log \left (a x + b \left (\frac {a}{b}\right )^{\frac {3}{4}}\right ) + {\left (b x \left (\frac {a}{b}\right )^{\frac {1}{4}} - a x\right )} \log \left (a x - b \left (\frac {a}{b}\right )^{\frac {3}{4}}\right ) + 4 \, b}{4 \, b x} \] Input:
integrate((a*x-b)/(a*x^6-b*x^2),x, algorithm="fricas")
Output:
-1/4*(4*a*x*log(x) + (b*x*sqrt(-sqrt(a/b)) - a*x)*log(a*x + b*sqrt(-sqrt(a /b))*sqrt(a/b)) - (b*x*sqrt(-sqrt(a/b)) + a*x)*log(a*x - b*sqrt(-sqrt(a/b) )*sqrt(a/b)) - (b*x*(a/b)^(1/4) + a*x)*log(a*x + b*(a/b)^(3/4)) + (b*x*(a/ b)^(1/4) - a*x)*log(a*x - b*(a/b)^(3/4)) + 4*b)/(b*x)
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (73) = 146\).
Time = 1.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.91 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=- \frac {a \log {\left (x \right )}}{b} + \operatorname {RootSum} {\left (256 t^{4} b^{4} - 256 t^{3} a b^{3} + 96 t^{2} a^{2} b^{2} - 16 t a^{3} b + a^{4} - a b^{3}, \left ( t \mapsto t \log {\left (x + \frac {32000 t^{4} a^{2} b^{4} + 8000 t^{3} a^{3} b^{3} - 64 t^{3} b^{6} - 18000 t^{2} a^{4} b^{2} + 48 t^{2} a b^{5} + 5500 t a^{5} b - 12 t a^{2} b^{4} - 500 a^{6} - 124 a^{3} b^{3}}{625 a^{4} b^{2} - a b^{5}} \right )} \right )\right )} - \frac {1}{x} \] Input:
integrate((a*x-b)/(a*x**6-b*x**2),x)
Output:
-a*log(x)/b + RootSum(256*_t**4*b**4 - 256*_t**3*a*b**3 + 96*_t**2*a**2*b* *2 - 16*_t*a**3*b + a**4 - a*b**3, Lambda(_t, _t*log(x + (32000*_t**4*a**2 *b**4 + 8000*_t**3*a**3*b**3 - 64*_t**3*b**6 - 18000*_t**2*a**4*b**2 + 48* _t**2*a*b**5 + 5500*_t*a**5*b - 12*_t*a**2*b**4 - 500*a**6 - 124*a**3*b**3 )/(625*a**4*b**2 - a*b**5)))) - 1/x
Leaf count of result is larger than twice the leaf count of optimal. 131 vs. \(2 (63) = 126\).
Time = 0.10 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.54 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=-\frac {a {\left (\frac {2 \, b \arctan \left (\frac {\sqrt {a} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {b \log \left (\frac {\sqrt {a} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {a} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} - \log \left (\sqrt {a} x^{2} + \sqrt {b}\right ) - \log \left (\sqrt {a} x^{2} - \sqrt {b}\right )\right )}}{4 \, b} - \frac {a \log \left (x\right )}{b} - \frac {1}{x} \] Input:
integrate((a*x-b)/(a*x^6-b*x^2),x, algorithm="maxima")
Output:
-1/4*a*(2*b*arctan(sqrt(a)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)* sqrt(b))) + b*log((sqrt(a)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*x + sqrt(sq rt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) - log(sqrt(a)*x^2 + sqrt( b)) - log(sqrt(a)*x^2 - sqrt(b)))/b - a*log(x)/b - 1/x
Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (63) = 126\).
