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

Optimal. Leaf size=37 \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c} \]

[Out]

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

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Rubi [A]  time = 0.0611698, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2512, 2302, 30} \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2512

Int[((a_.) + Log[((c_.)*Sqrt[(d_.) + (e_.)*(x_)])/Sqrt[(f_.) + (g_.)*(x_)]]*(b_.))^(n_.)/((A_.) + (C_.)*(x_)^2
), x_Symbol] :> Dist[g/(C*f), Subst[Int[(a + b*Log[c*x])^n/x, x], x, Sqrt[d + e*x]/Sqrt[f + g*x]], x] /; FreeQ
[{a, b, c, d, e, f, g, A, C, n}, x] && EqQ[C*d*f - A*e*g, 0] && EqQ[e*f + d*g, 0]

Rule 2302

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 30

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

Rubi steps

\begin{align*} \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{\operatorname{Subst}\left (\int \frac{(a+b \log (x))^3}{x} \, dx,x,\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )}{c}\\ &=-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{1+c x}}\right )\right )}{b c}\\ &=-\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]  time = 0.0089422, size = 37, normalized size = 1. \[ -\frac{\left (a+b \log \left (\frac{\sqrt{1-c x}}{\sqrt{c x+1}}\right )\right )^4}{4 b c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.411, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-{c}^{2}{x}^{2}+1} \left ( a+b\ln \left ({\sqrt{-cx+1}{\frac{1}{\sqrt{cx+1}}}} \right ) \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [B]  time = 1.35167, size = 710, normalized size = 19.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 2.01189, size = 248, normalized size = 6.7 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 47.0202, size = 65, normalized size = 1.76 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)