3.1.12 \(\int \frac {a+b \tanh ^{-1}(c+d x)}{c e+d e x} \, dx\) [12]

Optimal. Leaf size=54 \[ \frac {a \log (c+d x)}{d e}-\frac {b \text {PolyLog}(2,-c-d x)}{2 d e}+\frac {b \text {PolyLog}(2,c+d x)}{2 d e} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6242, 12, 6031} \begin {gather*} \frac {a \log (c+d x)}{d e}-\frac {b \text {Li}_2(-c-d x)}{2 d e}+\frac {b \text {Li}_2(c+d x)}{2 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c + d*x])/(c*e + d*e*x),x]

[Out]

(a*Log[c + d*x])/(d*e) - (b*PolyLog[2, -c - d*x])/(2*d*e) + (b*PolyLog[2, c + d*x])/(2*d*e)

Rule 12

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

Rule 6031

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b/2)*PolyLog[2, (-c)*x]
, x] + Simp[(b/2)*PolyLog[2, c*x], x]) /; FreeQ[{a, b, c}, x]

Rule 6242

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

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 288, normalized size = 5.33 \begin {gather*} \frac {a \log (c+d x)}{d e}-\frac {i b \left (i \tanh ^{-1}(c+d x) \left (-\log \left (\frac {1}{\sqrt {1-(c+d x)^2}}\right )+\log \left (\frac {i (c+d x)}{\sqrt {1-(c+d x)^2}}\right )\right )+\frac {1}{2} \left (-\frac {1}{4} i \left (\pi -2 i \tanh ^{-1}(c+d x)\right )^2+i \tanh ^{-1}(c+d x)^2+\left (\pi -2 i \tanh ^{-1}(c+d x)\right ) \log \left (1-e^{i \left (\pi -2 i \tanh ^{-1}(c+d x)\right )}\right )+2 i \tanh ^{-1}(c+d x) \log \left (1-e^{-2 \tanh ^{-1}(c+d x)}\right )-2 i \tanh ^{-1}(c+d x) \log \left (\frac {2 i (c+d x)}{\sqrt {1-(c+d x)^2}}\right )-\left (\pi -2 i \tanh ^{-1}(c+d x)\right ) \log \left (2 \sin \left (\frac {1}{2} \left (\pi -2 i \tanh ^{-1}(c+d x)\right )\right )\right )-i \text {PolyLog}\left (2,e^{i \left (\pi -2 i \tanh ^{-1}(c+d x)\right )}\right )-i \text {PolyLog}\left (2,e^{-2 \tanh ^{-1}(c+d x)}\right )\right )\right )}{d e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c + d*x])/(c*e + d*e*x),x]

[Out]

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

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Maple [A]
time = 0.79, size = 78, normalized size = 1.44

method	result	size
risch	\(\frac {b \dilog \left (-d x -c +1\right )}{2 d e}+\frac {a \ln \left (-d x -c \right )}{d e}-\frac {b \dilog \left (d x +c +1\right )}{2 e d}\)	\(54\)
derivativedivides	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \ln \left (d x +c \right ) \arctanh \left (d x +c \right )}{e}-\frac {b \dilog \left (d x +c +1\right )}{2 e}-\frac {b \ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{2 e}-\frac {b \dilog \left (d x +c \right )}{2 e}}{d}\)	\(78\)
default	\(\frac {\frac {a \ln \left (d x +c \right )}{e}+\frac {b \ln \left (d x +c \right ) \arctanh \left (d x +c \right )}{e}-\frac {b \dilog \left (d x +c +1\right )}{2 e}-\frac {b \ln \left (d x +c \right ) \ln \left (d x +c +1\right )}{2 e}-\frac {b \dilog \left (d x +c \right )}{2 e}}{d}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(d*x+c))/(d*e*x+c*e),x,method=_RETURNVERBOSE)

[Out]

1/d*(a/e*ln(d*x+c)+b/e*ln(d*x+c)*arctanh(d*x+c)-1/2*b/e*dilog(d*x+c+1)-1/2*b/e*ln(d*x+c)*ln(d*x+c+1)-1/2*b/e*d
ilog(d*x+c))

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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(d*x+c))/(d*e*x+c*e),x, algorithm="maxima")

[Out]

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

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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(d*x+c))/(d*e*x+c*e),x, algorithm="fricas")

[Out]

integral((b*arctanh(d*x + c) + a)*e^(-1)/(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(d*x+c))/(d*e*x+c*e),x)

[Out]

(Integral(a/(c + d*x), x) + Integral(b*atanh(c + d*x)/(c + d*x), x))/e

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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(d*x+c))/(d*e*x+c*e),x, algorithm="giac")

[Out]

integrate((b*arctanh(d*x + c) + a)/(d*e*x + c*e), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {atanh}\left (c+d\,x\right )}{c\,e+d\,e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c + d*x))/(c*e + d*e*x),x)

[Out]

int((a + b*atanh(c + d*x))/(c*e + d*e*x), x)

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