\(\int \frac {(a+b \log (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}))^3}{1-c^2 x^2} \, dx\) [44]

Optimal result

Integrand size = 40, antiderivative size = 37 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c} \]

[Out]

-1/4*(a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^4/b/c

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2573, 2576, 12, 2339, 30} \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {c x+1}}\right )\right )^4}{4 b c} \]

[In]

Int[(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]

[Out]

-1/4*(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(b*c)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2573

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)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

Rule 2576

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*(P2x_)^(m_.), x_Symbol]
:> With[{f = Coeff[P2x, x, 0], g = Coeff[P2x, x, 1], h = Coeff[P2x, x, 2]}, Dist[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]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\left (a+b \log \left (\sqrt {\frac {1-c x}{1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx,\sqrt {\frac {1-c x}{1+c x}},\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right ) \\ & = -\text {Subst}\left ((2 c) \text {Subst}\left (\int \frac {\left (a+b \log \left (\sqrt {x}\right )\right )^3}{4 c^2 x} \, dx,x,\frac {1-c x}{1+c x}\right ),\sqrt {\frac {1-c x}{1+c x}},\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right ) \\ & = -\text {Subst}\left (\frac {\text {Subst}\left (\int \frac {\left (a+b \log \left (\sqrt {x}\right )\right )^3}{x} \, dx,x,\frac {1-c x}{1+c x}\right )}{2 c},\sqrt {\frac {1-c x}{1+c x}},\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right ) \\ & = -\text {Subst}\left (\frac {\text {Subst}\left (\int x^3 \, dx,x,a+b \log \left (\sqrt {\frac {1-c x}{1+c x}}\right )\right )}{b c},\sqrt {\frac {1-c x}{1+c x}},\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right ) \\ & = -\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^4}{4 b c} \]

[In]

Integrate[(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^3/(1 - c^2*x^2),x]

[Out]

-1/4*(a + b*Log[Sqrt[1 - c*x]/Sqrt[1 + c*x]])^4/(b*c)

Maple [F]

\[\int \frac {\left (a +b \ln \left (\frac {\sqrt {-x c +1}}{\sqrt {x c +1}}\right )\right )^{3}}{-x^{2} c^{2}+1}d x\]

[In]

int((a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x)

[Out]

int((a+b*ln((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x)

Fricas [B] (verification not implemented)

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 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=-\frac {b^{3} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{4} + 4 \, a b^{2} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{3} + 6 \, a^{2} b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )^{2} + 4 \, a^{3} \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right )}{4 \, c} \]

[In]

integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="fricas")

[Out]

-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*log(sqrt(-c*x + 1)/sqrt(c*x + 1)))/c

Sympy [B] (verification not implemented)

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 \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\begin {cases} - \frac {a^{3} \operatorname {atan}{\left (\frac {x}{\sqrt {- \frac {1}{c^{2}}}} \right )}}{c^{2} \sqrt {- \frac {1}{c^{2}}}} & \text {for}\: b = 0 \\a^{3} x & \text {for}\: c = 0 \\- \frac {\left (a + b \log {\left (\frac {\sqrt {- c x + 1}}{\sqrt {c x + 1}} \right )}\right )^{4}}{4 b c} & \text {otherwise} \end {cases} \]

[In]

integrate((a+b*ln((-c*x+1)**(1/2)/(c*x+1)**(1/2)))**3/(-c**2*x**2+1),x)

[Out]

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))

Maxima [B] (verification not implemented)

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} \]

[In]

integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="maxima")

[Out]

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 - lo
g(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 - l
og(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*l
og(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 - lo
g(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

Giac [F]

\[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int { -\frac {{\left (b \log \left (\frac {\sqrt {-c x + 1}}{\sqrt {c x + 1}}\right ) + a\right )}^{3}}{c^{2} x^{2} - 1} \,d x } \]

[In]

integrate((a+b*log((-c*x+1)^(1/2)/(c*x+1)^(1/2)))^3/(-c^2*x^2+1),x, algorithm="giac")

[Out]

integrate(-(b*log(sqrt(-c*x + 1)/sqrt(c*x + 1)) + a)^3/(c^2*x^2 - 1), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )^3}{1-c^2 x^2} \, dx=\int -\frac {{\left (a+b\,\ln \left (\frac {\sqrt {1-c\,x}}{\sqrt {c\,x+1}}\right )\right )}^3}{c^2\,x^2-1} \,d x \]

[In]

int(-(a + b*log((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^3/(c^2*x^2 - 1),x)

[Out]

int(-(a + b*log((1 - c*x)^(1/2)/(c*x + 1)^(1/2)))^3/(c^2*x^2 - 1), x)