Integrand size = 22, antiderivative size = 61 \[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x+x^3}}{1+x^2}\right )}{\sqrt [4]{2}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x+x^3}}{1+x^2}\right )}{\sqrt [4]{2}} \]
-1/2*arctan(2^(1/4)*(x^3+x)^(1/2)/(x^2+1))*2^(3/4)-1/2*arctanh(2^(1/4)*(x^ 3+x)^(1/2)/(x^2+1))*2^(3/4)
Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.20 \[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=-\frac {\sqrt {x} \sqrt {1+x^2} \left (\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {1+x^2}}\right )+\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {1+x^2}}\right )\right )}{\sqrt [4]{2} \sqrt {x+x^3}} \]
-((Sqrt[x]*Sqrt[1 + x^2]*(ArcTan[(2^(1/4)*Sqrt[x])/Sqrt[1 + x^2]] + ArcTan h[(2^(1/4)*Sqrt[x])/Sqrt[1 + x^2]]))/(2^(1/4)*Sqrt[x + x^3]))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.09 (sec) , antiderivative size = 349, normalized size of antiderivative = 5.72, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2467, 25, 1388, 2035, 7276, 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.
| \(\displaystyle \int \frac {x^4-1}{\sqrt {x^3+x} \left (x^4+1\right )} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt {x^2+1} \int -\frac {1-x^4}{\sqrt {x} \sqrt {x^2+1} \left (x^4+1\right )}dx}{\sqrt {x^3+x}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2+1} \int \frac {1-x^4}{\sqrt {x} \sqrt {x^2+1} \left (x^4+1\right )}dx}{\sqrt {x^3+x}}\) |
| \(\Big \downarrow \) 1388 |
| \(\displaystyle -\frac {\sqrt {x} \sqrt {x^2+1} \int \frac {\left (1-x^2\right ) \sqrt {x^2+1}}{\sqrt {x} \left (x^4+1\right )}dx}{\sqrt {x^3+x}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+1} \int \frac {\left (1-x^2\right ) \sqrt {x^2+1}}{x^4+1}d\sqrt {x}}{\sqrt {x^3+x}}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+1} \int \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {x^2+1}}{i-x^2}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {x^2+1}}{x^2+i}\right )d\sqrt {x}}{\sqrt {x^3+x}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+1} \left (\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {x^2+1}}\right )}{2 \sqrt [4]{2}}-\frac {i \left (\sqrt {2}+(1+i)\right ) (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{8 \sqrt {x^2+1}}+\frac {i \left (\sqrt {2}+(-1-i)\right ) (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{8 \sqrt {x^2+1}}+\frac {\left ((1+i)+i \sqrt {2}\right ) (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{8 \sqrt {x^2+1}}+\frac {\left ((1+i)-i \sqrt {2}\right ) (x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{8 \sqrt {x^2+1}}-\frac {(x+1) \sqrt {\frac {x^2+1}{(x+1)^2}} \operatorname {EllipticF}\left (2 \arctan \left (\sqrt {x}\right ),\frac {1}{2}\right )}{2 \sqrt {x^2+1}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt {x^2+1}}\right )}{2 \sqrt [4]{2}}\right )}{\sqrt {x^3+x}}\) |
(-2*Sqrt[x]*Sqrt[1 + x^2]*(ArcTan[(2^(1/4)*Sqrt[x])/Sqrt[1 + x^2]]/(2*2^(1 /4)) + ArcTanh[(2^(1/4)*Sqrt[x])/Sqrt[1 + x^2]]/(2*2^(1/4)) - ((1 + x)*Sqr t[(1 + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(2*Sqrt[1 + x^2] ) + (((1 + I) - I*Sqrt[2])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticF[2*A rcTan[Sqrt[x]], 1/2])/(8*Sqrt[1 + x^2]) + (((1 + I) + I*Sqrt[2])*(1 + x)*S qrt[(1 + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/(8*Sqrt[1 + x^ 2]) + ((I/8)*((-1 - I) + Sqrt[2])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*Ellipt icF[2*ArcTan[Sqrt[x]], 1/2])/Sqrt[1 + x^2] - ((I/8)*((1 + I) + Sqrt[2])*(1 + x)*Sqrt[(1 + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/2])/Sqrt[1 + x^2]))/Sqrt[x + x^3]
3.8.96.3.1 Defintions of rubi rules used
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 5.00 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {\sqrt {\left (x^{2}+1\right ) x}\, 2^{\frac {3}{4}}}{2 x}\right )-\ln \left (\frac {2^{\frac {1}{4}} x +\sqrt {\left (x^{2}+1\right ) x}}{-2^{\frac {1}{4}} x +\sqrt {\left (x^{2}+1\right ) x}}\right )\right )}{4}\) | \(64\) |
| pseudoelliptic | \(\frac {2^{\frac {3}{4}} \left (2 \arctan \left (\frac {\sqrt {\left (x^{2}+1\right ) x}\, 2^{\frac {3}{4}}}{2 x}\right )-\ln \left (\frac {2^{\frac {1}{4}} x +\sqrt {\left (x^{2}+1\right ) x}}{-2^{\frac {1}{4}} x +\sqrt {\left (x^{2}+1\right ) x}}\right )\right )}{4}\) | \(64\) |
