Integrand size = 19, antiderivative size = 74 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-1-x^4}}\right )}{2 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {-1-x^4}} \] Output:
1/4*arctanh(2^(1/2)*x/(-x^4-1)^(1/2))*2^(1/2)+1/4*(x^2+1)*((x^4+1)/(x^2+1) ^2)^(1/2)*InverseJacobiAM(2*arctan(x),1/2*2^(1/2))/(-x^4-1)^(1/2)
Result contains complex when optimal does not.
Time = 10.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=-\frac {\sqrt [4]{-1} \sqrt {1+x^4} \operatorname {EllipticPi}\left (-i,i \text {arcsinh}\left (\sqrt [4]{-1} x\right ),-1\right )}{\sqrt {-1-x^4}} \] Input:
Integrate[1/((1 + x^2)*Sqrt[-1 - x^4]),x]
Output:
-(((-1)^(1/4)*Sqrt[1 + x^4]*EllipticPi[-I, I*ArcSinh[(-1)^(1/4)*x], -1])/S qrt[-1 - x^4])
Time = 0.43 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {1535, 761, 2213, 219}
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.
\(\displaystyle \int \frac {1}{\left (x^2+1\right ) \sqrt {-x^4-1}} \, dx\) |
\(\Big \downarrow \) 1535 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {-x^4-1}}dx+\frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {-x^4-1}}dx\) |
\(\Big \downarrow \) 761 |
\(\displaystyle \frac {1}{2} \int \frac {1-x^2}{\left (x^2+1\right ) \sqrt {-x^4-1}}dx+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {-x^4-1}}\) |
\(\Big \downarrow \) 2213 |
\(\displaystyle \frac {1}{2} \int \frac {1}{1-\frac {2 x^2}{-x^4-1}}d\frac {x}{\sqrt {-x^4-1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {-x^4-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan (x),\frac {1}{2}\right )}{4 \sqrt {-x^4-1}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} x}{\sqrt {-x^4-1}}\right )}{2 \sqrt {2}}\) |
Input:
Int[1/((1 + x^2)*Sqrt[-1 - x^4]),x]
Output:
ArcTanh[(Sqrt[2]*x)/Sqrt[-1 - x^4]]/(2*Sqrt[2]) + ((1 + x^2)*Sqrt[(1 + x^4 )/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/2])/(4*Sqrt[-1 - x^4])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[( 1 + q^2*x^2)*(Sqrt[(a + b*x^4)/(a*(1 + q^2*x^2)^2)]/(2*q*Sqrt[a + b*x^4]))* EllipticF[2*ArcTan[q*x], 1/2], x]] /; FreeQ[{a, b}, x] && PosQ[b/a]
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> Simp[ 1/(2*d) Int[1/Sqrt[a + c*x^4], x], x] + Simp[1/(2*d) Int[(d - e*x^2)/(( d + e*x^2)*Sqrt[a + c*x^4]), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[c*d^2 - a*e^2, 0]
Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]) , x_Symbol] :> Simp[A Subst[Int[1/(d + 2*a*e*x^2), x], x, x/Sqrt[a + c*x^ 4]], x] /; FreeQ[{a, c, d, e, A, B}, x] && EqQ[c*d^2 - a*e^2, 0] && EqQ[B*d + A*e, 0]
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.45
method | result | size |
default | \(\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \operatorname {EllipticPi}\left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}+\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \operatorname {EllipticPi}\left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\) | \(107\) |
elliptic | \(\frac {i \sqrt {-i}\, \sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \operatorname {EllipticPi}\left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-x^{4}-1}}+\frac {\sqrt {i x^{2}+1}\, \sqrt {-i x^{2}+1}\, \operatorname {EllipticPi}\left (\sqrt {-i}\, x , -i, \frac {\left (-1\right )^{\frac {1}{4}}}{\sqrt {-i}}\right )}{2 \sqrt {-i}\, \sqrt {-x^{4}-1}}\) | \(107\) |
Input:
int(1/(x^2+1)/(-x^4-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/2*I*(-I)^(1/2)*(1+I*x^2)^(1/2)*(1-I*x^2)^(1/2)/(-x^4-1)^(1/2)*EllipticPi ((-I)^(1/2)*x,-I,(-1)^(1/4)/(-I)^(1/2))+1/2/(-I)^(1/2)*(1+I*x^2)^(1/2)*(1- I*x^2)^(1/2)/(-x^4-1)^(1/2)*EllipticPi((-I)^(1/2)*x,-I,(-1)^(1/4)/(-I)^(1/ 2))
Result contains complex when optimal does not.
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.01 \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=-\frac {1}{2} \, \sqrt {i} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + \frac {1}{8} \, \sqrt {2} \log \left (\frac {\sqrt {2} x + \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) - \frac {1}{8} \, \sqrt {2} \log \left (-\frac {\sqrt {2} x - \sqrt {-x^{4} - 1}}{x^{2} + 1}\right ) \] Input:
integrate(1/(x^2+1)/(-x^4-1)^(1/2),x, algorithm="fricas")
Output:
-1/2*sqrt(I)*elliptic_f(arcsin(sqrt(I)*x), -1) + 1/8*sqrt(2)*log((sqrt(2)* x + sqrt(-x^4 - 1))/(x^2 + 1)) - 1/8*sqrt(2)*log(-(sqrt(2)*x - sqrt(-x^4 - 1))/(x^2 + 1))
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=\int \frac {1}{\left (x^{2} + 1\right ) \sqrt {- x^{4} - 1}}\, dx \] Input:
integrate(1/(x**2+1)/(-x**4-1)**(1/2),x)
Output:
Integral(1/((x**2 + 1)*sqrt(-x**4 - 1)), x)
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} - 1} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(1/(x^2+1)/(-x^4-1)^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(-x^4 - 1)*(x^2 + 1)), x)
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=\int { \frac {1}{\sqrt {-x^{4} - 1} {\left (x^{2} + 1\right )}} \,d x } \] Input:
integrate(1/(x^2+1)/(-x^4-1)^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(-x^4 - 1)*(x^2 + 1)), x)
Timed out. \[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=\int \frac {1}{\left (x^2+1\right )\,\sqrt {-x^4-1}} \,d x \] Input:
int(1/((x^2 + 1)*(- x^4 - 1)^(1/2)),x)
Output:
int(1/((x^2 + 1)*(- x^4 - 1)^(1/2)), x)
\[ \int \frac {1}{\left (1+x^2\right ) \sqrt {-1-x^4}} \, dx=-\left (\int \frac {\sqrt {x^{4}+1}}{x^{6}+x^{4}+x^{2}+1}d x \right ) i \] Input:
int(1/(x^2+1)/(-x^4-1)^(1/2),x)
Output:
- int(sqrt(x**4 + 1)/(x**6 + x**4 + x**2 + 1),x)*i