Integrand size = 9, antiderivative size = 139 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\frac {x}{a}+\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{2 \sqrt {2} a^{5/4}}-\frac {\sqrt [4]{b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {b}+\sqrt {a} x^2}\right )}{2 \sqrt {2} a^{5/4}} \] Output:
x/a-1/4*b^(1/4)*arctan(-1+2^(1/2)*a^(1/4)*x/b^(1/4))*2^(1/2)/a^(5/4)-1/4*b ^(1/4)*arctan(1+2^(1/2)*a^(1/4)*x/b^(1/4))*2^(1/2)/a^(5/4)-1/4*b^(1/4)*arc tanh(2^(1/2)*a^(1/4)*b^(1/4)*x/(b^(1/2)+a^(1/2)*x^2))*2^(1/2)/a^(5/4)
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.24 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\frac {8 \sqrt [4]{a} x+2 \sqrt {2} \sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )-2 \sqrt {2} \sqrt [4]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )+\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {b}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )-\sqrt {2} \sqrt [4]{b} \log \left (\sqrt {b}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2\right )}{8 a^{5/4}} \] Input:
Integrate[(a + b/x^4)^(-1),x]
Output:
(8*a^(1/4)*x + 2*Sqrt[2]*b^(1/4)*ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/b^(1/4)] - 2*Sqrt[2]*b^(1/4)*ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/b^(1/4)] + Sqrt[2]*b^(1/ 4)*Log[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2] - Sqrt[2]*b^(1/4 )*Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2])/(8*a^(5/4))
Time = 0.68 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.53, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {772, 843, 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.
| \(\displaystyle \int \frac {1}{a+\frac {b}{x^4}} \, dx\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \int \frac {x^4}{a x^4+b}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a x^4+b}dx}{a}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^2}{a x^4+b}dx}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {a} x^2+\sqrt {b}}{a x^4+b}dx}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx}{2 \sqrt {a}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx}{2 \sqrt {a}}}{2 \sqrt {b}}+\frac {\int \frac {\sqrt {b}-\sqrt {a} x^2}{a x^4+b}dx}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^2}{a x^4+b}dx}{2 \sqrt {b}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {\sqrt {b}-\sqrt {a} x^2}{a x^4+b}dx}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x}{\sqrt [4]{a} \left (x^2-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{a} x+\sqrt [4]{b}\right )}{\sqrt [4]{a} \left (x^2+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{b}-2 \sqrt [4]{a} x}{x^2-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx}{2 \sqrt {2} \sqrt {a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{a} x+\sqrt [4]{b}}{x^2+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+\frac {\sqrt {b}}{\sqrt {a}}}dx}{2 \sqrt {a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a} x^2+\sqrt {b}\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {b}}\right )}{a}\) |
Input:
Int[(a + b/x^4)^(-1),x]
Output:
x/a - (b*((-(ArcTan[1 - (Sqrt[2]*a^(1/4)*x)/b^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1 /4))) + ArcTan[1 + (Sqrt[2]*a^(1/4)*x)/b^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) /(2*Sqrt[b]) + (-1/2*Log[Sqrt[b] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[a]*x^2 ]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[b] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sq rt[a]*x^2]/(2*Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[b])))/a
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_)^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_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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 = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.24
| method | result | size |
| risch | \(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{4}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{4 a^{2}}\) | \(34\) |
| default | \(\frac {x}{a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {b}{a}}}{x^{2}-\left (\frac {b}{a}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {b}{a}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {b}{a}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}\) | \(108\) |
Input:
int(1/(a+b/x^4),x,method=_RETURNVERBOSE)
Output:
x/a-1/4/a^2*b*sum(1/_R^3*ln(x-_R),_R=RootOf(_Z^4*a+b))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.77 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=-\frac {a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) + i \, a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (i \, a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) - i \, a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) - a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} \log \left (-a \left (-\frac {b}{a^{5}}\right )^{\frac {1}{4}} + x\right ) - 4 \, x}{4 \, a} \] Input:
integrate(1/(a+b/x^4),x, algorithm="fricas")
Output:
-1/4*(a*(-b/a^5)^(1/4)*log(a*(-b/a^5)^(1/4) + x) + I*a*(-b/a^5)^(1/4)*log( I*a*(-b/a^5)^(1/4) + x) - I*a*(-b/a^5)^(1/4)*log(-I*a*(-b/a^5)^(1/4) + x) - a*(-b/a^5)^(1/4)*log(-a*(-b/a^5)^(1/4) + x) - 4*x)/a
Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.16 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{5} + b, \left ( t \mapsto t \log {\left (- 4 t a + x \right )} \right )\right )} + \frac {x}{a} \] Input:
integrate(1/(a+b/x**4),x)
Output:
RootSum(256*_t**4*a**5 + b, Lambda(_t, _t*log(-4*_t*a + x))) + x/a
Time = 0.12 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.29 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=-\frac {\frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {a} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {a} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {a} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {b}\right )}{a^{\frac {1}{4}}} - \frac {\sqrt {2} b^{\frac {1}{4}} \log \left (\sqrt {a} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {b}\right )}{a^{\frac {1}{4}}}}{8 \, a} + \frac {x}{a} \] Input:
integrate(1/(a+b/x^4),x, algorithm="maxima")
Output:
-1/8*(2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2)*(2*sqrt(a)*x + sqrt(2)*a^(1/4)* b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/sqrt(sqrt(a)*sqrt(b)) + 2*sqrt(2)*sqrt(b)* arctan(1/2*sqrt(2)*(2*sqrt(a)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sq rt(b)))/sqrt(sqrt(a)*sqrt(b)) + sqrt(2)*b^(1/4)*log(sqrt(a)*x^2 + sqrt(2)* a^(1/4)*b^(1/4)*x + sqrt(b))/a^(1/4) - sqrt(2)*b^(1/4)*log(sqrt(a)*x^2 - s qrt(2)*a^(1/4)*b^(1/4)*x + sqrt(b))/a^(1/4))/a + x/a
Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.24 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\frac {x}{a} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {b}{a}\right )^{\frac {1}{4}}}\right )}{4 \, a^{2}} - \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} x \left (\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{a}}\right )}{8 \, a^{2}} + \frac {\sqrt {2} \left (a^{3} b\right )^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} x \left (\frac {b}{a}\right )^{\frac {1}{4}} + \sqrt {\frac {b}{a}}\right )}{8 \, a^{2}} \] Input:
integrate(1/(a+b/x^4),x, algorithm="giac")
Output:
x/a - 1/4*sqrt(2)*(a^3*b)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(b/a)^(1 /4))/(b/a)^(1/4))/a^2 - 1/4*sqrt(2)*(a^3*b)^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(b/a)^(1/4))/(b/a)^(1/4))/a^2 - 1/8*sqrt(2)*(a^3*b)^(1/4)*log(x^ 2 + sqrt(2)*x*(b/a)^(1/4) + sqrt(b/a))/a^2 + 1/8*sqrt(2)*(a^3*b)^(1/4)*log (x^2 - sqrt(2)*x*(b/a)^(1/4) + sqrt(b/a))/a^2
Time = 0.37 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.35 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\frac {x}{a}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atan}\left (\frac {a^{1/4}\,x}{{\left (-b\right )}^{1/4}}\right )}{2\,a^{5/4}}-\frac {{\left (-b\right )}^{1/4}\,\mathrm {atanh}\left (\frac {a^{1/4}\,x}{{\left (-b\right )}^{1/4}}\right )}{2\,a^{5/4}} \] Input:
int(1/(a + b/x^4),x)
Output:
x/a - ((-b)^(1/4)*atan((a^(1/4)*x)/(-b)^(1/4)))/(2*a^(5/4)) - ((-b)^(1/4)* atanh((a^(1/4)*x)/(-b)^(1/4)))/(2*a^(5/4))
Time = 0.20 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.04 \[ \int \frac {1}{a+\frac {b}{x^4}} \, dx=\frac {2 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {a}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )-2 b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {a}\, x}{b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right )+b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {b}\right )-b^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (b^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}\, x^{2}+\sqrt {b}\right )+8 a x}{8 a^{2}} \] Input:
int(1/(a+b/x^4),x)
Output:
(2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(a)*x )/(b**(1/4)*a**(1/4)*sqrt(2))) - 2*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4 )*a**(1/4)*sqrt(2) + 2*sqrt(a)*x)/(b**(1/4)*a**(1/4)*sqrt(2))) + b**(1/4)* a**(3/4)*sqrt(2)*log( - b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a)*x**2 + sqrt( b)) - b**(1/4)*a**(3/4)*sqrt(2)*log(b**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a)* x**2 + sqrt(b)) + 8*a*x)/(8*a**2)