Integrand size = 18, antiderivative size = 60 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}+\text {arctanh}\left (1-2 x^2-2 \sqrt {1+x^4}\right )-\frac {1}{2} \log \left (1+x^2+\sqrt {1+x^4}\right ) \] Output:
1/2*(x^2+1)*(x^4+1)^(1/2)/x^2-arctanh(-1+2*x^2+2*(x^4+1)^(1/2))-1/2*ln(1+x ^2+(x^4+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {\left (1+x^2\right ) \sqrt {1+x^4}}{2 x^2}+\text {arctanh}\left (x^2-\sqrt {1+x^4}\right )+\frac {1}{2} \log \left (-x^2+\sqrt {1+x^4}\right ) \] Input:
Integrate[((-1 + x^2)*Sqrt[1 + x^4])/x^3,x]
Output:
((1 + x^2)*Sqrt[1 + x^4])/(2*x^2) + ArcTanh[x^2 - Sqrt[1 + x^4]] + Log[-x^ 2 + Sqrt[1 + x^4]]/2
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.68, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {1579, 25, 536, 538, 222, 243, 73, 220}
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.
\(\displaystyle \int \frac {\left (x^2-1\right ) \sqrt {x^4+1}}{x^3} \, dx\) |
\(\Big \downarrow \) 1579 |
\(\displaystyle \frac {1}{2} \int -\frac {\left (1-x^2\right ) \sqrt {x^4+1}}{x^4}dx^2\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{2} \int \frac {\left (1-x^2\right ) \sqrt {x^4+1}}{x^4}dx^2\) |
\(\Big \downarrow \) 536 |
\(\displaystyle \frac {1}{2} \left (\frac {\left (x^2+1\right ) \sqrt {x^4+1}}{x^2}-\int \frac {x^2-1}{x^2 \sqrt {x^4+1}}dx^2\right )\) |
\(\Big \downarrow \) 538 |
\(\displaystyle \frac {1}{2} \left (-\int \frac {1}{\sqrt {x^4+1}}dx^2+\int \frac {1}{x^2 \sqrt {x^4+1}}dx^2+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{x^2 \sqrt {x^4+1}}dx^2-\text {arcsinh}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right )\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2 \sqrt {x^4+1}}dx^4-\text {arcsinh}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{\sqrt {x^4+1}-1}d\sqrt {x^4+1}-\text {arcsinh}\left (x^2\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right )\) |
\(\Big \downarrow \) 220 |
\(\displaystyle \frac {1}{2} \left (-\text {arcsinh}\left (x^2\right )-\text {arctanh}\left (\sqrt {x^4+1}\right )+\frac {\sqrt {x^4+1} \left (x^2+1\right )}{x^2}\right )\) |
Input:
Int[((-1 + x^2)*Sqrt[1 + x^4])/x^3,x]
Output:
(((1 + x^2)*Sqrt[1 + x^4])/x^2 - ArcSinh[x^2] - ArcTanh[Sqrt[1 + x^4]])/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_))/(x_)^2, x_Symbol] :> S imp[(-(2*c*p - d*x))*((a + b*x^2)^p/(2*p*x)), x] + Int[(a*d + 2*b*c*p*x)*(( a + b*x^2)^(p - 1)/x), x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && Integer Q[2*p]
Int[((c_) + (d_.)*(x_))/((x_)*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> Simp [c Int[1/(x*Sqrt[a + b*x^2]), x], x] + Simp[d Int[1/Sqrt[a + b*x^2], x] , x] /; FreeQ[{a, b, c, d}, x]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.63
method | result | size |
trager | \(\frac {\left (x^{2}+1\right ) \sqrt {x^{4}+1}}{2 x^{2}}-\ln \left (\frac {1+x^{2}+\sqrt {x^{4}+1}}{x}\right )\) | \(38\) |
risch | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}\) | \(39\) |
elliptic | \(\frac {\sqrt {x^{4}+1}}{2 x^{2}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}\) | \(39\) |
