Integrand size = 21, antiderivative size = 265 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=-\frac {\left (\frac {a}{b^4}-\frac {b}{a^4}\right ) x}{12 \left (a^4+b^4 x^4\right )^3}+\frac {\left (a^5+11 b^5\right ) x}{96 a^8 b^4 \left (a^4+b^4 x^4\right )^2}+\frac {7 \left (a^5+11 b^5\right ) x}{384 a^{12} b^4 \left (a^4+b^4 x^4\right )}-\frac {7 \left (a^5+11 b^5\right ) \arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{256 \sqrt {2} a^{15} b^5}+\frac {7 \left (a^5+11 b^5\right ) \arctan \left (1+\frac {\sqrt {2} b x}{a}\right )}{256 \sqrt {2} a^{15} b^5}-\frac {7 \left (a^5+11 b^5\right ) \log \left (a^2-\sqrt {2} a b x+b^2 x^2\right )}{512 \sqrt {2} a^{15} b^5}+\frac {7 \left (a^5+11 b^5\right ) \log \left (a^2+\sqrt {2} a b x+b^2 x^2\right )}{512 \sqrt {2} a^{15} b^5} \] Output:
-1/12*(a/b^4-b/a^4)*x/(b^4*x^4+a^4)^3+1/96*(a^5+11*b^5)*x/a^8/b^4/(b^4*x^4 +a^4)^2+7/384*(a^5+11*b^5)*x/a^12/b^4/(b^4*x^4+a^4)-7/512*(a^5+11*b^5)*arc tan(1-2^(1/2)*b*x/a)*2^(1/2)/a^15/b^5+7/512*(a^5+11*b^5)*arctan(1+2^(1/2)* b*x/a)*2^(1/2)/a^15/b^5-7/1024*(a^5+11*b^5)*ln(a^2-2^(1/2)*a*b*x+b^2*x^2)* 2^(1/2)/a^15/b^5+7/1024*(a^5+11*b^5)*ln(a^2+2^(1/2)*a*b*x+b^2*x^2)*2^(1/2) /a^15/b^5
Timed out. \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=\text {\$Aborted} \] Input:
Integrate[(b + a*x^4)/(a^4 + b^4*x^4)^4,x]
Output:
$Aborted
Time = 0.53 (sec) , antiderivative size = 249, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {910, 749, 749, 755, 1476, 1082, 217, 1479, 25, 27, 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.
| \(\displaystyle \int \frac {a x^4+b}{\left (a^4+b^4 x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \int \frac {1}{\left (a^4+b^4 x^4\right )^3}dx-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \int \frac {1}{\left (a^4+b^4 x^4\right )^2}dx}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \int \frac {1}{a^4+b^4 x^4}dx}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {a^2-b^2 x^2}{a^4+b^4 x^4}dx}{2 a^2}+\frac {\int \frac {a^2+b^2 x^2}{a^4+b^4 x^4}dx}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {1}{\frac {a^2}{b^2}-\frac {\sqrt {2} x a}{b}+x^2}dx}{2 b^2}+\frac {\int \frac {1}{\frac {a^2}{b^2}+\frac {\sqrt {2} x a}{b}+x^2}dx}{2 b^2}}{2 a^2}+\frac {\int \frac {a^2-b^2 x^2}{a^4+b^4 x^4}dx}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} b x}{a}\right )^2-1}d\left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} b x}{a}+1\right )^2-1}d\left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}}{2 a^2}+\frac {\int \frac {a^2-b^2 x^2}{a^4+b^4 x^4}dx}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\int \frac {a^2-b^2 x^2}{a^4+b^4 x^4}dx}{2 a^2}+\frac {\frac {\arctan \left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}-\frac {\arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {-\frac {\int -\frac {\sqrt {2} a-2 b x}{b \left (\frac {a^2}{b^2}-\frac {\sqrt {2} x a}{b}+x^2\right )}dx}{2 \sqrt {2} a b}-\frac {\int -\frac {\sqrt {2} \left (a+\sqrt {2} b x\right )}{b \left (\frac {a^2}{b^2}+\frac {\sqrt {2} x a}{b}+x^2\right )}dx}{2 \sqrt {2} a b}}{2 a^2}+\frac {\frac {\arctan \left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}-\frac {\arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} a-2 b x}{b \left (\frac {a^2}{b^2}-\frac {\sqrt {2} x a}{b}+x^2\right )}dx}{2 \sqrt {2} a b}+\frac {\int \frac {\sqrt {2} \left (a+\sqrt {2} b x\right )}{b \left (\frac {a^2}{b^2}+\frac {\sqrt {2} x a}{b}+x^2\right )}dx}{2 \sqrt {2} a b}}{2 a^2}+\frac {\frac {\arctan \left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}-\frac {\arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {7 \left (\frac {3 \left (\frac {\frac {\int \frac {\sqrt {2} a-2 b x}{\frac {a^2}{b^2}-\frac {\sqrt {2} x a}{b}+x^2}dx}{2 \sqrt {2} a b^2}+\frac {\int \frac {a+\sqrt {2} b x}{\frac {a^2}{b^2}+\frac {\sqrt {2} x a}{b}+x^2}dx}{2 a b^2}}{2 a^2}+\frac {\frac {\arctan \left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}-\frac {\arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}}{2 a^2}\right )}{4 a^4}+\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}\right )}{8 a^4}+\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{12} \left (\frac {11 b}{a^4}+\frac {a}{b^4}\right ) \left (\frac {x}{8 a^4 \left (a^4+b^4 x^4\right )^2}+\frac {7 \left (\frac {x}{4 a^4 \left (a^4+b^4 x^4\right )}+\frac {3 \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} b x}{a}+1\right )}{\sqrt {2} a b}-\frac {\arctan \left (1-\frac {\sqrt {2} b x}{a}\right )}{\sqrt {2} a b}}{2 a^2}+\frac {\frac {\log \left (a^2+\sqrt {2} a b x+b^2 x^2\right )}{2 \sqrt {2} a b}-\frac {\log \left (a^2-\sqrt {2} a b x+b^2 x^2\right )}{2 \sqrt {2} a b}}{2 a^2}\right )}{4 a^4}\right )}{8 a^4}\right )-\frac {x \left (\frac {a}{b^4}-\frac {b}{a^4}\right )}{12 \left (a^4+b^4 x^4\right )^3}\) |
Input:
Int[(b + a*x^4)/(a^4 + b^4*x^4)^4,x]
Output:
-1/12*((a/b^4 - b/a^4)*x)/(a^4 + b^4*x^4)^3 + ((a/b^4 + (11*b)/a^4)*(x/(8* a^4*(a^4 + b^4*x^4)^2) + (7*(x/(4*a^4*(a^4 + b^4*x^4)) + (3*((-(ArcTan[1 - (Sqrt[2]*b*x)/a]/(Sqrt[2]*a*b)) + ArcTan[1 + (Sqrt[2]*b*x)/a]/(Sqrt[2]*a* b))/(2*a^2) + (-1/2*Log[a^2 - Sqrt[2]*a*b*x + b^2*x^2]/(Sqrt[2]*a*b) + Log [a^2 + Sqrt[2]*a*b*x + b^2*x^2]/(2*Sqrt[2]*a*b))/(2*a^2)))/(4*a^4)))/(8*a^ 4)))/12
