Integrand size = 26, antiderivative size = 132 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\frac {\left (c-\frac {\sqrt {a} e}{\sqrt {b}}\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt [4]{b}}+\frac {\left (\sqrt {b} c+\sqrt {a} e\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \] Output:
1/2*(c-a^(1/2)*e/b^(1/2))*arctan(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(1/4)+1/2*(b ^(1/2)*c+a^(1/2)*e)*arctanh(b^(1/4)*x/a^(1/4))/a^(3/4)/b^(3/4)+1/2*d*arcta nh(b^(1/2)*x^2/a^(1/2))/a^(1/2)/b^(1/2)-1/4*f*ln(-b*x^4+a)/b
Time = 0.05 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.62 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\frac {\left (\sqrt [4]{a} \sqrt {b} c-a^{3/4} e\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a b^{3/4}}-\frac {\left (\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d+a^{3/4} e\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{4 a b^{3/4}}-\frac {\left (-\sqrt [4]{a} \sqrt {b} c+\sqrt {a} \sqrt [4]{b} d-a^{3/4} e\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{4 a b^{3/4}}+\frac {d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b} \] Input:
Integrate[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4),x]
Output:
((a^(1/4)*Sqrt[b]*c - a^(3/4)*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a*b^(3/4) ) - ((a^(1/4)*Sqrt[b]*c + Sqrt[a]*b^(1/4)*d + a^(3/4)*e)*Log[a^(1/4) - b^( 1/4)*x])/(4*a*b^(3/4)) - ((-(a^(1/4)*Sqrt[b]*c) + Sqrt[a]*b^(1/4)*d - a^(3 /4)*e)*Log[a^(1/4) + b^(1/4)*x])/(4*a*b^(3/4)) + (d*Log[Sqrt[a] + Sqrt[b]* x^2])/(4*Sqrt[a]*Sqrt[b]) - (f*Log[a - b*x^4])/(4*b)
Time = 0.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2415, 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.
\(\displaystyle \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx\) |
\(\Big \downarrow \) 2415 |
\(\displaystyle \int \left (\frac {c+e x^2}{a-b x^4}+\frac {x \left (d+f x^2\right )}{a-b x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} c-\sqrt {a} e\right )}{2 a^{3/4} b^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e+\sqrt {b} c\right )}{2 a^{3/4} b^{3/4}}+\frac {d \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {f \log \left (a-b x^4\right )}{4 b}\) |
Input:
Int[(c + d*x + e*x^2 + f*x^3)/(a - b*x^4),x]
Output:
((Sqrt[b]*c - Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4)) + ((Sqrt[b]*c + Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(3/4 )) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]) - (f*Log[a - b *x^4])/(4*b)
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, 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 = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.33
method | result | size |
risch | \(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-a \right )}{\sum }\frac {\left (\textit {\_R}^{3} f +\textit {\_R}^{2} e +d \textit {\_R} +c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b}\) | \(44\) |
default | \(\frac {c \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {d \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {f \ln \left (-b \,x^{4}+a \right )}{4 b}\) | \(154\) |
Input:
int((f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x,method=_RETURNVERBOSE)
Output:
-1/4/b*sum((_R^3*f+_R^2*e+_R*d+c)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
Result contains complex when optimal does not.
Time = 5.04 (sec) , antiderivative size = 241149, normalized size of antiderivative = 1826.89 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\text {Too large to display} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\text {Timed out} \] Input:
integrate((f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.32 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\frac {{\left (\sqrt {b} c - \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{2 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} d + \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{4 \, \sqrt {a} b} - \frac {{\left (\sqrt {b} c + \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{4 \, \sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="maxima")
Output:
1/2*(sqrt(b)*c - sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt( a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) + 1/4*(sqrt(b)*d - sqrt(a)*f)*log(sqrt(b )*x^2 + sqrt(a))/(sqrt(a)*b) - 1/4*(sqrt(b)*d + sqrt(a)*f)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*b) - 1/4*(sqrt(b)*c + sqrt(a)*e)*log((sqrt(b)*x - sqr t(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqr t(a)*sqrt(b))*sqrt(b))
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (92) = 184\).