pseudoelliptic | \(\frac {\sqrt {x^{4}+1}\, x^{2}-\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right ) x^{2}-x^{2} \operatorname {arcsinh}\left (x^{2}\right )+\sqrt {x^{4}+1}}{2 x^{2}}\) | \(47\) |
default | \(\frac {\sqrt {x^{4}+1}}{2}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {x^{4}+1}}\right )}{2}+\frac {\left (x^{4}+1\right )^{\frac {3}{2}}}{2 x^{2}}-\frac {\sqrt {x^{4}+1}\, x^{2}}{2}-\frac {\operatorname {arcsinh}\left (x^{2}\right )}{2}\) | \(51\) |
meijerg | \(\frac {\frac {4 \sqrt {\pi }\, \sqrt {x^{4}+1}}{x^{2}}-4 \sqrt {\pi }\, \operatorname {arcsinh}\left (x^{2}\right )}{8 \sqrt {\pi }}-\frac {-2 \left (2-2 \ln \left (2\right )+4 \ln \left (x \right )\right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {x^{4}+1}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {x^{4}+1}}{2}\right )}{8 \sqrt {\pi }}\) | \(87\) |
Input:
int((x^2-1)*(x^4+1)^(1/2)/x^3,x,method=_RETURNVERBOSE)
Output:
1/2*(x^2+1)*(x^4+1)^(1/2)/x^2-ln((1+x^2+(x^4+1)^(1/2))/x)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.22 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {x^{2} \log \left (2 \, x^{4} + x^{2} - \sqrt {x^{4} + 1} {\left (2 \, x^{2} + 1\right )} + 1\right ) - x^{2} \log \left (-x^{2} + \sqrt {x^{4} + 1} + 1\right ) + x^{2} + \sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}}{2 \, x^{2}} \] Input:
integrate((x^2-1)*(x^4+1)^(1/2)/x^3,x, algorithm="fricas")
Output:
1/2*(x^2*log(2*x^4 + x^2 - sqrt(x^4 + 1)*(2*x^2 + 1) + 1) - x^2*log(-x^2 + sqrt(x^4 + 1) + 1) + x^2 + sqrt(x^4 + 1)*(x^2 + 1))/x^2
Time = 2.75 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.25 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {x^{2}}{2 \sqrt {x^{4} + 1}} + \frac {x^{2}}{2 \sqrt {1 + \frac {1}{x^{4}}}} - \frac {\operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{2} - \frac {\operatorname {asinh}{\left (x^{2} \right )}}{2} + \frac {1}{2 x^{2} \sqrt {x^{4} + 1}} + \frac {1}{2 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} \] Input:
integrate((x**2-1)*(x**4+1)**(1/2)/x**3,x)
Output:
x**2/(2*sqrt(x**4 + 1)) + x**2/(2*sqrt(1 + x**(-4))) - asinh(x**(-2))/2 - asinh(x**2)/2 + 1/(2*x**2*sqrt(x**4 + 1)) + 1/(2*x**2*sqrt(1 + x**(-4)))
Time = 0.12 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.30 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {\sqrt {x^{4} + 1}}{2 \, x^{2}} - \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) + \frac {1}{4} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) - \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) + \frac {1}{4} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \] Input:
integrate((x^2-1)*(x^4+1)^(1/2)/x^3,x, algorithm="maxima")
Output:
1/2*sqrt(x^4 + 1) + 1/2*sqrt(x^4 + 1)/x^2 - 1/4*log(sqrt(x^4 + 1) + 1) + 1 /4*log(sqrt(x^4 + 1) - 1) - 1/4*log(sqrt(x^4 + 1)/x^2 + 1) + 1/4*log(sqrt( x^4 + 1)/x^2 - 1)
Time = 0.36 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.35 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {1}{2} \, \sqrt {x^{4} + 1} - \frac {1}{{\left (x^{2} - \sqrt {x^{4} + 1}\right )}^{2} - 1} + \frac {1}{2} \, \log \left (x^{2} - \sqrt {x^{4} + 1} + 1\right ) - \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1} + 1\right ) + \frac {1}{2} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \] Input:
