Integrand size = 36, antiderivative size = 39 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=-\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+2 a \arctan \left (\sqrt {-1+x} \sqrt {1+x}\right ) \]
Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.95 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=-\frac {\sqrt {-1+x} \sqrt {1+x}}{x}+4 a \arctan \left (\sqrt {\frac {-1+x}{1+x}}\right ) \]
Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {206, 168, 27, 103, 216}
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.
\(\displaystyle \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {x-1} x^2 \sqrt {x+1}} \, dx\) |
\(\Big \downarrow \) 206 |
\(\displaystyle \int \frac {2 a x-1}{\sqrt {x-1} x^2 \sqrt {x+1}}dx\) |
\(\Big \downarrow \) 168 |
\(\displaystyle \int \frac {2 a}{\sqrt {x-1} x \sqrt {x+1}}dx-\frac {\sqrt {x-1} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 a \int \frac {1}{\sqrt {x-1} x \sqrt {x+1}}dx-\frac {\sqrt {x-1} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 103 |
\(\displaystyle 2 a \int \frac {1}{(x-1) (x+1)+1}d\left (\sqrt {x-1} \sqrt {x+1}\right )-\frac {\sqrt {x-1} \sqrt {x+1}}{x}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 2 a \arctan \left (\sqrt {x-1} \sqrt {x+1}\right )-\frac {\sqrt {x-1} \sqrt {x+1}}{x}\) |
3.1.32.3.1 Defintions of rubi rules used
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_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(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] + S imp[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*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
Int[(u_)^(m_.)*(v_)^(n_.)*(w_)^(p_.)*(z_)^(q_.), x_Symbol] :> Int[ExpandToS um[u, x]^m*ExpandToSum[v, x]^n*ExpandToSum[w, x]^p*ExpandToSum[z, x]^q, x] /; FreeQ[{m, n, p, q}, x] && LinearQ[{u, v, w, z}, x] && !LinearMatchQ[{u, v, w, z}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Time = 5.34 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {\left (-2 a x \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right )-\sqrt {x^{2}-1}\right ) \sqrt {-1+x}\, \sqrt {1+x}}{x \sqrt {x^{2}-1}}\) | \(44\) |
risch | \(-\frac {\sqrt {-1+x}\, \sqrt {1+x}}{x}-\frac {2 a \arctan \left (\frac {1}{\sqrt {x^{2}-1}}\right ) \sqrt {\left (-1+x \right ) \left (1+x \right )}}{\sqrt {-1+x}\, \sqrt {1+x}}\) | \(47\) |
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=\frac {4 \, a x \arctan \left (\sqrt {x + 1} \sqrt {x - 1} - x\right ) - \sqrt {x + 1} \sqrt {x - 1} - x}{x} \]
Result contains complex when optimal does not.
Time = 30.94 (sec) , antiderivative size = 117, normalized size of antiderivative = 3.00 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=- \frac {a {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {3}{4}, \frac {5}{4}, 1 & 1, 1, \frac {3}{2} \\\frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2} & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} + \frac {i a {G_{6, 6}^{2, 6}\left (\begin {matrix} 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, 1 & \\\frac {1}{4}, \frac {3}{4} & 0, \frac {1}{2}, \frac {1}{2}, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{2 \pi ^{\frac {3}{2}}} + \frac {{G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {5}{4}, \frac {7}{4}, 1 & \frac {3}{2}, \frac {3}{2}, 2 \\1, \frac {5}{4}, \frac {3}{2}, \frac {7}{4}, 2 & 0 \end {matrix} \middle | {\frac {1}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} + \frac {i {G_{6, 6}^{2, 6}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4}, \frac {3}{2}, 1 & \\\frac {3}{4}, \frac {5}{4} & \frac {1}{2}, 1, 1, 0 \end {matrix} \middle | {\frac {e^{2 i \pi }}{x^{2}}} \right )}}{4 \pi ^{\frac {3}{2}}} \]
