Integrand size = 20, antiderivative size = 53 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=-\frac {\sqrt {1-x^2}}{4 x^4}-\frac {3 \sqrt {1-x^2}}{8 x^2}-\frac {3}{8} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
-1/4*(-x^2+1)^(1/2)/x^4-3/8*(-x^2+1)^(1/2)/x^2-3/8*arctanh((-x^2+1)^(1/2))
Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\frac {\left (-2-3 x^2\right ) \sqrt {1-x^2}}{8 x^4}-\frac {3}{8} \text {arctanh}\left (\sqrt {1-x^2}\right ) \] Input:
Integrate[1/(Sqrt[1 - x]*x^5*Sqrt[1 + x]),x]
Output:
((-2 - 3*x^2)*Sqrt[1 - x^2])/(8*x^4) - (3*ArcTanh[Sqrt[1 - x^2]])/8
Time = 0.17 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {114, 27, 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.
\(\displaystyle \int \frac {1}{\sqrt {1-x} x^5 \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 114 |
\(\displaystyle -\frac {1}{4} \int -\frac {3}{\sqrt {1-x} x^3 \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{4} \int \frac {1}{\sqrt {1-x} x^3 \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {3}{4} \left (-\frac {1}{2} \int -\frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle \frac {3}{4} \left (-\frac {1}{2} \int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {3}{4} \left (-\frac {1}{2} \text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{2 x^2}\right )-\frac {\sqrt {1-x} \sqrt {x+1}}{4 x^4}\) |
Input:
Int[1/(Sqrt[1 - x]*x^5*Sqrt[1 + x]),x]
Output:
-1/4*(Sqrt[1 - x]*Sqrt[1 + x])/x^4 + (3*(-1/2*(Sqrt[1 - x]*Sqrt[1 + x])/x^ 2 - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]/2))/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 68, normalized size of antiderivative = 1.28
method | result | size |
default | \(-\frac {\sqrt {1-x}\, \sqrt {1+x}\, \left (3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x^{4}+3 \sqrt {-x^{2}+1}\, x^{2}+2 \sqrt {-x^{2}+1}\right )}{8 \sqrt {-x^{2}+1}\, x^{4}}\) | \(68\) |
risch | \(\frac {\sqrt {1+x}\, \left (-1+x \right ) \left (3 x^{2}+2\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8 x^{4} \sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}-\frac {3 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) \sqrt {\left (1+x \right ) \left (1-x \right )}}{8 \sqrt {1-x}\, \sqrt {1+x}}\) | \(85\) |
Input:
int(1/(1-x)^(1/2)/x^5/(1+x)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/8*(1-x)^(1/2)*(1+x)^(1/2)*(3*arctanh(1/(-x^2+1)^(1/2))*x^4+3*(-x^2+1)^( 1/2)*x^2+2*(-x^2+1)^(1/2))/(-x^2+1)^(1/2)/x^4
Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\frac {3 \, x^{4} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) - {\left (3 \, x^{2} + 2\right )} \sqrt {x + 1} \sqrt {-x + 1}}{8 \, x^{4}} \] Input:
integrate(1/(1-x)^(1/2)/x^5/(1+x)^(1/2),x, algorithm="fricas")
Output:
1/8*(3*x^4*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) - (3*x^2 + 2)*sqrt(x + 1) *sqrt(-x + 1))/x^4
Timed out. \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\text {Timed out} \] Input:
integrate(1/(1-x)**(1/2)/x**5/(1+x)**(1/2),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=-\frac {3 \, \sqrt {-x^{2} + 1}}{8 \, x^{2}} - \frac {\sqrt {-x^{2} + 1}}{4 \, x^{4}} - \frac {3}{8} \, \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate(1/(1-x)^(1/2)/x^5/(1+x)^(1/2),x, algorithm="maxima")
Output:
-3/8*sqrt(-x^2 + 1)/x^2 - 1/4*sqrt(-x^2 + 1)/x^4 - 3/8*log(2*sqrt(-x^2 + 1 )/abs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (41) = 82\).
