Integrand size = 20, antiderivative size = 35 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\frac {4 (1+x)}{\sqrt {1-x^2}}-\arcsin (x)-\text {arctanh}\left (\sqrt {1-x^2}\right ) \] Output:
4*(1+x)/(-x^2+1)^(1/2)-arcsin(x)-arctanh((-x^2+1)^(1/2))
Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=-\frac {4 \sqrt {1-x^2}}{-1+x}+4 \arctan \left (\frac {\sqrt {1+x}}{\sqrt {2}-\sqrt {1-x}}\right )+2 \text {arctanh}\left (\frac {\sqrt {1-x^2}}{-1+x}\right ) \] Input:
Integrate[(1 + x)^(3/2)/((1 - x)^(3/2)*x),x]
Output:
(-4*Sqrt[1 - x^2])/(-1 + x) + 4*ArcTan[Sqrt[1 + x]/(Sqrt[2] - Sqrt[1 - x]) ] + 2*ArcTanh[Sqrt[1 - x^2]/(-1 + x)]
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.23, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {109, 27, 140, 39, 103, 219, 223}
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.
\(\displaystyle \int \frac {(x+1)^{3/2}}{(1-x)^{3/2} x} \, dx\) |
\(\Big \downarrow \) 109 |
\(\displaystyle \frac {4 \sqrt {x+1}}{\sqrt {1-x}}-2 \int -\frac {\sqrt {1-x}}{2 x \sqrt {x+1}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \int \frac {\sqrt {1-x}}{x \sqrt {x+1}}dx+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
\(\Big \downarrow \) 140 |
\(\displaystyle -\int \frac {1}{\sqrt {1-x} \sqrt {x+1}}dx+\int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
\(\Big \downarrow \) 39 |
\(\displaystyle -\int \frac {1}{\sqrt {1-x^2}}dx+\int \frac {1}{\sqrt {1-x} x \sqrt {x+1}}dx+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle -\int \frac {1}{\sqrt {1-x^2}}dx-\int \frac {1}{1-(1-x) (x+1)}d\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\int \frac {1}{\sqrt {1-x^2}}dx-\text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\arcsin (x)-\text {arctanh}\left (\sqrt {1-x} \sqrt {x+1}\right )+\frac {4 \sqrt {x+1}}{\sqrt {1-x}}\) |
Input:
Int[(1 + x)^(3/2)/((1 - x)^(3/2)*x),x]
Output:
(4*Sqrt[1 + x])/Sqrt[1 - x] - ArcSin[x] - ArcTanh[Sqrt[1 - x]*Sqrt[1 + x]]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[( a*c + b*d*x^2)^m, x] /; FreeQ[{a, b, c, d, m}, x] && EqQ[b*c + a*d, 0] && ( IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))
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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((e + f *x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] + Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*d^(m + n)*f^p Int[(a + b*x)^(m - 1)/(c + d*x)^m, x] , x] + Int[(a + b*x)^(m - 1)*((e + f*x)^p/(c + d*x)^m)*ExpandToSum[(a + b*x )*(c + d*x)^(-p - 1) - (b*d^(-p - 1)*f^p)/(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m + n + p + 1, 0] && ILtQ[p, 0] && (GtQ[m, 0] || SumSimplerQ[m, -1] || !(GtQ[n, 0] || SumSimplerQ[n, -1]))
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_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(31)=62\).
Time = 0.06 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91
method | result | size |
default | \(\frac {\left (\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right ) x +\arcsin \left (x \right ) x -\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )-\arcsin \left (x \right )+4 \sqrt {-x^{2}+1}\right ) \sqrt {1+x}}{\sqrt {1-x}\, \sqrt {-x^{2}+1}}\) | \(67\) |
risch | \(\frac {4 \sqrt {1+x}\, \sqrt {\left (1-x \right ) \left (1+x \right )}}{\sqrt {-\left (1+x \right ) \left (-1+x \right )}\, \sqrt {1-x}}-\frac {\left (\arcsin \left (x \right )+\operatorname {arctanh}\left (\frac {1}{\sqrt {-x^{2}+1}}\right )\right ) \sqrt {\left (1-x \right ) \left (1+x \right )}}{\sqrt {1+x}\, \sqrt {1-x}}\) | \(75\) |
Input:
int((1+x)^(3/2)/(1-x)^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
(arctanh(1/(-x^2+1)^(1/2))*x+arcsin(x)*x-arctanh(1/(-x^2+1)^(1/2))-arcsin( x)+4*(-x^2+1)^(1/2))*(1+x)^(1/2)/(1-x)^(1/2)/(-x^2+1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (31) = 62\).
