Integrand size = 20, antiderivative size = 35 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=-\frac {\sqrt {1-x^2}}{2 x^2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
-1/2*(-x^2+1)^(1/2)/x^2-1/2*arctanh((-x^2+1)^(1/2))
Time = 0.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=-\frac {\sqrt {1-x^2}}{2 x^2}-\frac {1}{2} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Input:
Integrate[1/(Sqrt[1 - x]*x^3*Sqrt[1 + x]),x]
Output:
-1/2*Sqrt[1 - x^2]/x^2 - ArcTanh[Sqrt[1 - x^2]]/2
Time = 0.15 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.31, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {114, 25, 103, 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.
\(\displaystyle \int \frac {1}{\sqrt {1-x} x^3 \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{2} \int -\frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle -\frac {1}{2} \int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {1}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\) |
Input:
Int[1/(Sqrt[1 - x]*x^3*Sqrt[1 + x]),x]
Output:
-1/2*(Sqrt[1 - x]*Sqrt[1 + x])/x^2 - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]/2
Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_ ))), x_] :> Simp[b*f Subst[Int[1/(d*(b*e - a*f)^2 + b*f^2*x^2), x], x, Sq rt[a + b*x]*Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[2*b*d *e - f*(b*c + a*d), 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
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])
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.46
method | result | size |
default | \(-\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{2}+\sqrt {-x^{2}+1}\right )}{2 \sqrt {-x^{2}+1}\, x^{2}}\) | \(51\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 x^{2} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}-\frac {\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{2 \sqrt {1-x}\, \sqrt {1+x}}\) | \(78\) |
Input:
int(1/(1-x)^(1/2)/x^3/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*(1-x)^(1/2)*(1+x)^(1/2)*(arctanh(1/(-x^2+1)^(1/2))*x^2+(-x^2+1)^(1/2) )/(-x^2+1)^(1/2)/x^2
Time = 0.07 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=\frac {x^{2} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - \sqrt {x + 1} \sqrt {-x + 1}}{2 \, x^{2}} \] Input:
integrate(1/(1-x)^(1/2)/x^3/(1+x)^(1/2),x, algorithm="fricas")
Output:
1/2*(x^2*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - sqrt(x + 1)*sqrt(-x + 1)) /x^2
Timed out. \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=\text {Timed out} \] Input:
integrate(1/(1-x)**(1/2)/x**3/(1+x)**(1/2),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.14 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=-\frac {\sqrt {-x^{2} + 1}}{2 \, x^{2}} - \frac {1}{2} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate(1/(1-x)^(1/2)/x^3/(1+x)^(1/2),x, algorithm="maxima")
Output:
-1/2*sqrt(-x^2 + 1)/x^2 - 1/2*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (27) = 54\).
Time = 0.17 (sec) , antiderivative size = 232, normalized size of antiderivative = 6.63 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=\frac {2 \, {\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} + \frac {4 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {4 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}}{{\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{2}} - \frac {1}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {1}{2} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate(1/(1-x)^(1/2)/x^3/(1+x)^(1/2),x, algorithm="giac")
Output:
2*(((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^3 + 4*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 4*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sq rt(2) - sqrt(-x + 1)))^2 - 4)^2 - 1/2*log(abs(-(sqrt(2) - sqrt(-x + 1))/sq rt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + 1/2*log(abs(-(sqr t(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 2))
Time = 1.45 (sec) , antiderivative size = 186, normalized size of antiderivative = 5.31 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=\frac {\ln \left (\frac {{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}-1\right )}{2}-\frac {\ln \left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )}{2}+\frac {{\left (\sqrt {1-x}-1\right )}^2}{32\,{\left (\sqrt {x+1}-1\right )}^2}-\frac {\frac {{\left (\sqrt {1-x}-1\right )}^2}{16\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {15\,{\left (\sqrt {1-x}-1\right )}^4}{32\,{\left (\sqrt {x+1}-1\right )}^4}-\frac {1}{32}}{\frac {{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}-\frac {2\,{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}+\frac {{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}} \] Input:
int(1/(x^3*(1 - x)^(1/2)*(x + 1)^(1/2)),x)
Output:
log(((1 - x)^(1/2) - 1)^2/((x + 1)^(1/2) - 1)^2 - 1)/2 - log(((1 - x)^(1/2 ) - 1)/((x + 1)^(1/2) - 1))/2 + ((1 - x)^(1/2) - 1)^2/(32*((x + 1)^(1/2) - 1)^2) - (((1 - x)^(1/2) - 1)^2/(16*((x + 1)^(1/2) - 1)^2) + (15*((1 - x)^ (1/2) - 1)^4)/(32*((x + 1)^(1/2) - 1)^4) - 1/32)/(((1 - x)^(1/2) - 1)^2/(( x + 1)^(1/2) - 1)^2 - (2*((1 - x)^(1/2) - 1)^4)/((x + 1)^(1/2) - 1)^4 + (( 1 - x)^(1/2) - 1)^6/((x + 1)^(1/2) - 1)^6)
Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 3.43 \[ \int \frac {1}{\sqrt {1-x} x^3 \sqrt {1+x}} \, dx=\frac {-\sqrt {x +1}\, \sqrt {1-x}-\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}-\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{2}+\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{2}}{2 x^{2}} \] Input:
int(1/(1-x)^(1/2)/x^3/(1+x)^(1/2),x)
Output:
( - sqrt(x + 1)*sqrt( - x + 1) - log( - sqrt(2) + tan(asin(sqrt( - x + 1)/ sqrt(2))/2) - 1)*x**2 + log( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/ 2) + 1)*x**2 - log(sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**2 + log(sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**2)/(2*x**2)