| elliptic | \(\frac {i \sqrt {-i \left (i+x \right )}\, \sqrt {2}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i+x \right )}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}+x}}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-i \underline {\hspace {1.25 ex}}\alpha ^{2}-\underline {\hspace {1.25 ex}}\alpha +i\right ) \sqrt {-i \left (i+x \right )}\, \sqrt {i \left (-i+x \right )}\, \sqrt {i x}\, \operatorname {EllipticPi}\left (\sqrt {-i \left (i+x \right )}, -\frac {1}{2} i \underline {\hspace {1.25 ex}}\alpha ^{3}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {1}{2} i \underline {\hspace {1.25 ex}}\alpha +\frac {1}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {\left (x^{2}+1\right ) x}}\right )}{4}\) | \(151\) |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{5} x^{2}-2 x \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{5}-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x -16 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x^{2}+16 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) x +32 \sqrt {x^{3}+x}-16 \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+4 x^{2}-4 x +4}\right )}{4}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{4} x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{4} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{4}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x +16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x +32 \sqrt {x^{3}+x}+16 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 x \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}-4 x^{2}+4 x -4}\right )}{4}\) | \(351\) |
1/4*2^(3/4)*(2*arctan(1/2/x*((x^2+1)*x)^(1/2)*2^(3/4))-ln((2^(1/4)*x+((x^2 +1)*x)^(1/2))/(-2^(1/4)*x+((x^2+1)*x)^(1/2))))
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 239, normalized size of antiderivative = 3.92 \[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=-\frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {x^{4} + 4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{3} + x\right )} + 2 \, \sqrt {x^{3} + x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (x^{2} + 1\right )}\right )} + 1}{x^{4} + 1}\right ) + \frac {1}{8} \cdot 2^{\frac {3}{4}} \log \left (\frac {x^{4} + 4 \, x^{2} + 2 \, \sqrt {2} {\left (x^{3} + x\right )} - 2 \, \sqrt {x^{3} + x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (x^{2} + 1\right )}\right )} + 1}{x^{4} + 1}\right ) + \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {x^{4} + 4 \, x^{2} - 2 \, \sqrt {2} {\left (x^{3} + x\right )} - 2 \, \sqrt {x^{3} + x} {\left (i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (-i \, x^{2} - i\right )}\right )} + 1}{x^{4} + 1}\right ) - \frac {1}{8} i \cdot 2^{\frac {3}{4}} \log \left (\frac {x^{4} + 4 \, x^{2} - 2 \, \sqrt {2} {\left (x^{3} + x\right )} - 2 \, \sqrt {x^{3} + x} {\left (-i \cdot 2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (i \, x^{2} + i\right )}\right )} + 1}{x^{4} + 1}\right ) \]
-1/8*2^(3/4)*log((x^4 + 4*x^2 + 2*sqrt(2)*(x^3 + x) + 2*sqrt(x^3 + x)*(2^( 3/4)*x + 2^(1/4)*(x^2 + 1)) + 1)/(x^4 + 1)) + 1/8*2^(3/4)*log((x^4 + 4*x^2 + 2*sqrt(2)*(x^3 + x) - 2*sqrt(x^3 + x)*(2^(3/4)*x + 2^(1/4)*(x^2 + 1)) + 1)/(x^4 + 1)) + 1/8*I*2^(3/4)*log((x^4 + 4*x^2 - 2*sqrt(2)*(x^3 + x) - 2* sqrt(x^3 + x)*(I*2^(3/4)*x + 2^(1/4)*(-I*x^2 - I)) + 1)/(x^4 + 1)) - 1/8*I *2^(3/4)*log((x^4 + 4*x^2 - 2*sqrt(2)*(x^3 + x) - 2*sqrt(x^3 + x)*(-I*2^(3 /4)*x + 2^(1/4)*(I*x^2 + I)) + 1)/(x^4 + 1))
\[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt {x \left (x^{2} + 1\right )} \left (x^{4} + 1\right )}\, dx \]
\[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} + x}} \,d x } \]
\[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{4} + 1\right )} \sqrt {x^{3} + x}} \,d x } \]
Time = 5.85 (sec) , antiderivative size = 234, normalized size of antiderivative = 3.84 \[ \int \frac {-1+x^4}{\sqrt {x+x^3} \left (1+x^4\right )} \, dx=-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}}-\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,1{}\mathrm {i}}{\sqrt {x^3+x}}+\frac {\sqrt {1-x\,1{}\mathrm {i}}\,\sqrt {1+x\,1{}\mathrm {i}}\,\sqrt {-x\,1{}\mathrm {i}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x\,1{}\mathrm {i}}\right )\middle |-1\right )\,2{}\mathrm {i}}{\sqrt {x^3+x}} \]
((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i)^(1/2)*ellipticF(asin((-x*1i)^(1 /2)), -1)*2i)/(x + x^3)^(1/2) - ((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i) ^(1/2)*ellipticPi(2^(1/2)*(- 1/2 + 1i/2), asin((-x*1i)^(1/2)), -1)*1i)/(x + x^3)^(1/2) - ((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i)^(1/2)*ellipticPi (2^(1/2)*(1/2 - 1i/2), asin((-x*1i)^(1/2)), -1)*1i)/(x + x^3)^(1/2) - ((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i)^(1/2)*ellipticPi(2^(1/2)*(1/2 + 1i/ 2), asin((-x*1i)^(1/2)), -1)*1i)/(x + x^3)^(1/2) - ((1 - x*1i)^(1/2)*(x*1i + 1)^(1/2)*(-x*1i)^(1/2)*ellipticPi(2^(1/2)*(- 1/2 - 1i/2), asin((-x*1i)^ (1/2)), -1)*1i)/(x + x^3)^(1/2)