Integrand size = 21, antiderivative size = 33 \[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\frac {x \sqrt {1+x^2}}{\sqrt {1-x^2}}-E(\arcsin (x)|-1)+\operatorname {EllipticF}(\arcsin (x),-1) \] Output:
x*(x^2+1)^(1/2)/(-x^2+1)^(1/2)-EllipticE(x,I)+EllipticF(x,I)
Time = 1.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=-\frac {x \sqrt {1-x^4}}{-1+x^2}-E(\arcsin (x)|-1)+\operatorname {EllipticF}(\arcsin (x),-1) \] Input:
Integrate[Sqrt[1 + x^2]/(1 - x^2)^(3/2),x]
Output:
-((x*Sqrt[1 - x^4])/(-1 + x^2)) - EllipticE[ArcSin[x], -1] + EllipticF[Arc Sin[x], -1]
Time = 0.20 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {314, 335, 836, 762, 1388, 327}
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.
\(\displaystyle \int \frac {\sqrt {x^2+1}}{\left (1-x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 314 |
\(\displaystyle \frac {x \sqrt {x^2+1}}{\sqrt {1-x^2}}-\int \frac {x^2}{\sqrt {1-x^2} \sqrt {x^2+1}}dx\) |
\(\Big \downarrow \) 335 |
\(\displaystyle \frac {x \sqrt {x^2+1}}{\sqrt {1-x^2}}-\int \frac {x^2}{\sqrt {1-x^4}}dx\) |
\(\Big \downarrow \) 836 |
\(\displaystyle \int \frac {1}{\sqrt {1-x^4}}dx-\int \frac {x^2+1}{\sqrt {1-x^4}}dx+\frac {\sqrt {x^2+1} x}{\sqrt {1-x^2}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -\int \frac {x^2+1}{\sqrt {1-x^4}}dx+\operatorname {EllipticF}(\arcsin (x),-1)+\frac {\sqrt {x^2+1} x}{\sqrt {1-x^2}}\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -\int \frac {\sqrt {x^2+1}}{\sqrt {1-x^2}}dx+\operatorname {EllipticF}(\arcsin (x),-1)+\frac {\sqrt {x^2+1} x}{\sqrt {1-x^2}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \operatorname {EllipticF}(\arcsin (x),-1)-E(\arcsin (x)|-1)+\frac {\sqrt {x^2+1} x}{\sqrt {1-x^2}}\) |
Input:
Int[Sqrt[1 + x^2]/(1 - x^2)^(3/2),x]
Output:
(x*Sqrt[1 + x^2])/Sqrt[1 - x^2] - EllipticE[ArcSin[x], -1] + EllipticF[Arc Sin[x], -1]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* (p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d *(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(p _.), x_Symbol] :> Int[(e*x)^m*(a*c + b*d*x^4)^p, x] /; FreeQ[{a, b, c, d, e , m, p}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] ))
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
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]))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (29 ) = 58\).
Time = 1.85 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.33
method | result | size |
default | \(\frac {\left (-\operatorname {EllipticF}\left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}+\operatorname {EllipticE}\left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}-x^{3}-x \right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{x^{4}-1}\) | \(77\) |
risch | \(\frac {x \sqrt {x^{2}+1}\, \sqrt {\left (x^{2}+1\right ) \left (-x^{2}+1\right )}}{\sqrt {-\left (x^{2}-1\right ) \left (x^{2}+1\right )}\, \sqrt {-x^{2}+1}}+\frac {\left (\operatorname {EllipticF}\left (x , i\right )-\operatorname {EllipticE}\left (x , i\right )\right ) \sqrt {\left (x^{2}+1\right ) \left (-x^{2}+1\right )}}{\sqrt {-x^{4}+1}}\) | \(85\) |
elliptic | \(\frac {\sqrt {-x^{4}+1}\, \left (-\frac {\left (-x^{2}-1\right ) x}{\sqrt {\left (-x^{2}-1\right ) \left (x^{2}-1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , i\right )-\operatorname {EllipticE}\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\right )}{\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}\) | \(90\) |
Input:
int((x^2+1)^(1/2)/(-x^2+1)^(3/2),x,method=_RETURNVERBOSE)
Output:
(-EllipticF(x,I)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)+EllipticE(x,I)*(-x^2+1)^(1/2 )*(x^2+1)^(1/2)-x^3-x)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/(x^4-1)
Time = 0.08 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=-\frac {\sqrt {x^{2} + 1} \sqrt {-x^{2} + 1} x + {\left (x^{2} - 1\right )} E(\arcsin \left (x\right )\,|\,-1) - {\left (x^{2} - 1\right )} F(\arcsin \left (x\right )\,|\,-1)}{x^{2} - 1} \] Input:
integrate((x^2+1)^(1/2)/(-x^2+1)^(3/2),x, algorithm="fricas")
Output:
-(sqrt(x^2 + 1)*sqrt(-x^2 + 1)*x + (x^2 - 1)*elliptic_e(arcsin(x), -1) - ( x^2 - 1)*elliptic_f(arcsin(x), -1))/(x^2 - 1)
\[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x^{2} + 1}}{\left (- \left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((x**2+1)**(1/2)/(-x**2+1)**(3/2),x)
Output:
Integral(sqrt(x**2 + 1)/(-(x - 1)*(x + 1))**(3/2), x)
\[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((x^2+1)^(1/2)/(-x^2+1)^(3/2),x, algorithm="maxima")
Output:
integrate(sqrt(x^2 + 1)/(-x^2 + 1)^(3/2), x)
\[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {x^{2} + 1}}{{\left (-x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((x^2+1)^(1/2)/(-x^2+1)^(3/2),x, algorithm="giac")
Output:
integrate(sqrt(x^2 + 1)/(-x^2 + 1)^(3/2), x)
Timed out. \[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {x^2+1}}{{\left (1-x^2\right )}^{3/2}} \,d x \] Input:
int((x^2 + 1)^(1/2)/(1 - x^2)^(3/2),x)
Output:
int((x^2 + 1)^(1/2)/(1 - x^2)^(3/2), x)
\[ \int \frac {\sqrt {1+x^2}}{\left (1-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{x^{4}-2 x^{2}+1}d x \] Input:
int((x^2+1)^(1/2)/(-x^2+1)^(3/2),x)
Output:
int((sqrt( - x**2 + 1)*sqrt(x**2 + 1))/(x**4 - 2*x**2 + 1),x)