Integrand size = 40, antiderivative size = 37 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c} \] Output:
-1/3*(a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/b/c
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{3 b c} \] Input:
Integrate[(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]
Output:
-1/3*(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(b*c)
Time = 0.35 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2973, 2976, 27, 2739, 15}
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.
\(\displaystyle \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2} \, dx\) |
\(\Big \downarrow \) 2973 |
\(\displaystyle \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^2}{1-c^2 x^2}dx\) |
\(\Big \downarrow \) 2976 |
\(\displaystyle -2 c \int \frac {(c x+1) \left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )^2}{4 c^2 (1-c x)}d\frac {1-c x}{c x+1}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int \frac {(c x+1) \left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )^2}{1-c x}d\frac {1-c x}{c x+1}}{2 c}\) |
\(\Big \downarrow \) 2739 |
\(\displaystyle -\frac {\int \frac {(1-c x)^2}{(c x+1)^2}d\left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )}{b c}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {(1-c x)^3}{3 b c (c x+1)^3}\) |
Input:
Int[(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^2/(1 - c^2*x^2),x]
Output:
-1/3*(1 - c*x)^3/(b*c*(1 + c*x)^3)
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; Fr eeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] && !Intege rQ[n]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*(P2x_)^(m_.), x_Symbol] :> With[{f = Coeff[P2x, x, 0], g = Coef f[P2x, x, 1], h = Coeff[P2x, x, 2]}, Simp[(b*c - a*d) Subst[Int[(b^2*f - a*b*g + a^2*h - (2*b*d*f - b*c*g - a*d*g + 2*a*c*h)*x + (d^2*f - c*d*g + c^ 2*h)*x^2)^m*((A + B*Log[e*x^n])^p/(b - d*x)^(2*(m + 1))), x], x, (a + b*x)/ (c + d*x)], x]] /; FreeQ[{a, b, c, d, e, A, B, n}, x] && PolyQ[P2x, x, 2] & & NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]
\[\int \frac {\left (a +b \ln \left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )\right )^{2}}{-c^{2} x^{2}+1}d x\]
Input:
int((a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x)
Output:
int((a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x)
Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (31) = 62\).
Time = 0.15 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.00 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=-\frac {b^{2} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{3} + 3 \, a b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + 3 \, a^{2} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}{3 \, c} \] Input:
integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algori thm="fricas")
Output:
-1/3*(b^2*log(sqrt(-c*x + 1)/sqrt(c*x + 1))^3 + 3*a*b*log(sqrt(-c*x + 1)/s qrt(c*x + 1))^2 + 3*a^2*log(sqrt(-c*x + 1)/sqrt(c*x + 1)))/c
Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (29) = 58\).
Time = 3.42 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\begin {cases} - \frac {a^{2} \operatorname {atan}{\left (\frac {x}{\sqrt {- \frac {1}{c^{2}}}} \right )}}{c^{2} \sqrt {- \frac {1}{c^{2}}}} & \text {for}\: b = 0 \\a^{2} x & \text {for}\: c = 0 \\- \frac {\left (a + b \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{3}}{3 b c} & \text {otherwise} \end {cases} \] Input:
integrate((a+b*ln((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**2/(-c**2*x**2+1),x)
Output:
Piecewise((-a**2*atan(x/sqrt(-1/c**2))/(c**2*sqrt(-1/c**2)), Eq(b, 0)), (a **2*x, Eq(c, 0)), (-(a + b*log(sqrt(-c*x + 1)/sqrt(c*x + 1)))**3/(3*b*c), True))
Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (31) = 62\).
Time = 0.06 (sec) , antiderivative size = 268, normalized size of antiderivative = 7.24 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {1}{2} \, b^{2} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + a b {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + \frac {1}{24} \, b^{2} {\left (\frac {6 \, {\left (\log \left (c x + 1\right )^{2} - 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right ) + \log \left (c x - 1\right )^{2}\right )} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}{c} + \frac {\log \left (c x + 1\right )^{3} - 3 \, \log \left (c x + 1\right )^{2} \log \left (c x - 1\right ) + 3 \, \log \left (c x + 1\right ) \log \left (c x - 1\right )^{2} - \log \left (c x - 1\right )^{3}}{c}\right )} + \frac {1}{2} \, a^{2} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} + \frac {{\left (\log \left (c x + 1\right )^{2} - 2 \, \log \left (c x + 1\right ) \log \left (c x - 1\right ) + \log \left (c x - 1\right )^{2}\right )} a b}{4 \, c} \] Input:
integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algori thm="maxima")
Output:
1/2*b^2*(log(c*x + 1)/c - log(c*x - 1)/c)*log(sqrt(-c*x + 1)/sqrt(c*x + 1) )^2 + a*b*(log(c*x + 1)/c - log(c*x - 1)/c)*log(sqrt(-c*x + 1)/sqrt(c*x + 1)) + 1/24*b^2*(6*(log(c*x + 1)^2 - 2*log(c*x + 1)*log(c*x - 1) + log(c*x - 1)^2)*log(sqrt(-c*x + 1)/sqrt(c*x + 1))/c + (log(c*x + 1)^3 - 3*log(c*x + 1)^2*log(c*x - 1) + 3*log(c*x + 1)*log(c*x - 1)^2 - log(c*x - 1)^3)/c) + 1/2*a^2*(log(c*x + 1)/c - log(c*x - 1)/c) + 1/4*(log(c*x + 1)^2 - 2*log(c *x + 1)*log(c*x - 1) + log(c*x - 1)^2)*a*b/c
\[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{2}}{c^{2} x^{2} - 1} \,d x } \] Input:
integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x, algori thm="giac")
Output:
integrate(-(b*log(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^2/(c^2*x^2 - 1), x)
Timed out. \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^2}{c^2\,x^2-1} \,d x \] Input:
int(-(a + b*log((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1),x)
Output:
int(-(a + b*log((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^2/(c^2*x^2 - 1), x)
Time = 0.20 (sec) , antiderivative size = 129, normalized size of antiderivative = 3.49 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^2}{1-c^2 x^2} \, dx=\frac {-3 \,\mathrm {log}\left (c^{2} x -c \right ) a^{2}+3 \,\mathrm {log}\left (c^{2} x +c \right ) a^{2}-2 \mathrm {log}\left (-\frac {2 \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )}{\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )^{2}-1}\right )^{3} b^{2}-6 \mathrm {log}\left (-\frac {2 \tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )}{\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right )}{2}\right )^{2}-1}\right )^{2} a b}{6 c} \] Input:
int((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^2/(-c^2*x^2+1),x)
Output:
( - 3*log(c**2*x - c)*a**2 + 3*log(c**2*x + c)*a**2 - 2*log(( - 2*tan(asin (sqrt( - c*x + 1)/sqrt(2))/2))/(tan(asin(sqrt( - c*x + 1)/sqrt(2))/2)**2 - 1))**3*b**2 - 6*log(( - 2*tan(asin(sqrt( - c*x + 1)/sqrt(2))/2))/(tan(asi n(sqrt( - c*x + 1)/sqrt(2))/2)**2 - 1))**2*a*b)/(6*c)