Integrand size = 40, antiderivative size = 37 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c} \]
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c} \]
Time = 0.33 (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.
| \(\displaystyle \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{1-c^2 x^2} \, dx\) |
| \(\Big \downarrow \) 2973 |
| \(\displaystyle \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^3}{1-c^2 x^2}dx\) |
| \(\Big \downarrow \) 2976 |
| \(\displaystyle -2 c \int \frac {(c x+1) \left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )^3}{4 c^2 (1-c x)}d\frac {1-c x}{c x+1}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(c x+1) \left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )^3}{1-c x}d\frac {1-c x}{c x+1}}{2 c}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle -\frac {\int \frac {(1-c x)^3}{(c x+1)^3}d\left (a+b \log \left (\sqrt {\frac {1-c x}{c x+1}}\right )\right )}{b c}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle -\frac {(1-c x)^4}{4 b c (c x+1)^4}\) |
3.1.44.3.1 Defintions of rubi rules used
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]
\[\int \frac {\left (a +b \ln \left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )\right )^{3}}{-x^{2} c^{2}+1}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (31) = 62\).
Time = 0.31 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.73 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {b^{3} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{4} + 4 \, a b^{2} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{3} + 6 \, a^{2} b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + 4 \, a^{3} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}{4 \, c} \]
-1/4*(b^3*log(sqrt(-c*x + 1)/sqrt(c*x + 1))^4 + 4*a*b^2*log(sqrt(-c*x + 1) /sqrt(c*x + 1))^3 + 6*a^2*b*log(sqrt(-c*x + 1)/sqrt(c*x + 1))^2 + 4*a^3*lo g(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 = 4.51 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.76 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\begin {cases} - \frac {a^{3} \operatorname {atan}{\left (\frac {x}{\sqrt {- \frac {1}{c^{2}}}} \right )}}{c^{2} \sqrt {- \frac {1}{c^{2}}}} & \text {for}\: b = 0 \\a^{3} x & \text {for}\: c = 0 \\- \frac {\left (a + b \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{4}}{4 b c} & \text {otherwise} \end {cases} \]
Piecewise((-a**3*atan(x/sqrt(-1/c**2))/(c**2*sqrt(-1/c**2)), Eq(b, 0)), (a **3*x, Eq(c, 0)), (-(a + b*log(sqrt(-c*x + 1)/sqrt(c*x + 1)))**4/(4*b*c), True))
Leaf count of result is larger than twice the leaf count of optimal. 526 vs. \(2 (31) = 62\).
Time = 0.24 (sec) , antiderivative size = 526, normalized size of antiderivative = 14.22 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\frac {1}{2} \, b^{3} {\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 )^{3} + \frac {3}{2} \, a 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} + \frac {3}{2} \, a^{2} 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}{64} \, {\left (\frac {24 \, {\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 )^{2}}{c} + \frac {8 \, {\left (\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}\right )} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}{c} + \frac {\log \left (c x + 1\right )^{4} - 4 \, \log \left (c x + 1\right )^{3} \log \left (c x - 1\right ) + 6 \, \log \left (c x + 1\right )^{2} \log \left (c x - 1\right )^{2} - 4 \, \log \left (c x + 1\right ) \log \left (c x - 1\right )^{3} + \log \left (c x - 1\right )^{4}}{c}\right )} b^{3} + \frac {1}{8} \, a 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^{3} {\left (\frac {\log \left (c x + 1\right )}{c} - \frac {\log \left (c x - 1\right )}{c}\right )} + \frac {3 \, {\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^{2} b}{8 \, c} \]
1/2*b^3*(log(c*x + 1)/c - log(c*x - 1)/c)*log(sqrt(-c*x + 1)/sqrt(c*x + 1) )^3 + 3/2*a*b^2*(log(c*x + 1)/c - log(c*x - 1)/c)*log(sqrt(-c*x + 1)/sqrt( c*x + 1))^2 + 3/2*a^2*b*(log(c*x + 1)/c - log(c*x - 1)/c)*log(sqrt(-c*x + 1)/sqrt(c*x + 1)) + 1/64*(24*(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))^2/c + 8*(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)*log(sqrt(-c*x + 1)/sqrt(c*x + 1))/c + (log(c*x + 1)^4 - 4*log(c *x + 1)^3*log(c*x - 1) + 6*log(c*x + 1)^2*log(c*x - 1)^2 - 4*log(c*x + 1)* log(c*x - 1)^3 + log(c*x - 1)^4)/c)*b^3 + 1/8*a*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^3*(log(c*x + 1)/c - log(c*x - 1)/c) + 3/8*(log(c*x + 1)^2 - 2*log(c*x + 1)*log(c*x - 1) + log(c*x - 1)^2 )*a^2*b/c
\[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]