-a*meijerg(((3/4, 5/4, 1), (1, 1, 3/2)), ((1/2, 3/4, 1, 5/4, 3/2), (0,)), x**(-2))/(2*pi**(3/2)) + I*a*meijerg(((0, 1/4, 1/2, 3/4, 1, 1), ()), ((1/4 , 3/4), (0, 1/2, 1/2, 0)), exp_polar(2*I*pi)/x**2)/(2*pi**(3/2)) + meijerg (((5/4, 7/4, 1), (3/2, 3/2, 2)), ((1, 5/4, 3/2, 7/4, 2), (0,)), x**(-2))/( 4*pi**(3/2)) + I*meijerg(((1/2, 3/4, 1, 5/4, 3/2, 1), ()), ((3/4, 5/4), (1 /2, 1, 1, 0)), exp_polar(2*I*pi)/x**2)/(4*pi**(3/2))
Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.54 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=-2 \, a \arcsin \left (\frac {1}{{\left | x \right |}}\right ) - \frac {\sqrt {x^{2} - 1}}{x} \]
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.10 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=-4 \, a \arctan \left (\frac {1}{2} \, {\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{2}\right ) - \frac {8}{{\left (\sqrt {x + 1} - \sqrt {x - 1}\right )}^{4} + 4} \]
Time = 6.01 (sec) , antiderivative size = 444, normalized size of antiderivative = 11.38 \[ \int \frac {a^2 x^2-(1-a x)^2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx=a\,\ln \left (\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}\right )\,2{}\mathrm {i}-a^2\,\mathrm {atan}\left (\frac {1024\,a^6}{1024\,a^5+1024\,a^7+\frac {a^6\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}+\frac {a^8\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}}+\frac {1024\,a^8}{1024\,a^5+1024\,a^7+\frac {a^6\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}+\frac {a^8\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}}-\frac {a^5\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\left (\sqrt {x+1}-1\right )\,\left (1024\,a^5+1024\,a^7+\frac {a^6\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}+\frac {a^8\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}\right )}-\frac {a^7\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\left (\sqrt {x+1}-1\right )\,\left (1024\,a^5+1024\,a^7+\frac {a^6\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}+\frac {a^8\,\left (\sqrt {x-1}-\mathrm {i}\right )\,1024{}\mathrm {i}}{\sqrt {x+1}-1}\right )}\right )\,4{}\mathrm {i}-a\,\ln \left (\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {x+1}-1\right )}^2}+1\right )\,2{}\mathrm {i}-\frac {\sqrt {x-1}-\mathrm {i}}{4\,\left (\sqrt {x+1}-1\right )}+a^2\,\mathrm {acosh}\left (x\right )-\frac {\frac {5\,{\left (\sqrt {x-1}-\mathrm {i}\right )}^2}{4\,{\left (\sqrt {x+1}-1\right )}^2}+\frac {1}{4}}{\frac {{\left (\sqrt {x-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {x+1}-1\right )}^3}+\frac {\sqrt {x-1}-\mathrm {i}}{\sqrt {x+1}-1}} \]
a*log(((x - 1)^(1/2) - 1i)/((x + 1)^(1/2) - 1))*2i - a^2*atan((1024*a^6)/( 1024*a^5 + 1024*a^7 + (a^6*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1) + (a^8*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1)) + (1024*a^8)/(102 4*a^5 + 1024*a^7 + (a^6*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1) + (a^8*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1)) - (a^5*((x - 1)^(1/2 ) - 1i)*1024i)/(((x + 1)^(1/2) - 1)*(1024*a^5 + 1024*a^7 + (a^6*((x - 1)^( 1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1) + (a^8*((x - 1)^(1/2) - 1i)*1024i)/( (x + 1)^(1/2) - 1))) - (a^7*((x - 1)^(1/2) - 1i)*1024i)/(((x + 1)^(1/2) - 1)*(1024*a^5 + 1024*a^7 + (a^6*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1) + (a^8*((x - 1)^(1/2) - 1i)*1024i)/((x + 1)^(1/2) - 1))))*4i - a*log( ((x - 1)^(1/2) - 1i)^2/((x + 1)^(1/2) - 1)^2 + 1)*2i - ((x - 1)^(1/2) - 1i )/(4*((x + 1)^(1/2) - 1)) + a^2*acosh(x) - ((5*((x - 1)^(1/2) - 1i)^2)/(4* ((x + 1)^(1/2) - 1)^2) + 1/4)/(((x - 1)^(1/2) - 1i)^3/((x + 1)^(1/2) - 1)^ 3 + ((x - 1)^(1/2) - 1i)/((x + 1)^(1/2) - 1))