Integrand size = 27, antiderivative size = 96 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {a \log (x)-a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}^4+\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}-2 \text {$\#$1}^5}\&\right ]}{8 b} \]
Time = 0.00 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\frac {\text {RootSum}\left [b+2 a \text {$\#$1}^4-2 \text {$\#$1}^8\&,\frac {-a \log (x)+a \log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{b+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-a \text {$\#$1}+2 \text {$\#$1}^5}\&\right ]}{8 b} \]
-1/8*RootSum[b + 2*a*#1^4 - 2*#1^8 & , (-(a*Log[x]) + a*Log[(b + a*x^4)^(1 /4) - x*#1] + Log[x]*#1^4 - Log[(b + a*x^4)^(1/4) - x*#1]*#1^4)/(-(a*#1) + 2*#1^5) & ]/b
Leaf count is larger than twice the leaf count of optimal. \(429\) vs. \(2(96)=192\).
Time = 0.56 (sec) , antiderivative size = 429, normalized size of antiderivative = 4.47, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1758, 27, 902, 756, 218, 221}
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.
\(\displaystyle \int \frac {1}{\sqrt [4]{a x^4+b} \left (-2 a x^4-2 b+x^8\right )} \, dx\) |
\(\Big \downarrow \) 1758 |
\(\displaystyle \frac {\int -\frac {1}{2 \left (-x^4+a+\sqrt {a^2+2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2+2 b}}-\frac {\int -\frac {1}{2 \left (-x^4+a-\sqrt {a^2+2 b}\right ) \sqrt [4]{a x^4+b}}dx}{\sqrt {a^2+2 b}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {1}{\left (-x^4+a-\sqrt {a^2+2 b}\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {a^2+2 b}}-\frac {\int \frac {1}{\left (-x^4+a+\sqrt {a^2+2 b}\right ) \sqrt [4]{a x^4+b}}dx}{2 \sqrt {a^2+2 b}}\) |
\(\Big \downarrow \) 902 |
\(\displaystyle \frac {\int \frac {1}{-\frac {\left (b+a \left (a-\sqrt {a^2+2 b}\right )\right ) x^4}{a x^4+b}+a-\sqrt {a^2+2 b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a^2+2 b}}-\frac {\int \frac {1}{-\frac {\left (b+a \left (a+\sqrt {a^2+2 b}\right )\right ) x^4}{a x^4+b}+a+\sqrt {a^2+2 b}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a^2+2 b}}\) |
\(\Big \downarrow \) 756 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}-\frac {\sqrt {a^2-\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2+2 b}}}+\frac {\int \frac {1}{\frac {\sqrt {a^2-\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}+\sqrt {a-\sqrt {a^2+2 b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2+2 b}}}}{2 \sqrt {a^2+2 b}}-\frac {\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}-\frac {\sqrt {a^2+\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2+2 b}+a}}+\frac {\int \frac {1}{\frac {\sqrt {a^2+\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}+\sqrt {a+\sqrt {a^2+2 b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2+2 b}+a}}}{2 \sqrt {a^2+2 b}}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {a-\sqrt {a^2+2 b}}-\frac {\sqrt {a^2-\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {a-\sqrt {a^2+2 b}}}+\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}}{2 \sqrt {a^2+2 b}}-\frac {\frac {\int \frac {1}{\sqrt {a+\sqrt {a^2+2 b}}-\frac {\sqrt {a^2+\sqrt {a^2+2 b} a+b} x^2}{\sqrt {a x^4+b}}}d\frac {x}{\sqrt [4]{a x^4+b}}}{2 \sqrt {\sqrt {a^2+2 b}+a}}+\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}}{2 \sqrt {a^2+2 b}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {\arctan \left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{a-\sqrt {a^2+2 b}} \sqrt [4]{a x^4+b}}\right )}{2 \left (a-\sqrt {a^2+2 b}\right )^{3/4} \sqrt [4]{-a \sqrt {a^2+2 b}+a^2+b}}}{2 \sqrt {a^2+2 b}}-\frac {\frac {\arctan \left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}+\frac {\text {arctanh}\left (\frac {x \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}{\sqrt [4]{\sqrt {a^2+2 b}+a} \sqrt [4]{a x^4+b}}\right )}{2 \left (\sqrt {a^2+2 b}+a\right )^{3/4} \sqrt [4]{a \sqrt {a^2+2 b}+a^2+b}}}{2 \sqrt {a^2+2 b}}\) |
(ArcTan[((a^2 + b - a*Sqrt[a^2 + 2*b])^(1/4)*x)/((a - Sqrt[a^2 + 2*b])^(1/ 4)*(b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 + 2*b])^(3/4)*(a^2 + b - a*Sqrt[a^ 2 + 2*b])^(1/4)) + ArcTanh[((a^2 + b - a*Sqrt[a^2 + 2*b])^(1/4)*x)/((a - S qrt[a^2 + 2*b])^(1/4)*(b + a*x^4)^(1/4))]/(2*(a - Sqrt[a^2 + 2*b])^(3/4)*( a^2 + b - a*Sqrt[a^2 + 2*b])^(1/4)))/(2*Sqrt[a^2 + 2*b]) - (ArcTan[((a^2 + b + a*Sqrt[a^2 + 2*b])^(1/4)*x)/((a + Sqrt[a^2 + 2*b])^(1/4)*(b + a*x^4)^ (1/4))]/(2*(a + Sqrt[a^2 + 2*b])^(3/4)*(a^2 + b + a*Sqrt[a^2 + 2*b])^(1/4) ) + ArcTanh[((a^2 + b + a*Sqrt[a^2 + 2*b])^(1/4)*x)/((a + Sqrt[a^2 + 2*b]) ^(1/4)*(b + a*x^4)^(1/4))]/(2*(a + Sqrt[a^2 + 2*b])^(3/4)*(a^2 + b + a*Sqr t[a^2 + 2*b])^(1/4)))/(2*Sqrt[a^2 + 2*b])
3.14.38.3.1 Defintions of rubi rules used
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/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Su bst[Int[1/(c - (b*c - a*d)*x^n), x], x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b , c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]
Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_ )), x_Symbol] :> With[{r = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/r) Int[(d + e*x ^n)^q/(b - r + 2*c*x^n), x], x] - Simp[2*(c/r) Int[(d + e*x^n)^q/(b + r + 2*c*x^n), x], x]] /; FreeQ[{a, b, c, d, e, n, q}, x] && EqQ[n2, 2*n] && Ne Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !IntegerQ[q]
Time = 0.00 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69
method | result | size |
pseudoelliptic | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-2 \textit {\_Z}^{4} a -b \right )}{\sum }\frac {\left (\textit {\_R}^{4}-a \right ) \ln \left (\frac {-\textit {\_R} x +\left (a \,x^{4}+b \right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R} \left (2 \textit {\_R}^{4}-a \right )}}{8 b}\) | \(66\) |
1/8*sum(1/_R*(_R^4-a)*ln((-_R*x+(a*x^4+b)^(1/4))/x)/(2*_R^4-a),_R=RootOf(2 *_Z^8-2*_Z^4*a-b))/b
Timed out. \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\text {Timed out} \]
Not integrable
Time = 6.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.27 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int \frac {1}{\sqrt [4]{a x^{4} + b} \left (- 2 a x^{4} - 2 b + x^{8}\right )}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 2 \, a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 0.92 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=\int { \frac {1}{{\left (x^{8} - 2 \, a x^{4} - 2 \, b\right )} {\left (a x^{4} + b\right )}^{\frac {1}{4}}} \,d x } \]
Not integrable
Time = 0.00 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.32 \[ \int \frac {1}{\sqrt [4]{b+a x^4} \left (-2 b-2 a x^4+x^8\right )} \, dx=-\int \frac {1}{{\left (a\,x^4+b\right )}^{1/4}\,\left (-x^8+2\,a\,x^4+2\,b\right )} \,d x \]