integrate((x^2-1)*(x^4+1)^(1/2)/x^3,x, algorithm="giac")
Output:
1/2*sqrt(x^4 + 1) - 1/((x^2 - sqrt(x^4 + 1))^2 - 1) + 1/2*log(x^2 - sqrt(x ^4 + 1) + 1) - 1/2*log(-x^2 + sqrt(x^4 + 1) + 1) + 1/2*log(-x^2 + sqrt(x^4 + 1))
Time = 8.56 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {\sqrt {x^4+1}}{2}-\frac {\mathrm {asinh}\left (x^2\right )}{2}+\frac {\sqrt {x^4+1}}{2\,x^2}+\frac {\mathrm {atan}\left (\sqrt {x^4+1}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2} \] Input:
int(((x^2 - 1)*(x^4 + 1)^(1/2))/x^3,x)
Output:
(atan((x^4 + 1)^(1/2)*1i)*1i)/2 - asinh(x^2)/2 + (x^4 + 1)^(1/2)/2 + (x^4 + 1)^(1/2)/(2*x^2)
Time = 0.16 (sec) , antiderivative size = 422, normalized size of antiderivative = 7.03 \[ \int \frac {\left (-1+x^2\right ) \sqrt {1+x^4}}{x^3} \, dx=\frac {-4 \sqrt {x^{4}+1}\, \mathrm {log}\left (\sqrt {x^{4}+1}+x^{2}\right ) x^{4}+2 \sqrt {x^{4}+1}\, \mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}-\sqrt {x^{4}+1}+x^{4}-x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{4}-2 \sqrt {x^{4}+1}\, \mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}+\sqrt {x^{4}+1}+x^{4}+x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{4}+4 \sqrt {x^{4}+1}\, x^{6}+8 \sqrt {x^{4}+1}\, x^{4}+2 \sqrt {x^{4}+1}\, x^{2}+2 \sqrt {x^{4}+1}-4 \,\mathrm {log}\left (\sqrt {x^{4}+1}+x^{2}\right ) x^{6}-2 \,\mathrm {log}\left (\sqrt {x^{4}+1}+x^{2}\right ) x^{2}+2 \,\mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}-\sqrt {x^{4}+1}+x^{4}-x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{6}+\mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}-\sqrt {x^{4}+1}+x^{4}-x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{2}-2 \,\mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}+\sqrt {x^{4}+1}+x^{4}+x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{6}-\mathrm {log}\left (\frac {\sqrt {x^{4}+1}\, x^{2}+\sqrt {x^{4}+1}+x^{4}+x^{2}+1}{\sqrt {x^{4}+1}+x^{2}}\right ) x^{2}+4 x^{8}+8 x^{6}+4 x^{4}+6 x^{2}}{4 x^{2} \left (2 \sqrt {x^{4}+1}\, x^{2}+2 x^{4}+1\right )} \] Input:
int((x^2-1)*(x^4+1)^(1/2)/x^3,x)
Output:
( - 4*sqrt(x**4 + 1)*log(sqrt(x**4 + 1) + x**2)*x**4 + 2*sqrt(x**4 + 1)*lo g((sqrt(x**4 + 1)*x**2 - sqrt(x**4 + 1) + x**4 - x**2 + 1)/(sqrt(x**4 + 1) + x**2))*x**4 - 2*sqrt(x**4 + 1)*log((sqrt(x**4 + 1)*x**2 + sqrt(x**4 + 1 ) + x**4 + x**2 + 1)/(sqrt(x**4 + 1) + x**2))*x**4 + 4*sqrt(x**4 + 1)*x**6 + 8*sqrt(x**4 + 1)*x**4 + 2*sqrt(x**4 + 1)*x**2 + 2*sqrt(x**4 + 1) - 4*lo g(sqrt(x**4 + 1) + x**2)*x**6 - 2*log(sqrt(x**4 + 1) + x**2)*x**2 + 2*log( (sqrt(x**4 + 1)*x**2 - sqrt(x**4 + 1) + x**4 - x**2 + 1)/(sqrt(x**4 + 1) + x**2))*x**6 + log((sqrt(x**4 + 1)*x**2 - sqrt(x**4 + 1) + x**4 - x**2 + 1 )/(sqrt(x**4 + 1) + x**2))*x**2 - 2*log((sqrt(x**4 + 1)*x**2 + sqrt(x**4 + 1) + x**4 + x**2 + 1)/(sqrt(x**4 + 1) + x**2))*x**6 - log((sqrt(x**4 + 1) *x**2 + sqrt(x**4 + 1) + x**4 + x**2 + 1)/(sqrt(x**4 + 1) + x**2))*x**2 + 4*x**8 + 8*x**6 + 4*x**4 + 6*x**2)/(4*x**2*(2*sqrt(x**4 + 1)*x**2 + 2*x**4 + 1))