3.40 \(\int \frac{a+b \tanh ^{-1}(c \sqrt{x})}{x (1-c^2 x)} \, dx\)

Optimal. Leaf size=69 \[ -b \text{PolyLog}\left (2,\frac{2}{c \sqrt{x}+1}-1\right )+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b}+2 \log \left (2-\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \]

[Out]

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

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Rubi [A]  time = 0.24363, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {36, 29, 31, 1593, 5988, 5932, 2447} \[ -b \text{PolyLog}\left (2,\frac{2}{c \sqrt{x}+1}-1\right )+\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b}+2 \log \left (2-\frac{2}{c \sqrt{x}+1}\right ) \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 5988

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c
*x])^(p + 1)/(b*d*(p + 1)), x] + Dist[1/d, Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c,
d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]

Rule 5932

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Simp[((a + b*ArcTanh[c*
x])^p*Log[2 - 2/(1 + (e*x)/d)])/d, x] - Dist[(b*c*p)/d, Int[((a + b*ArcTanh[c*x])^(p - 1)*Log[2 - 2/(1 + (e*x)
/d)])/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c^2*d^2 - e^2, 0]

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{x \left (1-c^2 x\right )} \, dx &=2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x-c^2 x^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b}+2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x (1+c x)} \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b}+2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (2-\frac{2}{1+c \sqrt{x}}\right )-(2 b c) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx,x,\sqrt{x}\right )\\ &=\frac{\left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right )^2}{b}+2 \left (a+b \tanh ^{-1}\left (c \sqrt{x}\right )\right ) \log \left (2-\frac{2}{1+c \sqrt{x}}\right )-b \text{Li}_2\left (-1+\frac{2}{1+c \sqrt{x}}\right )\\ \end{align*}

Mathematica [A]  time = 0.124835, size = 72, normalized size = 1.04 \[ -b \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )-a \log \left (1-c^2 x\right )+a \log (x)+b \tanh ^{-1}\left (c \sqrt{x}\right ) \left (\tanh ^{-1}\left (c \sqrt{x}\right )+2 \log \left (1-e^{-2 \tanh ^{-1}\left (c \sqrt{x}\right )}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

b*ArcTanh[c*Sqrt[x]]*(ArcTanh[c*Sqrt[x]] + 2*Log[1 - E^(-2*ArcTanh[c*Sqrt[x]])]) + a*Log[x] - a*Log[1 - c^2*x]
- b*PolyLog[2, E^(-2*ArcTanh[c*Sqrt[x]])]

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Maple [B]  time = 0.053, size = 217, normalized size = 3.1 \begin{align*} -a\ln \left ( c\sqrt{x}-1 \right ) +2\,a\ln \left ( c\sqrt{x} \right ) -a\ln \left ( 1+c\sqrt{x} \right ) -b{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x}-1 \right ) +2\,b{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( c\sqrt{x} \right ) -b{\it Artanh} \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) -b{\it dilog} \left ( c\sqrt{x} \right ) -b{\it dilog} \left ( 1+c\sqrt{x} \right ) -b\ln \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) -{\frac{b}{4} \left ( \ln \left ( c\sqrt{x}-1 \right ) \right ) ^{2}}+b{\it dilog} \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) +{\frac{b}{2}\ln \left ( c\sqrt{x}-1 \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }+{\frac{b}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ({\frac{1}{2}}+{\frac{c}{2}\sqrt{x}} \right ) }-{\frac{b}{2}\ln \left ( -{\frac{c}{2}\sqrt{x}}+{\frac{1}{2}} \right ) \ln \left ( 1+c\sqrt{x} \right ) }+{\frac{b}{4} \left ( \ln \left ( 1+c\sqrt{x} \right ) \right ) ^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.62442, size = 215, normalized size = 3.12 \begin{align*} -\frac{1}{4} \, b \log \left (c \sqrt{x} + 1\right )^{2} + \frac{1}{2} \, b \log \left (c \sqrt{x} + 1\right ) \log \left (-c \sqrt{x} + 1\right ) + \frac{1}{4} \, b \log \left (-c \sqrt{x} + 1\right )^{2} -{\left (\log \left (c \sqrt{x} + 1\right ) \log \left (-\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right ) +{\rm Li}_2\left (\frac{1}{2} \, c \sqrt{x} + \frac{1}{2}\right )\right )} b -{\left (\log \left (c \sqrt{x}\right ) \log \left (-c \sqrt{x} + 1\right ) +{\rm Li}_2\left (-c \sqrt{x} + 1\right )\right )} b +{\left (\log \left (c \sqrt{x} + 1\right ) \log \left (-c \sqrt{x}\right ) +{\rm Li}_2\left (c \sqrt{x} + 1\right )\right )} b - a{\left (\log \left (c \sqrt{x} + 1\right ) + \log \left (c \sqrt{x} - 1\right ) - \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*b*log(c*sqrt(x) + 1)^2 + 1/2*b*log(c*sqrt(x) + 1)*log(-c*sqrt(x) + 1) + 1/4*b*log(-c*sqrt(x) + 1)^2 - (lo
g(c*sqrt(x) + 1)*log(-1/2*c*sqrt(x) + 1/2) + dilog(1/2*c*sqrt(x) + 1/2))*b - (log(c*sqrt(x))*log(-c*sqrt(x) +
1) + dilog(-c*sqrt(x) + 1))*b + (log(c*sqrt(x) + 1)*log(-c*sqrt(x)) + dilog(c*sqrt(x) + 1))*b - a*(log(c*sqrt(
x) + 1) + log(c*sqrt(x) - 1) - log(x))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{c^{2} x^{2} - x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(b*arctanh(c*sqrt(x)) + a)/(c^2*x^2 - x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{a}{c^{2} x^{2} - x}\, dx - \int \frac{b \operatorname{atanh}{\left (c \sqrt{x} \right )}}{c^{2} x^{2} - x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Integral(a/(c**2*x**2 - x), x) - Integral(b*atanh(c*sqrt(x))/(c**2*x**2 - x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \operatorname{artanh}\left (c \sqrt{x}\right ) + a}{{\left (c^{2} x - 1\right )} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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