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[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{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_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\frac {7 \left (a^{5}+11 b^{5}\right ) b^{4} x^{9}}{384 a^{12}}+\frac {3 \left (a^{5}+11 b^{5}\right ) x^{5}}{64 a^{8}}-\frac {\left (7 a^{5}-51 b^{5}\right ) x}{128 a^{4} b^{4}}}{\left (b^{4} x^{4}+a^{4}\right )^{3}}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (b^{4} \textit {\_Z}^{4}+a^{4}\right )}{\sum }\frac {\left (a^{5}+11 b^{5}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{512 a^{12} b^{8}}\) | \(116\) |
| default | \(\frac {\frac {7 \left (a^{5}+11 b^{5}\right ) b^{4} x^{9}}{384 a^{12}}+\frac {3 \left (a^{5}+11 b^{5}\right ) x^{5}}{64 a^{8}}-\frac {\left (7 a^{5}-51 b^{5}\right ) x}{128 a^{4} b^{4}}}{\left (b^{4} x^{4}+a^{4}\right )^{3}}+\frac {7 \left (a^{5}+11 b^{5}\right ) \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a^{4}}{b^{4}}}}{x^{2}-\left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a^{4}}{b^{4}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}}-1\right )\right )}{1024 a^{16} b^{4}}\) | \(201\) |
Input:
int((a*x^4+b)/(b^4*x^4+a^4)^4,x,method=_RETURNVERBOSE)
Output:
(7/384*(a^5+11*b^5)/a^12*b^4*x^9+3/64/a^8*(a^5+11*b^5)*x^5-1/128*(7*a^5-51 *b^5)/a^4/b^4*x)/(b^4*x^4+a^4)^3+7/512/a^12/b^8*sum((a^5+11*b^5)/_R^3*ln(x -_R),_R=RootOf(_Z^4*b^4+a^4))
Leaf count of result is larger than twice the leaf count of optimal. 476 vs. \(2 (235) = 470\).
Time = 0.08 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.80 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=\frac {56 \, {\left (a^{8} b^{9} + 11 \, a^{3} b^{14}\right )} x^{9} + 144 \, {\left (a^{12} b^{5} + 11 \, a^{7} b^{10}\right )} x^{5} + 42 \, \sqrt {2} {\left (a^{17} + 11 \, a^{12} b^{5} + {\left (a^{5} b^{12} + 11 \, b^{17}\right )} x^{12} + 3 \, {\left (a^{9} b^{8} + 11 \, a^{4} b^{13}\right )} x^{8} + 3 \, {\left (a^{13} b^{4} + 11 \, a^{8} b^{9}\right )} x^{4}\right )} \arctan \left (\frac {\sqrt {2} b x + a}{a}\right ) + 42 \, \sqrt {2} {\left (a^{17} + 11 \, a^{12} b^{5} + {\left (a^{5} b^{12} + 11 \, b^{17}\right )} x^{12} + 3 \, {\left (a^{9} b^{8} + 11 \, a^{4} b^{13}\right )} x^{8} + 3 \, {\left (a^{13} b^{4} + 11 \, a^{8} b^{9}\right )} x^{4}\right )} \arctan \left (\frac {\sqrt {2} b x - a}{a}\right ) + 21 \, \sqrt {2} {\left (a^{17} + 11 \, a^{12} b^{5} + {\left (a^{5} b^{12} + 11 \, b^{17}\right )} x^{12} + 3 \, {\left (a^{9} b^{8} + 11 \, a^{4} b^{13}\right )} x^{8} + 3 \, {\left (a^{13} b^{4} + 11 \, a^{8} b^{9}\right )} x^{4}\right )} \log \left (b^{2} x^{2} + \sqrt {2} a b x + a^{2}\right ) - 21 \, \sqrt {2} {\left (a^{17} + 11 \, a^{12} b^{5} + {\left (a^{5} b^{12} + 11 \, b^{17}\right )} x^{12} + 3 \, {\left (a^{9} b^{8} + 11 \, a^{4} b^{13}\right )} x^{8} + 3 \, {\left (a^{13} b^{4} + 11 \, a^{8} b^{9}\right )} x^{4}\right )} \log \left (b^{2} x^{2} - \sqrt {2} a b x + a^{2}\right ) - 24 \, {\left (7 \, a^{16} b - 51 \, a^{11} b^{6}\right )} x}{3072 \, {\left (a^{15} b^{17} x^{12} + 3 \, a^{19} b^{13} x^{8} + 3 \, a^{23} b^{9} x^{4} + a^{27} b^{5}\right )}} \] Input:
integrate((a*x^4+b)/(b^4*x^4+a^4)^4,x, algorithm="fricas")
Output:
1/3072*(56*(a^8*b^9 + 11*a^3*b^14)*x^9 + 144*(a^12*b^5 + 11*a^7*b^10)*x^5 + 42*sqrt(2)*(a^17 + 11*a^12*b^5 + (a^5*b^12 + 11*b^17)*x^12 + 3*(a^9*b^8 + 11*a^4*b^13)*x^8 + 3*(a^13*b^4 + 11*a^8*b^9)*x^4)*arctan((sqrt(2)*b*x + a)/a) + 42*sqrt(2)*(a^17 + 11*a^12*b^5 + (a^5*b^12 + 11*b^17)*x^12 + 3*(a^ 9*b^8 + 11*a^4*b^13)*x^8 + 3*(a^13*b^4 + 11*a^8*b^9)*x^4)*arctan((sqrt(2)* b*x - a)/a) + 21*sqrt(2)*(a^17 + 11*a^12*b^5 + (a^5*b^12 + 11*b^17)*x^12 + 3*(a^9*b^8 + 11*a^4*b^13)*x^8 + 3*(a^13*b^4 + 11*a^8*b^9)*x^4)*log(b^2*x^ 2 + sqrt(2)*a*b*x + a^2) - 21*sqrt(2)*(a^17 + 11*a^12*b^5 + (a^5*b^12 + 11 *b^17)*x^12 + 3*(a^9*b^8 + 11*a^4*b^13)*x^8 + 3*(a^13*b^4 + 11*a^8*b^9)*x^ 4)*log(b^2*x^2 - sqrt(2)*a*b*x + a^2) - 24*(7*a^16*b - 51*a^11*b^6)*x)/(a^ 15*b^17*x^12 + 3*a^19*b^13*x^8 + 3*a^23*b^9*x^4 + a^27*b^5)
Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.63 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=\frac {x^{9} \cdot \left (7 a^{5} b^{8} + 77 b^{13}\right ) + x^{5} \cdot \left (18 a^{9} b^{4} + 198 a^{4} b^{9}\right ) + x \left (- 21 a^{13} + 153 a^{8} b^{5}\right )}{384 a^{24} b^{4} + 1152 a^{20} b^{8} x^{4} + 1152 a^{16} b^{12} x^{8} + 384 a^{12} b^{16} x^{12}} + \operatorname {RootSum} {\left (68719476736 t^{4} a^{60} b^{20} + 2401 a^{20} + 105644 a^{15} b^{5} + 1743126 a^{10} b^{10} + 12782924 a^{5} b^{15} + 35153041 b^{20}, \left ( t \mapsto t \log {\left (\frac {512 t a^{16} b^{4}}{7 a^{5} + 77 b^{5}} + x \right )} \right )\right )} \] Input:
integrate((a*x**4+b)/(b**4*x**4+a**4)**4,x)
Output:
(x**9*(7*a**5*b**8 + 77*b**13) + x**5*(18*a**9*b**4 + 198*a**4*b**9) + x*( -21*a**13 + 153*a**8*b**5))/(384*a**24*b**4 + 1152*a**20*b**8*x**4 + 1152* a**16*b**12*x**8 + 384*a**12*b**16*x**12) + RootSum(68719476736*_t**4*a**6 0*b**20 + 2401*a**20 + 105644*a**15*b**5 + 1743126*a**10*b**10 + 12782924* a**5*b**15 + 35153041*b**20, Lambda(_t, _t*log(512*_t*a**16*b**4/(7*a**5 + 77*b**5) + x)))