Time = 0.13 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.09 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {f \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} \] Input:
integrate((f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="giac")
Output:
-1/4*sqrt(2)*(b^2*c - sqrt(2)*(-a*b^3)^(1/4)*b*d + sqrt(-a*b)*b*e)*arctan( 1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/ 4*sqrt(2)*(b^2*c + sqrt(2)*(-a*b^3)^(1/4)*b*d - sqrt(-a*b)*b*e)*arctan(1/2 *sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/8*s qrt(2)*(b^2*c - sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a /b))/(-a*b^3)^(3/4) + 1/8*sqrt(2)*(b^2*c - sqrt(-a*b)*b*e)*log(x^2 - sqrt( 2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4) - 1/4*f*log(abs(b*x^4 - a)) /b
Time = 7.40 (sec) , antiderivative size = 1970, normalized size of antiderivative = 14.92 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\text {Too large to display} \] Input:
int((c + d*x + e*x^2 + f*x^3)/(a - b*x^4),x)
Output:
symsum(log(b^2*c^2*e - b^2*c*d^2 - b^2*d^3*x - a*b*e^3 - a*b*c*f^2 - 16*ro ot(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f ^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16 *a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4 *a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a *b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)^2*a*b^3*c - 4*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3 *b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f*z - 16*a^2*b^2*d^2*f* z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b*f^3*z + 4*a^2*b*d*e^2 *f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)*b^3* c^2*x - b^2*c^2*f*x + 16*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2 *b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^2*c*e*f* z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 16*a^3*b* f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4*a*b^2*c*d^ 2*e - 2*a^2*b*d^2*f^2 + 2*a*b^2*c^2*e^2 + a*b^2*d^4 + a^3*f^4 - a^2*b*e^4 - b^3*c^4, z, k)^2*a*b^3*d*x - 4*root(256*a^3*b^4*z^4 + 256*a^3*b^3*f*z^3 - 64*a^2*b^3*c*e*z^2 + 96*a^3*b^2*f^2*z^2 - 32*a^2*b^3*d^2*z^2 - 32*a^2*b^ 2*c*e*f*z - 16*a^2*b^2*d^2*f*z + 16*a^2*b^2*d*e^2*z + 16*a*b^3*c^2*d*z + 1 6*a^3*b*f^3*z + 4*a^2*b*d*e^2*f - 4*a^2*b*c*e*f^2 + 4*a*b^2*c^2*d*f - 4...
Time = 0.19 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.66 \[ \int \frac {c+d x+e x^2+f x^3}{a-b x^4} \, dx=\frac {-2 b^{\frac {1}{4}} a^{\frac {3}{4}} \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) e +2 b^{\frac {3}{4}} a^{\frac {1}{4}} \mathit {atan} \left (\frac {\sqrt {b}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}}}\right ) c -b^{\frac {1}{4}} a^{\frac {3}{4}} \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) e +b^{\frac {1}{4}} a^{\frac {3}{4}} \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) e -b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) c +b^{\frac {3}{4}} a^{\frac {1}{4}} \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) c -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) d -\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) d +\sqrt {b}\, \sqrt {a}\, \mathrm {log}\left (\sqrt {a}+\sqrt {b}\, x^{2}\right ) d -\mathrm {log}\left (a^{\frac {1}{4}}-b^{\frac {1}{4}} x \right ) a f -\mathrm {log}\left (a^{\frac {1}{4}}+b^{\frac {1}{4}} x \right ) a f -\mathrm {log}\left (\sqrt {a}+\sqrt {b}\, x^{2}\right ) a f}{4 a b} \] Input:
int((f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)
Output:
( - 2*b**(1/4)*a**(3/4)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*e + 2*b**(3/ 4)*a**(1/4)*atan((sqrt(b)*x)/(b**(1/4)*a**(1/4)))*c - b**(1/4)*a**(3/4)*lo g(a**(1/4) - b**(1/4)*x)*e + b**(1/4)*a**(3/4)*log(a**(1/4) + b**(1/4)*x)* e - b**(3/4)*a**(1/4)*log(a**(1/4) - b**(1/4)*x)*c + b**(3/4)*a**(1/4)*log (a**(1/4) + b**(1/4)*x)*c - sqrt(b)*sqrt(a)*log(a**(1/4) - b**(1/4)*x)*d - sqrt(b)*sqrt(a)*log(a**(1/4) + b**(1/4)*x)*d + sqrt(b)*sqrt(a)*log(sqrt(a ) + sqrt(b)*x**2)*d - log(a**(1/4) - b**(1/4)*x)*a*f - log(a**(1/4) + b**( 1/4)*x)*a*f - log(sqrt(a) + sqrt(b)*x**2)*a*f)/(4*a*b)