Time = 0.09 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.11 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\frac {2 \, {\left (x - 1\right )} \arctan \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + {\left (x - 1\right )} \log \left (\frac {\sqrt {x + 1} \sqrt {-x + 1} - 1}{x}\right ) + 4 \, x - 4 \, \sqrt {x + 1} \sqrt {-x + 1} - 4}{x - 1} \] Input:
integrate((1+x)^(3/2)/(1-x)^(3/2)/x,x, algorithm="fricas")
Output:
(2*(x - 1)*arctan((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + (x - 1)*log((sqrt(x + 1)*sqrt(-x + 1) - 1)/x) + 4*x - 4*sqrt(x + 1)*sqrt(-x + 1) - 4)/(x - 1)
\[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\int \frac {\left (x + 1\right )^{\frac {3}{2}}}{x \left (1 - x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((1+x)**(3/2)/(1-x)**(3/2)/x,x)
Output:
Integral((x + 1)**(3/2)/(x*(1 - x)**(3/2)), x)
Time = 0.10 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.51 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\frac {4 \, x}{\sqrt {-x^{2} + 1}} + \frac {4}{\sqrt {-x^{2} + 1}} - \arcsin \left (x\right ) - \log \left (\frac {2 \, \sqrt {-x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(3/2)/x,x, algorithm="maxima")
Output:
4*x/sqrt(-x^2 + 1) + 4/sqrt(-x^2 + 1) - arcsin(x) - log(2*sqrt(-x^2 + 1)/a bs(x) + 2/abs(x))
Leaf count of result is larger than twice the leaf count of optimal. 163 vs. \(2 (31) = 62\).
Time = 0.17 (sec) , antiderivative size = 163, normalized size of antiderivative = 4.66 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=-\pi - \frac {4 \, \sqrt {x + 1} \sqrt {-x + 1}}{x - 1} - 2 \, \arctan \left (\frac {\sqrt {x + 1} {\left (\frac {{\left (\sqrt {2} - \sqrt {-x + 1}\right )}^{2}}{x + 1} - 1\right )}}{2 \, {\left (\sqrt {2} - \sqrt {-x + 1}\right )}}\right ) - \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} + 2 \right |}\right ) + \log \left ({\left | -\frac {\sqrt {2} - \sqrt {-x + 1}}{\sqrt {x + 1}} + \frac {\sqrt {x + 1}}{\sqrt {2} - \sqrt {-x + 1}} - 2 \right |}\right ) \] Input:
integrate((1+x)^(3/2)/(1-x)^(3/2)/x,x, algorithm="giac")
Output:
-pi - 4*sqrt(x + 1)*sqrt(-x + 1)/(x - 1) - 2*arctan(1/2*sqrt(x + 1)*((sqrt (2) - sqrt(-x + 1))^2/(x + 1) - 1)/(sqrt(2) - sqrt(-x + 1))) - log(abs(-(s qrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt(2) - sqrt(-x + 1)) + 2)) + log(abs(-(sqrt(2) - sqrt(-x + 1))/sqrt(x + 1) + sqrt(x + 1)/(sqrt( 2) - sqrt(-x + 1)) - 2))
Timed out. \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\int \frac {{\left (x+1\right )}^{3/2}}{x\,{\left (1-x\right )}^{3/2}} \,d x \] Input:
int((x + 1)^(3/2)/(x*(1 - x)^(3/2)),x)
Output:
int((x + 1)^(3/2)/(x*(1 - x)^(3/2)), x)
Time = 0.22 (sec) , antiderivative size = 150, normalized size of antiderivative = 4.29 \[ \int \frac {(1+x)^{3/2}}{(1-x)^{3/2} x} \, dx=\frac {2 \sqrt {1-x}\, \mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )-\sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {1-x}\, \mathrm {log}\left (-\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right )-\sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )-1\right )+\sqrt {1-x}\, \mathrm {log}\left (\sqrt {2}+\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {1-x}}{\sqrt {2}}\right )}{2}\right )+1\right )+4 \sqrt {x +1}}{\sqrt {1-x}} \] Input:
int((1+x)^(3/2)/(1-x)^(3/2)/x,x)
Output:
(2*sqrt( - x + 1)*asin(sqrt( - x + 1)/sqrt(2)) - sqrt( - x + 1)*log( - sqr t(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1) + sqrt( - x + 1)*log( - sq rt(2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1) - sqrt( - x + 1)*log(sqrt (2) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) - 1) + sqrt( - x + 1)*log(sqrt(2 ) + tan(asin(sqrt( - x + 1)/sqrt(2))/2) + 1) + 4*sqrt(x + 1))/sqrt( - x + 1)