3.1.11 \(\int (d+e x) (a+b \tanh ^{-1}(c x))^2 \, dx\) [11]

Optimal. Leaf size=160 \[ \frac {a b e x}{c}+\frac {b^2 e x \tanh ^{-1}(c x)}{c}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {\left (d^2+\frac {e^2}{c^2}\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}-\frac {2 b d \left (a+b \tanh ^{-1}(c x)\right ) \log \left (\frac {2}{1-c x}\right )}{c}+\frac {b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \text {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{c} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {6065, 6021, 266, 6195, 6095, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {\left (\frac {e^2}{c^2}+d^2\right ) \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {(d+e x)^2 \left (a+b \tanh ^{-1}(c x)\right )^2}{2 e}+\frac {d \left (a+b \tanh ^{-1}(c x)\right )^2}{c}-\frac {2 b d \log \left (\frac {2}{1-c x}\right ) \left (a+b \tanh ^{-1}(c x)\right )}{c}+\frac {a b e x}{c}+\frac {b^2 e \log \left (1-c^2 x^2\right )}{2 c^2}-\frac {b^2 d \text {Li}_2\left (1-\frac {2}{1-c x}\right )}{c}+\frac {b^2 e x \tanh ^{-1}(c x)}{c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)*(a + b*ArcTanh[c*x])^2,x]

[Out]

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

Rule 266

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

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 6021

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

Rule 6055

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^p)
*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[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 6065

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

Rule 6095

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

Rule 6131

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

Rule 6195

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>
Int[ExpandIntegrand[(a + b*ArcTanh[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
&& IGtQ[p, 0] && EqQ[c^2*d + e, 0] && IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.26, size = 174, normalized size = 1.09 \begin {gather*} \frac {2 a^2 c^2 d x+2 a b c e x+a^2 c^2 e x^2+b^2 (-1+c x) (2 c d+e+c e x) \tanh ^{-1}(c x)^2+2 b c \tanh ^{-1}(c x) \left (b e x+a c x (2 d+e x)-2 b d \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right )\right )+a b e \log (1-c x)-a b e \log (1+c x)+2 a b c d \log \left (1-c^2 x^2\right )+b^2 e \log \left (1-c^2 x^2\right )+2 b^2 c d \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right )}{2 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)*(a + b*ArcTanh[c*x])^2,x]

[Out]

(2*a^2*c^2*d*x + 2*a*b*c*e*x + a^2*c^2*e*x^2 + b^2*(-1 + c*x)*(2*c*d + e + c*e*x)*ArcTanh[c*x]^2 + 2*b*c*ArcTa
nh[c*x]*(b*e*x + a*c*x*(2*d + e*x) - 2*b*d*Log[1 + E^(-2*ArcTanh[c*x])]) + a*b*e*Log[1 - c*x] - a*b*e*Log[1 +
c*x] + 2*a*b*c*d*Log[1 - c^2*x^2] + b^2*e*Log[1 - c^2*x^2] + 2*b^2*c*d*PolyLog[2, -E^(-2*ArcTanh[c*x])])/(2*c^
2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(441\) vs. \(2(154)=308\).
time = 0.28, size = 442, normalized size = 2.76

