Integrand size = 43, antiderivative size = 67 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=-\frac {5 x \sqrt {-1+c x}}{2 \sqrt {1-c x} \left (1-c^2 x^2\right )}+\frac {3 \sqrt {-1+c x} \text {arctanh}(c x)}{2 c \sqrt {1-c x}} \] Output:
-5/2*x*(c*x-1)^(1/2)/(-c*x+1)^(1/2)/(-c^2*x^2+1)+3/2*(c*x-1)^(1/2)*arctanh (c*x)/c/(-c*x+1)^(1/2)
Time = 0.40 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.16 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=\frac {10 c x+\left (3-3 c^2 x^2\right ) \log (-1+c x)+3 \left (-1+c^2 x^2\right ) \log (1+c x)}{4 c \sqrt {-1+c x} \sqrt {1+c x} \sqrt {1-c^2 x^2}} \] Input:
Integrate[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2 )),x]
Output:
(10*c*x + (3 - 3*c^2*x^2)*Log[-1 + c*x] + 3*(-1 + c^2*x^2)*Log[1 + c*x])/( 4*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*Sqrt[1 - c^2*x^2])
Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.73, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2003, 37, 643, 298, 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.
\(\displaystyle \int \frac {4 c^2 x^2+1}{\sqrt {c x-1} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 2003 |
\(\displaystyle \int \frac {4 c^2 x^2+1}{(1-c x)^{3/2} \sqrt {c x-1} (c x+1)^2}dx\) |
\(\Big \downarrow \) 37 |
\(\displaystyle \frac {(c x-1)^{3/2} \int \frac {4 c^2 x^2+1}{(1-c x)^2 (c x+1)^2}dx}{(1-c x)^{3/2}}\) |
\(\Big \downarrow \) 643 |
\(\displaystyle \frac {(c x-1)^{3/2} \int \frac {4 c^2 x^2+1}{\left (1-c^2 x^2\right )^2}dx}{(1-c x)^{3/2}}\) |
\(\Big \downarrow \) 298 |
\(\displaystyle \frac {(c x-1)^{3/2} \left (\frac {5 x}{2 \left (1-c^2 x^2\right )}-\frac {3}{2} \int \frac {1}{1-c^2 x^2}dx\right )}{(1-c x)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {(c x-1)^{3/2} \left (\frac {5 x}{2 \left (1-c^2 x^2\right )}-\frac {3 \text {arctanh}(c x)}{2 c}\right )}{(1-c x)^{3/2}}\) |
Input:
Int[(1 + 4*c^2*x^2)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(3/2)),x]
Output:
((-1 + c*x)^(3/2)*((5*x)/(2*(1 - c^2*x^2)) - (3*ArcTanh[c*x])/(2*c)))/(1 - c*x)^(3/2)
Int[(u_.)*((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> S imp[(a + b*x)^m/(c + d*x)^m Int[u*(c + d*x)^(m + n), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && EqQ[b*c - a*d, 0] && !SimplerQ[a + b*x, c + d*x]
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-( b*c - a*d))*x*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] - Simp[(a*d - b*c*( 2*p + 3))/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/2 + p, 0])
Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) ^2)^(p_.), x_Symbol] :> Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x] /; FreeQ[{a , b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && (Integer Q[m] || (GtQ[c, 0] && GtQ[e, 0]))
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 = 81, normalized size of antiderivative = 1.21
method | result | size |
default | \(-\frac {3 \ln \left (c x -1\right ) c^{2} x^{2}-3 \ln \left (c x +1\right ) c^{2} x^{2}-10 c x -3 \ln \left (c x -1\right )+3 \ln \left (c x +1\right )}{4 \sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-c^{2} x^{2}+1}\, c}\) | \(81\) |
risch | \(\frac {\sqrt {\frac {-c^{2} x^{2}+1}{\left (c x -1\right ) \left (c x +1\right )}}\, \sqrt {c x -1}\, \sqrt {c x +1}\, \left (-\frac {5 i x}{2 \left (c x -1\right ) \left (c x +1\right )}+\frac {3 i \ln \left (c x -1\right )}{4 c}-\frac {3 i \ln \left (-c x -1\right )}{4 c}\right )}{\sqrt {-c^{2} x^{2}+1}}\) | \(99\) |
Input:
int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^(3/2),x,method= _RETURNVERBOSE)
Output:
-1/4/(c*x-1)^(1/2)/(c*x+1)^(1/2)*(3*ln(c*x-1)*c^2*x^2-3*ln(c*x+1)*c^2*x^2- 10*c*x-3*ln(c*x-1)+3*ln(c*x+1))/(-c^2*x^2+1)^(1/2)/c
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (54) = 108\).
Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.69 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=-\frac {10 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {c x - 1} c x + 3 \, {\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \arctan \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} \sqrt {c x + 1} \sqrt {c x - 1} c x}{c^{4} x^{4} - 1}\right )}{4 \, {\left (c^{5} x^{4} - 2 \, c^{3} x^{2} + c\right )}} \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
-1/4*(10*sqrt(-c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1)*c*x + 3*(c^4*x^4 - 2*c^2*x^2 + 1)*arctan(2*sqrt(-c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(c*x - 1)*c* x/(c^4*x^4 - 1)))/(c^5*x^4 - 2*c^3*x^2 + c)
\[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=\int \frac {4 c^{2} x^{2} + 1}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \sqrt {c x - 1} \sqrt {c x + 1}}\, dx \] Input:
integrate((4*c**2*x**2+1)/(c*x-1)**(1/2)/(c*x+1)**(1/2)/(-c**2*x**2+1)**(3 /2),x)
Output:
Integral((4*c**2*x**2 + 1)/((-(c*x - 1)*(c*x + 1))**(3/2)*sqrt(c*x - 1)*sq rt(c*x + 1)), x)
\[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=\int { \frac {4 \, c^{2} x^{2} + 1}{{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} \sqrt {c x + 1} \sqrt {c x - 1}} \,d x } \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
integrate((4*c^2*x^2 + 1)/((-c^2*x^2 + 1)^(3/2)*sqrt(c*x + 1)*sqrt(c*x - 1 )), x)
Result contains complex when optimal does not.
Time = 0.14 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.66 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=-\frac {5 \, x}{2 \, {\left (-i \, {\left (c x + 1\right )}^{2} + 2 i \, c x + 2 i\right )}} - \frac {3 i \, \log \left (c x + 1\right )}{4 \, c} + \frac {3 i \, \log \left ({\left | c x - 1 \right |}\right )}{4 \, c} \] Input:
integrate((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
-5/2*x/(-I*(c*x + 1)^2 + 2*I*c*x + 2*I) - 3/4*I*log(c*x + 1)/c + 3/4*I*log (abs(c*x - 1))/c
Timed out. \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=\int \frac {4\,c^2\,x^2+1}{{\left (1-c^2\,x^2\right )}^{3/2}\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}} \,d x \] Input:
int((4*c^2*x^2 + 1)/((1 - c^2*x^2)^(3/2)*(c*x - 1)^(1/2)*(c*x + 1)^(1/2)), x)
Output:
int((4*c^2*x^2 + 1)/((1 - c^2*x^2)^(3/2)*(c*x - 1)^(1/2)*(c*x + 1)^(1/2)), x)
Time = 0.17 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.87 \[ \int \frac {1+4 c^2 x^2}{\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}} \, dx=\frac {i \left (3 \,\mathrm {log}\left (\sqrt {-c x +1}-\sqrt {2}\right ) c^{2} x^{2}-3 \,\mathrm {log}\left (\sqrt {-c x +1}-\sqrt {2}\right )+3 \,\mathrm {log}\left (\sqrt {-c x +1}+\sqrt {2}\right ) c^{2} x^{2}-3 \,\mathrm {log}\left (\sqrt {-c x +1}+\sqrt {2}\right )-6 \,\mathrm {log}\left (\sqrt {-c x +1}\right ) c^{2} x^{2}+6 \,\mathrm {log}\left (\sqrt {-c x +1}\right )-5 c^{2} x^{2}+10 c x +5\right )}{4 c \left (c^{2} x^{2}-1\right )} \] Input:
int((4*c^2*x^2+1)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(-c^2*x^2+1)^(3/2),x)
Output:
(i*(3*log(sqrt( - c*x + 1) - sqrt(2))*c**2*x**2 - 3*log(sqrt( - c*x + 1) - sqrt(2)) + 3*log(sqrt( - c*x + 1) + sqrt(2))*c**2*x**2 - 3*log(sqrt( - c* x + 1) + sqrt(2)) - 6*log(sqrt( - c*x + 1))*c**2*x**2 + 6*log(sqrt( - c*x + 1)) - 5*c**2*x**2 + 10*c*x + 5))/(4*c*(c**2*x**2 - 1))