Integrand size = 41, antiderivative size = 93 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=-\frac {5 x}{7 (-1+c x)^{7/2} (1+c x)^{7/2}}+\frac {2 x}{35 (-1+c x)^{5/2} (1+c x)^{5/2}}-\frac {8 x}{105 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {16 x}{105 \sqrt {-1+c x} \sqrt {1+c x}} \] Output:
-5/7*x/(c*x-1)^(7/2)/(c*x+1)^(7/2)+2/35*x/(c*x-1)^(5/2)/(c*x+1)^(5/2)-8/10 5*x/(c*x-1)^(3/2)/(c*x+1)^(3/2)+16/105*x/(c*x-1)^(1/2)/(c*x+1)^(1/2)
Time = 0.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.53 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\frac {x \left (-105+70 c^2 x^2-56 c^4 x^4+16 c^6 x^6\right )}{105 (-1+c x)^{7/2} (1+c x)^{7/2}} \] Input:
Integrate[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^4),x ]
Output:
(x*(-105 + 70*c^2*x^2 - 56*c^4*x^4 + 16*c^6*x^6))/(105*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2))
Time = 0.39 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.11, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {2003, 35, 645, 42, 42, 41}
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.
\(\displaystyle \int \frac {4 c^2 x^2+1}{\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^4} \, dx\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle \int \frac {4 c^2 x^2+1}{(-c x-1)^4 (c x-1)^{9/2} \sqrt {c x+1}}dx\) |
\(\Big \downarrow \) 35 |
\(\displaystyle \int \frac {4 c^2 x^2+1}{(c x-1)^{9/2} (c x+1)^{9/2}}dx\) |
\(\Big \downarrow \) 645 |
\(\displaystyle -\frac {2}{7} \int \frac {1}{(c x-1)^{7/2} (c x+1)^{7/2}}dx-\frac {5 x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\) |
\(\Big \downarrow \) 42 |
\(\displaystyle -\frac {2}{7} \left (-\frac {4}{5} \int \frac {1}{(c x-1)^{5/2} (c x+1)^{5/2}}dx-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {5 x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\) |
\(\Big \downarrow \) 42 |
\(\displaystyle -\frac {2}{7} \left (-\frac {4}{5} \left (-\frac {2}{3} \int \frac {1}{(c x-1)^{3/2} (c x+1)^{3/2}}dx-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}\right )-\frac {5 x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}\) |
\(\Big \downarrow \) 41 |
\(\displaystyle -\frac {5 x}{7 (c x-1)^{7/2} (c x+1)^{7/2}}-\frac {2}{7} \left (-\frac {x}{5 (c x-1)^{5/2} (c x+1)^{5/2}}-\frac {4}{5} \left (\frac {2 x}{3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {x}{3 (c x-1)^{3/2} (c x+1)^{3/2}}\right )\right )\) |
Input:
Int[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^4),x]
Output:
(-5*x)/(7*(-1 + c*x)^(7/2)*(1 + c*x)^(7/2)) - (2*(-1/5*x/((-1 + c*x)^(5/2) *(1 + c*x)^(5/2)) - (4*(-1/3*x/((-1 + c*x)^(3/2)*(1 + c*x)^(3/2)) + (2*x)/ (3*Sqrt[-1 + c*x]*Sqrt[1 + c*x])))/5))/7
Int[(u_.)*((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.), x_Symbol] :> Simp[(b/d)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n} , x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && !(IntegerQ[n] && SimplerQ[a + b*x, c + d*x])
Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 3)/(2*a*c*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 0]