method	result	size
derivativedivides	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b^{2} c \arctanh \left (c x \right )^{2} e \,x^{2}}{2}+a b e x +\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) e}{2 c}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) e}{2 c}+b^{2} \arctanh \left (c x \right )^{2} d c x -\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d}{2}+\frac {b^{2} \ln \left (c x -1\right ) e}{2 c}+\frac {b^{2} \ln \left (c x +1\right ) e}{2 c}+a b \ln \left (c x -1\right ) d +a b \ln \left (c x +1\right ) d +\frac {b^{2} \ln \left (c x +1\right )^{2} e}{8 c}+\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{2}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{2}+\frac {b^{2} \ln \left (c x -1\right )^{2} e}{8 c}-\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}+\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}+\frac {a b \ln \left (c x -1\right ) e}{2 c}-\frac {a b \ln \left (c x +1\right ) e}{2 c}-\frac {b^{2} \ln \left (c x +1\right )^{2} d}{4}+\frac {b^{2} \ln \left (c x -1\right )^{2} d}{4}-b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right ) d +a b c \arctanh \left (c x \right ) e \,x^{2}+2 a b \arctanh \left (c x \right ) d c x +b^{2} \arctanh \left (c x \right ) e x +b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) d +b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) d}{c}\)	\(442\)
default	\(\frac {\frac {a^{2} \left (d \,c^{2} x +\frac {1}{2} e \,c^{2} x^{2}\right )}{c}+\frac {b^{2} c \arctanh \left (c x \right )^{2} e \,x^{2}}{2}+a b e x +\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) e}{2 c}-\frac {b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) e}{2 c}+b^{2} \arctanh \left (c x \right )^{2} d c x -\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d}{2}+\frac {b^{2} \ln \left (c x -1\right ) e}{2 c}+\frac {b^{2} \ln \left (c x +1\right ) e}{2 c}+a b \ln \left (c x -1\right ) d +a b \ln \left (c x +1\right ) d +\frac {b^{2} \ln \left (c x +1\right )^{2} e}{8 c}+\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{2}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{2}+\frac {b^{2} \ln \left (c x -1\right )^{2} e}{8 c}-\frac {b^{2} \ln \left (c x +1\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}+\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}-\frac {b^{2} \ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) e}{4 c}+\frac {a b \ln \left (c x -1\right ) e}{2 c}-\frac {a b \ln \left (c x +1\right ) e}{2 c}-\frac {b^{2} \ln \left (c x +1\right )^{2} d}{4}+\frac {b^{2} \ln \left (c x -1\right )^{2} d}{4}-b^{2} \dilog \left (\frac {c x}{2}+\frac {1}{2}\right ) d +a b c \arctanh \left (c x \right ) e \,x^{2}+2 a b \arctanh \left (c x \right ) d c x +b^{2} \arctanh \left (c x \right ) e x +b^{2} \arctanh \left (c x \right ) \ln \left (c x -1\right ) d +b^{2} \arctanh \left (c x \right ) \ln \left (c x +1\right ) d}{c}\)	\(442\)
risch	\(\frac {b^{2} \ln \left (-\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) d}{c}-\frac {b^{2} \ln \left (\frac {c x}{2}+\frac {1}{2}\right ) \ln \left (-c x +1\right ) d}{c}+\frac {b \ln \left (-c x -1\right ) a d}{c}-\frac {b \ln \left (-c x -1\right ) a e}{2 c^{2}}-\frac {b a e}{c^{2}}-\frac {b^{2} e \ln \left (-c x +1\right )^{2}}{8 c^{2}}+\frac {3 b^{2} e \ln \left (-c x +1\right )}{8 c^{2}}-\frac {\ln \left (-c x +1\right )^{2} b^{2} d}{4 c}+\frac {\ln \left (-c x +1\right ) b^{2} d}{2 c}-\frac {\ln \left (-c x +1\right ) x \,b^{2} d}{2}+\frac {b^{2} e \ln \left (-c x +1\right )^{2} x^{2}}{8}-\frac {b^{2} e \ln \left (-c x +1\right ) x^{2}}{8}+\frac {\ln \left (-c x +1\right )^{2} x \,b^{2} d}{4}+a^{2} d x +\left (-\frac {b^{2} x \left (e x +2 d \right ) \ln \left (-c x +1\right )}{4}+\frac {b \left (2 a \,c^{2} e \,x^{2}+4 a \,c^{2} d x +2 \ln \left (-c x +1\right ) b c d +2 b c e x +\ln \left (-c x +1\right ) b e \right )}{4 c^{2}}\right ) \ln \left (c x +1\right )+\frac {\ln \left (-c x +1\right ) a b d}{c}+\frac {b a e \ln \left (-c x +1\right )}{2 c^{2}}-\frac {b^{2} e \ln \left (-c x +1\right ) x}{4 c}-\ln \left (-c x +1\right ) x a b d -\frac {b a e \ln \left (-c x +1\right ) x^{2}}{2}+\frac {b^{2} \left (-c x +1\right )^{2} \ln \left (-c x +1\right ) e}{8 c^{2}}-\frac {b^{2} \left (-c x +1\right ) \ln \left (-c x +1\right ) d}{2 c}-\frac {e \,a^{2}}{2 c^{2}}-\frac {d \,a^{2}}{c}+\frac {e \,a^{2} x^{2}}{2}+\frac {b^{2} \dilog \left (-\frac {c x}{2}+\frac {1}{2}\right ) d}{c}+\frac {b^{2} \ln \left (-c x -1\right ) e}{2 c^{2}}+\frac {b^{2} \left (e \,c^{2} x^{2}+2 d \,c^{2} x +2 d c -e \right ) \ln \left (c x +1\right )^{2}}{8 c^{2}}+\frac {a b e x}{c}\)	\(528\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(a+b*arctanh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(a^2/c*(d*c^2*x+1/2*e*c^2*x^2)+a*b*c*arctanh(c*x)*e*x^2+2*a*b*arctanh(c*x)*d*c*x+1/2*b^2*c*arctanh(c*x)^2*
e*x^2+b^2*arctanh(c*x)*e*x+a*b*e*x-1/2*b^2*ln(c*x-1)*ln(1/2*c*x+1/2)*d+1/2*b^2/c*ln(c*x-1)*e+1/2*b^2/c*ln(c*x+
1)*e+a*b*ln(c*x-1)*d+a*b*ln(c*x+1)*d+b^2*arctanh(c*x)*ln(c*x-1)*d+b^2*arctanh(c*x)*ln(c*x+1)*d+1/8*b^2/c*ln(c*
x+1)^2*e+1/2*b^2*ln(c*x+1)*ln(-1/2*c*x+1/2)*d-1/2*b^2*ln(1/2*c*x+1/2)*ln(-1/2*c*x+1/2)*d+1/8*b^2/c*ln(c*x-1)^2
*e-1/2*b^2/c*arctanh(c*x)*ln(c*x+1)*e-1/4*b^2/c*ln(c*x+1)*ln(-1/2*c*x+1/2)*e+1/4*b^2/c*ln(1/2*c*x+1/2)*ln(-1/2
*c*x+1/2)*e-1/4*b^2/c*ln(c*x-1)*ln(1/2*c*x+1/2)*e+1/2*a*b/c*ln(c*x-1)*e-1/2*a*b/c*ln(c*x+1)*e-1/4*b^2*ln(c*x+1
)^2*d-b^2*dilog(1/2*c*x+1/2)*d+1/4*b^2*ln(c*x-1)^2*d+b^2*arctanh(c*x)^2*d*c*x+1/2*b^2/c*arctanh(c*x)*ln(c*x-1)
*e)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (152) = 304\).
time = 0.41, size = 329, normalized size = 2.06 \begin {gather*} \frac {1}{2} \, a^{2} x^{2} e + a^{2} d x + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (c x\right ) + c {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )}\right )} a b e + \frac {{\left (2 \, c x \operatorname {artanh}\left (c x\right ) + \log \left (-c^{2} x^{2} + 1\right )\right )} a b d}{c} + \frac {{\left (\log \left (c x + 1\right ) \log \left (-\frac {1}{2} \, c x + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} \, c x + \frac {1}{2}\right )\right )} b^{2} d}{c} + \frac {b^{2} e \log \left (c x + 1\right )}{2 \, c^{2}} + \frac {b^{2} e \log \left (c x - 1\right )}{2 \, c^{2}} + \frac {4 \, b^{2} c x e \log \left (c x + 1\right ) + {\left (b^{2} c^{2} x^{2} e + 2 \, b^{2} c^{2} d x + 2 \, b^{2} c d - b^{2} e\right )} \log \left (c x + 1\right )^{2} + {\left (b^{2} c^{2} x^{2} e + 2 \, b^{2} c^{2} d x - 2 \, b^{2} c d - b^{2} e\right )} \log \left (-c x + 1\right )^{2} - 2 \, {\left (2 \, b^{2} c x e + {\left (b^{2} c^{2} x^{2} e + 2 \, b^{2} c^{2} d x + 2 \, b^{2} c d - b^{2} e\right )} \log \left (c x + 1\right )\right )} \log \left (-c x + 1\right )}{8 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="maxima")