Time = 0.11 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.06 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=\frac {7 \, {\left (a^{5} b^{8} + 11 \, b^{13}\right )} x^{9} + 18 \, {\left (a^{9} b^{4} + 11 \, a^{4} b^{9}\right )} x^{5} - 3 \, {\left (7 \, a^{13} - 51 \, a^{8} b^{5}\right )} x}{384 \, {\left (a^{12} b^{16} x^{12} + 3 \, a^{16} b^{12} x^{8} + 3 \, a^{20} b^{8} x^{4} + a^{24} b^{4}\right )}} + \frac {7 \, {\left (\frac {2 \, \sqrt {2} {\left (a^{5} + 11 \, b^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, b^{2} x + \sqrt {2} a b\right )}}{2 \, a b}\right )}{a^{3} b} + \frac {2 \, \sqrt {2} {\left (a^{5} + 11 \, b^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, b^{2} x - \sqrt {2} a b\right )}}{2 \, a b}\right )}{a^{3} b} + \frac {\sqrt {2} {\left (a^{5} + 11 \, b^{5}\right )} \log \left (b^{2} x^{2} + \sqrt {2} a b x + a^{2}\right )}{a^{3} b} - \frac {\sqrt {2} {\left (a^{5} + 11 \, b^{5}\right )} \log \left (b^{2} x^{2} - \sqrt {2} a b x + a^{2}\right )}{a^{3} b}\right )}}{1024 \, a^{12} b^{4}} \] Input:
integrate((a*x^4+b)/(b^4*x^4+a^4)^4,x, algorithm="maxima")
Output:
1/384*(7*(a^5*b^8 + 11*b^13)*x^9 + 18*(a^9*b^4 + 11*a^4*b^9)*x^5 - 3*(7*a^ 13 - 51*a^8*b^5)*x)/(a^12*b^16*x^12 + 3*a^16*b^12*x^8 + 3*a^20*b^8*x^4 + a ^24*b^4) + 7/1024*(2*sqrt(2)*(a^5 + 11*b^5)*arctan(1/2*sqrt(2)*(2*b^2*x + sqrt(2)*a*b)/(a*b))/(a^3*b) + 2*sqrt(2)*(a^5 + 11*b^5)*arctan(1/2*sqrt(2)* (2*b^2*x - sqrt(2)*a*b)/(a*b))/(a^3*b) + sqrt(2)*(a^5 + 11*b^5)*log(b^2*x^ 2 + sqrt(2)*a*b*x + a^2)/(a^3*b) - sqrt(2)*(a^5 + 11*b^5)*log(b^2*x^2 - sq rt(2)*a*b*x + a^2)/(a^3*b))/(a^12*b^4)
Time = 0.11 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.12 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx=\frac {7 \, a^{5} b^{8} x^{9} + 77 \, b^{13} x^{9} + 18 \, a^{9} b^{4} x^{5} + 198 \, a^{4} b^{9} x^{5} - 21 \, a^{13} x + 153 \, a^{8} b^{5} x}{384 \, {\left (b^{4} x^{4} + a^{4}\right )}^{3} a^{12} b^{4}} + \frac {7 \, {\left (\sqrt {2} a^{5} + 11 \, \sqrt {2} b^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}}\right )}{512 \, a^{15} b^{5}} + \frac {7 \, {\left (\sqrt {2} a^{5} + 11 \, \sqrt {2} b^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}}}\right )}{512 \, a^{15} b^{5}} + \frac {7 \, {\left (\sqrt {2} a^{5} + 11 \, \sqrt {2} b^{5}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {a^{4}}{b^{4}}}\right )}{1024 \, a^{15} b^{5}} - \frac {7 \, {\left (\sqrt {2} a^{5} + 11 \, \sqrt {2} b^{5}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a^{4}}{b^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {a^{4}}{b^{4}}}\right )}{1024 \, a^{15} b^{5}} \] Input:
integrate((a*x^4+b)/(b^4*x^4+a^4)^4,x, algorithm="giac")
Output:
1/384*(7*a^5*b^8*x^9 + 77*b^13*x^9 + 18*a^9*b^4*x^5 + 198*a^4*b^9*x^5 - 21 *a^13*x + 153*a^8*b^5*x)/((b^4*x^4 + a^4)^3*a^12*b^4) + 7/512*(sqrt(2)*a^5 + 11*sqrt(2)*b^5)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a^4/b^4)^(1/4))/(a^4 /b^4)^(1/4))/(a^15*b^5) + 7/512*(sqrt(2)*a^5 + 11*sqrt(2)*b^5)*arctan(1/2* sqrt(2)*(2*x - sqrt(2)*(a^4/b^4)^(1/4))/(a^4/b^4)^(1/4))/(a^15*b^5) + 7/10 24*(sqrt(2)*a^5 + 11*sqrt(2)*b^5)*log(x^2 + sqrt(2)*x*(a^4/b^4)^(1/4) + sq rt(a^4/b^4))/(a^15*b^5) - 7/1024*(sqrt(2)*a^5 + 11*sqrt(2)*b^5)*log(x^2 - sqrt(2)*x*(a^4/b^4)^(1/4) + sqrt(a^4/b^4))/(a^15*b^5)
Time = 0.35 (sec) , antiderivative size = 795, normalized size of antiderivative = 3.00 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx =\text {Too large to display} \] Input:
int((b + a*x^4)/(a^4 + b^4*x^4)^4,x)
Output:
((3*x^5*(a^5 + 11*b^5))/(64*a^8) - (x*(7*a^5 - 51*b^5))/(128*a^4*b^4) + (7 *b^4*x^9*(a^5 + 11*b^5))/(384*a^12))/(a^12 + b^12*x^12 + 3*a^8*b^4*x^4 + 3 *a^4*b^8*x^8) + ((-1)^(1/4)*atan((((-1)^(1/4)*((49*x*(121*b^14 + 22*a^5*b^ 9 + a^10*b^4))/(4096*a^24) - (49*(-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^5* b^8))/(4096*a^23*b^5))*(a^5 + 11*b^5)*7i)/(512*a^15*b^5) + ((-1)^(1/4)*((4 9*x*(121*b^14 + 22*a^5*b^9 + a^10*b^4))/(4096*a^24) + (49*(-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^5*b^8))/(4096*a^23*b^5))*(a^5 + 11*b^5)*7i)/(512*a^ 15*b^5))/((7*(-1)^(1/4)*((49*x*(121*b^14 + 22*a^5*b^9 + a^10*b^4))/(4096*a ^24) - (49*(-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^5*b^8))/(4096*a^23*b^5)) *(a^5 + 11*b^5))/(512*a^15*b^5) - (7*(-1)^(1/4)*((49*x*(121*b^14 + 22*a^5* b^9 + a^10*b^4))/(4096*a^24) + (49*(-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^ 5*b^8))/(4096*a^23*b^5))*(a^5 + 11*b^5))/(512*a^15*b^5)))*(a^5 + 11*b^5)*7 i)/(256*a^15*b^5) + (7*(-1)^(1/4)*atan(((7*(-1)^(1/4)*((49*x*(121*b^14 + 2 2*a^5*b^9 + a^10*b^4))/(4096*a^24) - ((-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^5*b^8)*49i)/(4096*a^23*b^5))*(a^5 + 11*b^5))/(512*a^15*b^5) + (7*(-1)^( 1/4)*((49*x*(121*b^14 + 22*a^5*b^9 + a^10*b^4))/(4096*a^24) + ((-1)^(1/4)* (a^5 + 11*b^5)*(11*b^13 + a^5*b^8)*49i)/(4096*a^23*b^5))*(a^5 + 11*b^5))/( 512*a^15*b^5))/(((-1)^(1/4)*((49*x*(121*b^14 + 22*a^5*b^9 + a^10*b^4))/(40 96*a^24) - ((-1)^(1/4)*(a^5 + 11*b^5)*(11*b^13 + a^5*b^8)*49i)/(4096*a^23* b^5))*(a^5 + 11*b^5)*7i)/(512*a^15*b^5) - ((-1)^(1/4)*((49*x*(121*b^14 ...