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2), x_Symbol] :> Simp[(b*c*e - a*d*f)*x*(c + d*x)^(m + 1)*((e + f*x)^(n + 1)/(2*c*d*e*f*(m + 1))), x] - Simp[(b*c*e - a*d*f*(2*m + 3))/(2*c*d*e*f*(m + 1)) Int[(c + d*x)^(m + 1)*(e + f*x)^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && LtQ[m, -1]
Int[(u_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] : > Int[u*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; FreeQ[{a, b, c, d, n, p} , x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.59
method | result | size |
gosper | \(\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, x \left (16 c^{6} x^{6}-56 c^{4} x^{4}+70 c^{2} x^{2}-105\right )}{105 \left (c^{2} x^{2}-1\right )^{4}}\) | \(55\) |
default | \(\frac {\left (16 c^{6} x^{6}-56 c^{4} x^{4}+70 c^{2} x^{2}-105\right ) x}{105 \left (c^{2} x^{2}-1\right )^{3} \sqrt {c x +1}\, \sqrt {c x -1}}\) | \(55\) |
orering | \(\frac {x \left (16 c^{6} x^{6}-56 c^{4} x^{4}+70 c^{2} x^{2}-105\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{105 \left (-c^{2} x^{2}+1\right )^{4}}\) | \(56\) |
Input:
int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^4,x,method=_RET URNVERBOSE)
Output:
1/105*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*(16*c^6*x^6-56*c^4*x^4+70*c^2*x^2-105) /(c^2*x^2-1)^4
Time = 0.09 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.24 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\frac {16 \, c^{8} x^{8} - 64 \, c^{6} x^{6} + 96 \, c^{4} x^{4} - 64 \, c^{2} x^{2} + {\left (16 \, c^{7} x^{7} - 56 \, c^{5} x^{5} + 70 \, c^{3} x^{3} - 105 \, c x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + 16}{105 \, {\left (c^{9} x^{8} - 4 \, c^{7} x^{6} + 6 \, c^{5} x^{4} - 4 \, c^{3} x^{2} + c\right )}} \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^4,x, algo rithm="fricas")
Output:
1/105*(16*c^8*x^8 - 64*c^6*x^6 + 96*c^4*x^4 - 64*c^2*x^2 + (16*c^7*x^7 - 5 6*c^5*x^5 + 70*c^3*x^3 - 105*c*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + 16)/(c^9*x ^8 - 4*c^7*x^6 + 6*c^5*x^4 - 4*c^3*x^2 + c)
\[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\int \frac {4 c^{2} x^{2} + 1}{\left (c x - 1\right )^{\frac {9}{2}} \left (c x + 1\right )^{\frac {9}{2}}}\, dx \] Input:
integrate((4*c**2*x**2+1)/(c*x-1)**(1/2)/(c*x+1)**(1/2)/(-c**2*x**2+1)**4, x)
Output:
Integral((4*c**2*x**2 + 1)/((c*x - 1)**(9/2)*(c*x + 1)**(9/2)), x)
Time = 0.03 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\frac {16 \, x}{105 \, \sqrt {c^{2} x^{2} - 1}} - \frac {8 \, x}{105 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {3}{2}}} + \frac {2 \, x}{35 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {5}{2}}} - \frac {5 \, x}{7 \, {\left (c^{2} x^{2} - 1\right )}^{\frac {7}{2}}} \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^4,x, algo rithm="maxima")
Output:
16/105*x/sqrt(c^2*x^2 - 1) - 8/105*x/(c^2*x^2 - 1)^(3/2) + 2/35*x/(c^2*x^2 - 1)^(5/2) - 5/7*x/(c^2*x^2 - 1)^(7/2)
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (69) = 138\).
Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.29 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\frac {{\left ({\left (c x + 1\right )} {\left ({\left (c x + 1\right )} {\left (\frac {256 \, {\left (c x + 1\right )}}{c} - \frac {1687}{c}\right )} + \frac {3850}{c}\right )} - \frac {3150}{c}\right )} \sqrt {c x + 1}}{3360 \, {\left (c x - 1\right )}^{\frac {7}{2}}} + \frac {105 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{12} + 1470 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{10} + 10360 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{8} + 50960 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{6} + 74256 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{4} + 53984 \, {\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 16384}{840 \, {\left ({\left (\sqrt {c x + 1} - \sqrt {c x - 1}\right )}^{2} + 2\right )}^{7} c} \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^4,x, algo rithm="giac")
Output:
1/3360*((c*x + 1)*((c*x + 1)*(256*(c*x + 1)/c - 1687/c) + 3850/c) - 3150/c )*sqrt(c*x + 1)/(c*x - 1)^(7/2) + 1/840*(105*(sqrt(c*x + 1) - sqrt(c*x - 1 ))^12 + 1470*(sqrt(c*x + 1) - sqrt(c*x - 1))^10 + 10360*(sqrt(c*x + 1) - s qrt(c*x - 1))^8 + 50960*(sqrt(c*x + 1) - sqrt(c*x - 1))^6 + 74256*(sqrt(c* x + 1) - sqrt(c*x - 1))^4 + 53984*(sqrt(c*x + 1) - sqrt(c*x - 1))^2 + 1638 4)/(((sqrt(c*x + 1) - sqrt(c*x - 1))^2 + 2)^7*c)
Time = 22.70 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.03 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=-\frac {105\,x\,\sqrt {c\,x-1}-70\,c^2\,x^3\,\sqrt {c\,x-1}+56\,c^4\,x^5\,\sqrt {c\,x-1}-16\,c^6\,x^7\,\sqrt {c\,x-1}}{\left (\left (c\,x-1\right )\,\left (\left (c\,x-1\right )\,\left (105\,c\,x+525\right )+1260\right )+840\right )\,{\left (c\,x-1\right )}^4\,\sqrt {c\,x+1}} \] Input:
int((4*c^2*x^2 + 1)/((c^2*x^2 - 1)^4*(c*x - 1)^(1/2)*(c*x + 1)^(1/2)),x)
Output:
-(105*x*(c*x - 1)^(1/2) - 70*c^2*x^3*(c*x - 1)^(1/2) + 56*c^4*x^5*(c*x - 1 )^(1/2) - 16*c^6*x^7*(c*x - 1)^(1/2))/(((c*x - 1)*((c*x - 1)*(105*c*x + 52 5) + 1260) + 840)*(c*x - 1)^4*(c*x + 1)^(1/2))
Time = 0.20 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.39 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^4} \, dx=\frac {-16 \sqrt {c x -1}\, c^{7} x^{7}-16 \sqrt {c x -1}\, c^{6} x^{6}+48 \sqrt {c x -1}\, c^{5} x^{5}+48 \sqrt {c x -1}\, c^{4} x^{4}-48 \sqrt {c x -1}\, c^{3} x^{3}-48 \sqrt {c x -1}\, c^{2} x^{2}+16 \sqrt {c x -1}\, c x +16 \sqrt {c x -1}+16 \sqrt {c x +1}\, c^{7} x^{7}-56 \sqrt {c x +1}\, c^{5} x^{5}+70 \sqrt {c x +1}\, c^{3} x^{3}-105 \sqrt {c x +1}\, c x}{105 \sqrt {c x -1}\, c \left (c^{7} x^{7}+c^{6} x^{6}-3 c^{5} x^{5}-3 c^{4} x^{4}+3 c^{3} x^{3}+3 c^{2} x^{2}-c x -1\right )} \] Input:
int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^4,x)
Output:
( - 16*sqrt(c*x - 1)*c**7*x**7 - 16*sqrt(c*x - 1)*c**6*x**6 + 48*sqrt(c*x - 1)*c**5*x**5 + 48*sqrt(c*x - 1)*c**4*x**4 - 48*sqrt(c*x - 1)*c**3*x**3 - 48*sqrt(c*x - 1)*c**2*x**2 + 16*sqrt(c*x - 1)*c*x + 16*sqrt(c*x - 1) + 16 *sqrt(c*x + 1)*c**7*x**7 - 56*sqrt(c*x + 1)*c**5*x**5 + 70*sqrt(c*x + 1)*c **3*x**3 - 105*sqrt(c*x + 1)*c*x)/(105*sqrt(c*x - 1)*c*(c**7*x**7 + c**6*x **6 - 3*c**5*x**5 - 3*c**4*x**4 + 3*c**3*x**3 + 3*c**2*x**2 - c*x - 1))