[Out]

1/2*a^2*x^2*e + a^2*d*x + 1/2*(2*x^2*arctanh(c*x) + c*(2*x/c^2 - log(c*x + 1)/c^3 + log(c*x - 1)/c^3))*a*b*e +
(2*c*x*arctanh(c*x) + log(-c^2*x^2 + 1))*a*b*d/c + (log(c*x + 1)*log(-1/2*c*x + 1/2) + dilog(1/2*c*x + 1/2))*
b^2*d/c + 1/2*b^2*e*log(c*x + 1)/c^2 + 1/2*b^2*e*log(c*x - 1)/c^2 + 1/8*(4*b^2*c*x*e*log(c*x + 1) + (b^2*c^2*x
^2*e + 2*b^2*c^2*d*x + 2*b^2*c*d - b^2*e)*log(c*x + 1)^2 + (b^2*c^2*x^2*e + 2*b^2*c^2*d*x - 2*b^2*c*d - b^2*e)
*log(-c*x + 1)^2 - 2*(2*b^2*c*x*e + (b^2*c^2*x^2*e + 2*b^2*c^2*d*x + 2*b^2*c*d - b^2*e)*log(c*x + 1))*log(-c*x
+ 1))/c^2

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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((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="fricas")

[Out]

integral(a^2*x*e + a^2*d + (b^2*x*e + b^2*d)*arctanh(c*x)^2 + 2*(a*b*x*e + a*b*d)*arctanh(c*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(a+b*atanh(c*x))**2,x)

[Out]

Integral((a + b*atanh(c*x))**2*(d + e*x), x)

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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((e*x+d)*(a+b*arctanh(c*x))^2,x, algorithm="giac")

[Out]

integrate((e*x + d)*(b*arctanh(c*x) + a)^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*atanh(c*x))^2*(d + e*x),x)

[Out]

int((a + b*atanh(c*x))^2*(d + e*x), x)

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