Time = 0.18 (sec) , antiderivative size = 1056, normalized size of antiderivative = 3.98 \[ \int \frac {b+a x^4}{\left (a^4+b^4 x^4\right )^4} \, dx =\text {Too large to display} \] Input:
int((a*x^4+b)/(b^4*x^4+a^4)^4,x)
Output:
( - 42*sqrt(2)*atan((sqrt(2)*a - 2*b*x)/(sqrt(2)*a))*a**17 - 126*sqrt(2)*a tan((sqrt(2)*a - 2*b*x)/(sqrt(2)*a))*a**13*b**4*x**4 - 462*sqrt(2)*atan((s qrt(2)*a - 2*b*x)/(sqrt(2)*a))*a**12*b**5 - 126*sqrt(2)*atan((sqrt(2)*a - 2*b*x)/(sqrt(2)*a))*a**9*b**8*x**8 - 1386*sqrt(2)*atan((sqrt(2)*a - 2*b*x) /(sqrt(2)*a))*a**8*b**9*x**4 - 42*sqrt(2)*atan((sqrt(2)*a - 2*b*x)/(sqrt(2 )*a))*a**5*b**12*x**12 - 1386*sqrt(2)*atan((sqrt(2)*a - 2*b*x)/(sqrt(2)*a) )*a**4*b**13*x**8 - 462*sqrt(2)*atan((sqrt(2)*a - 2*b*x)/(sqrt(2)*a))*b**1 7*x**12 + 42*sqrt(2)*atan((sqrt(2)*a + 2*b*x)/(sqrt(2)*a))*a**17 + 126*sqr t(2)*atan((sqrt(2)*a + 2*b*x)/(sqrt(2)*a))*a**13*b**4*x**4 + 462*sqrt(2)*a tan((sqrt(2)*a + 2*b*x)/(sqrt(2)*a))*a**12*b**5 + 126*sqrt(2)*atan((sqrt(2 )*a + 2*b*x)/(sqrt(2)*a))*a**9*b**8*x**8 + 1386*sqrt(2)*atan((sqrt(2)*a + 2*b*x)/(sqrt(2)*a))*a**8*b**9*x**4 + 42*sqrt(2)*atan((sqrt(2)*a + 2*b*x)/( sqrt(2)*a))*a**5*b**12*x**12 + 1386*sqrt(2)*atan((sqrt(2)*a + 2*b*x)/(sqrt (2)*a))*a**4*b**13*x**8 + 462*sqrt(2)*atan((sqrt(2)*a + 2*b*x)/(sqrt(2)*a) )*b**17*x**12 - 21*sqrt(2)*log( - sqrt(2)*a*b*x + a**2 + b**2*x**2)*a**17 - 63*sqrt(2)*log( - sqrt(2)*a*b*x + a**2 + b**2*x**2)*a**13*b**4*x**4 - 23 1*sqrt(2)*log( - sqrt(2)*a*b*x + a**2 + b**2*x**2)*a**12*b**5 - 63*sqrt(2) *log( - sqrt(2)*a*b*x + a**2 + b**2*x**2)*a**9*b**8*x**8 - 693*sqrt(2)*log ( - sqrt(2)*a*b*x + a**2 + b**2*x**2)*a**8*b**9*x**4 - 21*sqrt(2)*log( - s qrt(2)*a*b*x + a**2 + b**2*x**2)*a**5*b**12*x**12 - 693*sqrt(2)*log( - ...