3.1.21 \(\int \frac {a+b \tanh ^{-1}(c x)}{1+2 c x} \, dx\) [21]

Optimal. Leaf size=67 \[ \frac {\left (a-b \tanh ^{-1}\left (\frac {1}{2}\right )\right ) \log \left (-\frac {1+2 c x}{2 d}\right )}{2 c}-\frac {b \text {PolyLog}(2,-1-2 c x)}{4 c}+\frac {b \text {PolyLog}\left (2,\frac {1}{3} (1+2 c x)\right )}{4 c} \]

[Out]

1/2*(a-b*arctanh(1/2))*ln(1/2*(-2*c*x-1)/d)/c-1/4*b*polylog(2,-2*c*x-1)/c+1/4*b*polylog(2,2/3*c*x+1/3)/c

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.63, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6057, 2449, 2352, 2497} \begin {gather*} -\frac {\log \left (\frac {2}{c x+1}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {\log \left (\frac {2 (2 c x+1)}{3 (c x+1)}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{2 c}+\frac {b \text {Li}_2\left (1-\frac {2}{c x+1}\right )}{4 c}-\frac {b \text {Li}_2\left (1-\frac {2 (2 c x+1)}{3 (c x+1)}\right )}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/c + ((a + b*ArcTanh[c*x])*Log[(2*(1 + 2*c*x))/(3*(1 + c*x))])/(2*
c) + (b*PolyLog[2, 1 - 2/(1 + c*x)])/(4*c) - (b*PolyLog[2, 1 - (2*(1 + 2*c*x))/(3*(1 + c*x))])/(4*c)

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

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]]

Rule 6057

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

Rubi steps

\begin {align*} \int \frac {a+b \tanh ^{-1}(c x)}{1+2 c x} \, dx &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{2 c}+\frac {1}{2} b \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx-\frac {1}{2} b \int \frac {\log \left (\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{1-c^2 x^2} \, dx\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{2 c}-\frac {b \text {Li}_2\left (1-\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{4 c}+\frac {b \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 c}\\ &=-\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1+c x}\right )}{2 c}+\frac {\left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{2 c}+\frac {b \text {Li}_2\left (1-\frac {2}{1+c x}\right )}{4 c}-\frac {b \text {Li}_2\left (1-\frac {2 (1+2 c x)}{3 (1+c x)}\right )}{4 c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 240, normalized size = 3.58 \begin {gather*} \frac {a \log (1+2 c x)+b \tanh ^{-1}(c x) \left (\frac {1}{2} \log \left (1-c^2 x^2\right )+\log \left (i \sinh \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right )\right )\right )-\frac {1}{2} i b \left (-\frac {1}{4} i \left (\pi -2 i \tanh ^{-1}(c x)\right )^2+i \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right )^2+\left (\pi -2 i \tanh ^{-1}(c x)\right ) \log \left (1+e^{2 \tanh ^{-1}(c x)}\right )+2 i \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right ) \log \left (1-e^{-2 \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right )}\right )-\left (\pi -2 i \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{\sqrt {1-c^2 x^2}}\right )-2 i \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right ) \log \left (2 i \sinh \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right )\right )-i \text {PolyLog}\left (2,-e^{2 \tanh ^{-1}(c x)}\right )-i \text {PolyLog}\left (2,e^{-2 \left (\tanh ^{-1}\left (\frac {1}{2}\right )+\tanh ^{-1}(c x)\right )}\right )\right )}{2 c} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a*Log[1 + 2*c*x] + b*ArcTanh[c*x]*(Log[1 - c^2*x^2]/2 + Log[I*Sinh[ArcTanh[1/2] + ArcTanh[c*x]]]) - (I/2)*b*(
(-1/4*I)*(Pi - (2*I)*ArcTanh[c*x])^2 + I*(ArcTanh[1/2] + ArcTanh[c*x])^2 + (Pi - (2*I)*ArcTanh[c*x])*Log[1 + E
^(2*ArcTanh[c*x])] + (2*I)*(ArcTanh[1/2] + ArcTanh[c*x])*Log[1 - E^(-2*(ArcTanh[1/2] + ArcTanh[c*x]))] - (Pi -
(2*I)*ArcTanh[c*x])*Log[2/Sqrt[1 - c^2*x^2]] - (2*I)*(ArcTanh[1/2] + ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[1/2
] + ArcTanh[c*x]]] - I*PolyLog[2, -E^(2*ArcTanh[c*x])] - I*PolyLog[2, E^(-2*(ArcTanh[1/2] + ArcTanh[c*x]))]))/
(2*c)