Time = 0.20 (sec) , antiderivative size = 326, normalized size of antiderivative = 6.15 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\frac {5 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{7} + 12 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{5} + 48 \, {\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{3} + \frac {320 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}{\sqrt {x + 1}} - \frac {320 \, \sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}}{2 \, {\left ({\left (\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} - \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}}\right )}^{2} - 4\right )}^{4}} - \frac {3}{8} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \frac {3}{8} \, \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate(1/(1-x)^(1/2)/x^5/(1+x)^(1/2),x, algorithm="giac")
Output:
1/2*(5*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt (-x + 1)))^7 + 12*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt(x + 1)/(sqr t(2) - sqrt(-x + 1)))^5 + 48*((sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - sqrt( x + 1)/(sqrt(2) - sqrt(-x + 1)))^3 + 320*(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) - 320*sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))/(((sqrt(2) - sqrt(-x + 1)) /sqrt(x + 1) - sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)))^2 - 4)^4 - 3/8*log(ab s(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + 3/8*log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) - 2))
Time = 1.67 (sec) , antiderivative size = 317, normalized size of antiderivative = 5.98 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\frac {3\,\ln \left (\frac {{\left (\sqrt {1-x}-1\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}-1\right )}{8}-\frac {3\,\ln \left (\frac {\sqrt {1-x}-1}{\sqrt {x+1}-1}\right )}{8}+\frac {\frac {3\,{\left (\sqrt {1-x}-1\right )}^2}{128\,{\left (\sqrt {x+1}-1\right )}^2}-\frac {53\,{\left (\sqrt {1-x}-1\right )}^4}{512\,{\left (\sqrt {x+1}-1\right )}^4}-\frac {87\,{\left (\sqrt {1-x}-1\right )}^6}{256\,{\left (\sqrt {x+1}-1\right )}^6}+\frac {657\,{\left (\sqrt {1-x}-1\right )}^8}{1024\,{\left (\sqrt {x+1}-1\right )}^8}-\frac {121\,{\left (\sqrt {1-x}-1\right )}^{10}}{256\,{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {1}{1024}}{\frac {{\left (\sqrt {1-x}-1\right )}^4}{{\left (\sqrt {x+1}-1\right )}^4}-\frac {4\,{\left (\sqrt {1-x}-1\right )}^6}{{\left (\sqrt {x+1}-1\right )}^6}+\frac {6\,{\left (\sqrt {1-x}-1\right )}^8}{{\left (\sqrt {x+1}-1\right )}^8}-\frac {4\,{\left (\sqrt {1-x}-1\right )}^{10}}{{\left (\sqrt {x+1}-1\right )}^{10}}+\frac {{\left (\sqrt {1-x}-1\right )}^{12}}{{\left (\sqrt {x+1}-1\right )}^{12}}}+\frac {7\,{\left (\sqrt {1-x}-1\right )}^2}{256\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {{\left (\sqrt {1-x}-1\right )}^4}{1024\,{\left (\sqrt {x+1}-1\right )}^4} \] Input:
int(1/(x^5*(1 - x)^(1/2)*(x + 1)^(1/2)),x)
Output:
(3*log(((1 - x)^(1/2) - 1)^2/((x + 1)^(1/2) - 1)^2 - 1))/8 - (3*log(((1 - x)^(1/2) - 1)/((x + 1)^(1/2) - 1)))/8 + ((3*((1 - x)^(1/2) - 1)^2)/(128*(( x + 1)^(1/2) - 1)^2) - (53*((1 - x)^(1/2) - 1)^4)/(512*((x + 1)^(1/2) - 1) ^4) - (87*((1 - x)^(1/2) - 1)^6)/(256*((x + 1)^(1/2) - 1)^6) + (657*((1 - x)^(1/2) - 1)^8)/(1024*((x + 1)^(1/2) - 1)^8) - (121*((1 - x)^(1/2) - 1)^1 0)/(256*((x + 1)^(1/2) - 1)^10) + 1/1024)/(((1 - x)^(1/2) - 1)^4/((x + 1)^ (1/2) - 1)^4 - (4*((1 - x)^(1/2) - 1)^6)/((x + 1)^(1/2) - 1)^6 + (6*((1 - x)^(1/2) - 1)^8)/((x + 1)^(1/2) - 1)^8 - (4*((1 - x)^(1/2) - 1)^10)/((x + 1)^(1/2) - 1)^10 + ((1 - x)^(1/2) - 1)^12/((x + 1)^(1/2) - 1)^12) + (7*((1 - x)^(1/2) - 1)^2)/(256*((x + 1)^(1/2) - 1)^2) + ((1 - x)^(1/2) - 1)^4/(1 024*((x + 1)^(1/2) - 1)^4)
Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.58 \[ \int \frac {1}{\sqrt {1-x} x^5 \sqrt {1+x}} \, dx=\frac {-3 \sqrt {x +1}\, \sqrt {1-x}\, x^{2}-2 \sqrt {x +1}\, \sqrt {1-x}-3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{4}+3 \,\mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{4}-3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right ) x^{4}+3 \,\mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right ) x^{4}}{8 x^{4}} \] Input:
int(1/(1-x)^(1/2)/x^5/(1+x)^(1/2),x)
Output:
( - 3*sqrt(x + 1)*sqrt( - x + 1)*x**2 - 2*sqrt(x + 1)*sqrt( - x + 1) - 3*l og( - sqrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**4 + 3*log( - s qrt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1)*x**4 - 3*log(sqrt(2) + t an(asin(sqrt( - x + 1)/sqrt(2))/2) - 1)*x**4 + 3*log(sqrt(2) + tan(asin(sq rt( - x + 1)/sqrt(2))/2) + 1)*x**4)/(8*x**4)