Time = 0.16 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.61 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=\frac {a \log \left ({\left | a x^{4} - b \right |}\right )}{4 \, b} - \frac {a \log \left ({\left | x \right |}\right )}{b} + \frac {\sqrt {2} \left (-a^{3} b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2} b} + \frac {\sqrt {2} \left (-a^{3} b\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2} b} - \frac {\sqrt {2} \left (-a^{3} b\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (-\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {-\frac {b}{a}}\right )}{8 \, a^{2} b} + \frac {\sqrt {2} \left (-a^{3} b\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (-\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {-\frac {b}{a}}\right )}{8 \, a^{2} b} - \frac {1}{x} \] Input:
integrate((a*x-b)/(a*x^6-b*x^2),x, algorithm="giac")
Output:
1/4*a*log(abs(a*x^4 - b))/b - a*log(abs(x))/b + 1/4*sqrt(2)*(-a^3*b)^(3/4) *arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-b/a)^(1/4))/(-b/a)^(1/4))/(a^2*b) + 1 /4*sqrt(2)*(-a^3*b)^(3/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-b/a)^(1/4))/ (-b/a)^(1/4))/(a^2*b) - 1/8*sqrt(2)*(-a^3*b)^(3/4)*log(x^2 + sqrt(2)*x*(-b /a)^(1/4) + sqrt(-b/a))/(a^2*b) + 1/8*sqrt(2)*(-a^3*b)^(3/4)*log(x^2 - sqr t(2)*x*(-b/a)^(1/4) + sqrt(-b/a))/(a^2*b) - 1/x
Time = 0.99 (sec) , antiderivative size = 306, normalized size of antiderivative = 3.60 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=\frac {\ln \left (b\,{\left (\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}+5\,a\,\sqrt {a\,b^5}+5\,a\,x\,{\left (\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}+a\,b^3\,x\right )\,\left (a+\sqrt {\frac {\sqrt {a\,b^5}}{b}}\right )}{4\,b}-\frac {a\,\ln \left (x\right )}{b}-\frac {1}{x}+\frac {\ln \left (b\,{\left (-\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}-5\,a\,\sqrt {a\,b^5}+5\,a\,x\,{\left (-\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}+a\,b^3\,x\right )\,\left (a+\sqrt {-\frac {\sqrt {a\,b^5}}{b}}\right )}{4\,b}+\frac {\ln \left (b\,{\left (\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}-5\,a\,\sqrt {a\,b^5}+5\,a\,x\,{\left (\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}-a\,b^3\,x\right )\,\left (a-\sqrt {\frac {\sqrt {a\,b^5}}{b}}\right )}{4\,b}+\frac {\ln \left (b\,{\left (-\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}+5\,a\,\sqrt {a\,b^5}+5\,a\,x\,{\left (-\frac {\sqrt {a\,b^5}}{b}\right )}^{3/2}-a\,b^3\,x\right )\,\left (a-\sqrt {-\frac {\sqrt {a\,b^5}}{b}}\right )}{4\,b} \] Input:
int(-(b - a*x)/(a*x^6 - b*x^2),x)
Output:
(log(b*((a*b^5)^(1/2)/b)^(3/2) + 5*a*(a*b^5)^(1/2) + 5*a*x*((a*b^5)^(1/2)/ b)^(3/2) + a*b^3*x)*(a + ((a*b^5)^(1/2)/b)^(1/2)))/(4*b) - (a*log(x))/b - 1/x + (log(b*(-(a*b^5)^(1/2)/b)^(3/2) - 5*a*(a*b^5)^(1/2) + 5*a*x*(-(a*b^5 )^(1/2)/b)^(3/2) + a*b^3*x)*(a + (-(a*b^5)^(1/2)/b)^(1/2)))/(4*b) + (log(b *((a*b^5)^(1/2)/b)^(3/2) - 5*a*(a*b^5)^(1/2) + 5*a*x*((a*b^5)^(1/2)/b)^(3/ 2) - a*b^3*x)*(a - ((a*b^5)^(1/2)/b)^(1/2)))/(4*b) + (log(b*(-(a*b^5)^(1/2 )/b)^(3/2) + 5*a*(a*b^5)^(1/2) + 5*a*x*(-(a*b^5)^(1/2)/b)^(3/2) - a*b^3*x) *(a - (-(a*b^5)^(1/2)/b)^(1/2)))/(4*b)
Time = 0.17 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.39 \[ \int \frac {-b+a x}{-b x^2+a x^6} \, dx=\frac {-2 b^{\frac {3}{4}} a^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {a}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) x +b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}} x +b^{\frac {1}{4}}\right ) x -b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}} x -b^{\frac {1}{4}}\right ) x +\mathrm {log}\left (a^{\frac {1}{4}} x +b^{\frac {1}{4}}\right ) a x +\mathrm {log}\left (a^{\frac {1}{4}} x -b^{\frac {1}{4}}\right ) a x +\mathrm {log}\left (\sqrt {a}\, x^{2}+\sqrt {b}\right ) a x -4 \,\mathrm {log}\left (x \right ) a x -4 b}{4 b x} \] Input:
int((a*x-b)/(a*x^6-b*x^2),x)
Output:
( - 2*b**(3/4)*a**(1/4)*atan((sqrt(a)*x)/(b**(1/4)*a**(1/4)))*x + b**(3/4) *a**(1/4)*log(a**(1/4)*x + b**(1/4))*x - b**(3/4)*a**(1/4)*log(a**(1/4)*x - b**(1/4))*x + log(a**(1/4)*x + b**(1/4))*a*x + log(a**(1/4)*x - b**(1/4) )*a*x + log(sqrt(a)*x**2 + sqrt(b))*a*x - 4*log(x)*a*x - 4*b)/(4*b*x)