________________________________________________________________________________________

Maple [A]
time = 0.26, size = 101, normalized size = 1.51

method	result	size
derivativedivides	\(\frac {\frac {a \ln \left (2 c x +1\right )}{2}+\frac {b \ln \left (2 c x +1\right ) \arctanh \left (c x \right )}{2}+\frac {b \ln \left (\frac {2}{3}-\frac {2 c x}{3}\right ) \ln \left (2 c x +1\right )}{4}-\frac {b \ln \left (\frac {2}{3}-\frac {2 c x}{3}\right ) \ln \left (\frac {2 c x}{3}+\frac {1}{3}\right )}{4}-\frac {b \dilog \left (\frac {2 c x}{3}+\frac {1}{3}\right )}{4}-\frac {b \dilog \left (2 c x +2\right )}{4}-\frac {b \ln \left (2 c x +1\right ) \ln \left (2 c x +2\right )}{4}}{c}\)	\(101\)
default	\(\frac {\frac {a \ln \left (2 c x +1\right )}{2}+\frac {b \ln \left (2 c x +1\right ) \arctanh \left (c x \right )}{2}+\frac {b \ln \left (\frac {2}{3}-\frac {2 c x}{3}\right ) \ln \left (2 c x +1\right )}{4}-\frac {b \ln \left (\frac {2}{3}-\frac {2 c x}{3}\right ) \ln \left (\frac {2 c x}{3}+\frac {1}{3}\right )}{4}-\frac {b \dilog \left (\frac {2 c x}{3}+\frac {1}{3}\right )}{4}-\frac {b \dilog \left (2 c x +2\right )}{4}-\frac {b \ln \left (2 c x +1\right ) \ln \left (2 c x +2\right )}{4}}{c}\)	\(101\)
risch	\(\frac {b \ln \left (\frac {2 c x}{3}+\frac {1}{3}\right ) \ln \left (\frac {2}{3}-\frac {2 c x}{3}\right )}{4 c}-\frac {b \ln \left (\frac {2 c x}{3}+\frac {1}{3}\right ) \ln \left (-c x +1\right )}{4 c}+\frac {b \dilog \left (\frac {2}{3}-\frac {2 c x}{3}\right )}{4 c}+\frac {a \ln \left (-2 c x -1\right )}{2 c}-\frac {b \ln \left (-2 c x -1\right ) \ln \left (2 c x +2\right )}{4 c}+\frac {b \ln \left (-2 c x -1\right ) \ln \left (c x +1\right )}{4 c}-\frac {b \dilog \left (2 c x +2\right )}{4 c}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x))/(2*c*x+1),x,method=_RETURNVERBOSE)

[Out]

1/c*(1/2*a*ln(2*c*x+1)+1/2*b*ln(2*c*x+1)*arctanh(c*x)+1/4*b*ln(2/3-2/3*c*x)*ln(2*c*x+1)-1/4*b*ln(2/3-2/3*c*x)*
ln(2/3*c*x+1/3)-1/4*b*dilog(2/3*c*x+1/3)-1/4*b*dilog(2*c*x+2)-1/4*b*ln(2*c*x+1)*ln(2*c*x+2))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c*x) + a)/(2*c*x + 1), x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \operatorname {atanh}{\left (c x \right )}}{2 c x + 1}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c\,x\right )}{2\,c\,x+1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))/(2*c*x + 1),x)

[Out]

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

________________________________________________________________________________________