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

Optimal. Leaf size=179 \[ -\frac{b^2 e^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac{e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac{2 b e^2 \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac{b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac{1}{3} b^2 e^2 x \]

[Out]

(b^2*e^2*x)/3 - (b^2*e^2*ArcTanh[c + d*x])/(3*d) + (b*e^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/(3*d) + (e^2*(
a + b*ArcTanh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcTanh[c + d*x])^2)/(3*d) - (2*b*e^2*(a + b*ArcTan
h[c + d*x])*Log[2/(1 - c - d*x)])/(3*d) - (b^2*e^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(3*d)

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Rubi [A]  time = 0.237224, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {6107, 12, 5916, 5980, 321, 206, 5984, 5918, 2402, 2315} \[ -\frac{b^2 e^2 \text{PolyLog}\left (2,-\frac{c+d x+1}{-c-d x+1}\right )}{3 d}+\frac{e^2 (c+d x)^3 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}+\frac{b e^2 (c+d x)^2 \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}+\frac{e^2 \left (a+b \tanh ^{-1}(c+d x)\right )^2}{3 d}-\frac{2 b e^2 \log \left (\frac{2}{-c-d x+1}\right ) \left (a+b \tanh ^{-1}(c+d x)\right )}{3 d}-\frac{b^2 e^2 \tanh ^{-1}(c+d x)}{3 d}+\frac{1}{3} b^2 e^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*e^2*x)/3 - (b^2*e^2*ArcTanh[c + d*x])/(3*d) + (b*e^2*(c + d*x)^2*(a + b*ArcTanh[c + d*x]))/(3*d) + (e^2*(
a + b*ArcTanh[c + d*x])^2)/(3*d) + (e^2*(c + d*x)^3*(a + b*ArcTanh[c + d*x])^2)/(3*d) - (2*b*e^2*(a + b*ArcTan
h[c + d*x])*Log[2/(1 - c - d*x)])/(3*d) - (b^2*e^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(3*d)

Rule 6107

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]

Rule 12

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

Rule 5916

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

Rule 5980

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

Rule 321

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 5984

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 5918

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 2402

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 2315

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

Rubi steps

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

Mathematica [A]  time = 0.450277, size = 150, normalized size = 0.84 \[ \frac{e^2 \left (b^2 \left (\text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c+d x)}\right )+(c+d x)^3 \tanh ^{-1}(c+d x)^2+(c+d x)^2 \tanh ^{-1}(c+d x)-\tanh ^{-1}(c+d x)^2-\tanh ^{-1}(c+d x)-2 \tanh ^{-1}(c+d x) \log \left (e^{-2 \tanh ^{-1}(c+d x)}+1\right )+c+d x\right )+a^2 (c+d x)^3+a b \left ((c+d x)^2+\log \left ((c+d x)^2-1\right )+2 (c+d x)^3 \tanh ^{-1}(c+d x)\right )\right )}{3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(e^2*(a^2*(c + d*x)^3 + a*b*((c + d*x)^2 + 2*(c + d*x)^3*ArcTanh[c + d*x] + Log[-1 + (c + d*x)^2]) + b^2*(c +
d*x - ArcTanh[c + d*x] + (c + d*x)^2*ArcTanh[c + d*x] - ArcTanh[c + d*x]^2 + (c + d*x)^3*ArcTanh[c + d*x]^2 -
2*ArcTanh[c + d*x]*Log[1 + E^(-2*ArcTanh[c + d*x])] + PolyLog[2, -E^(-2*ArcTanh[c + d*x])])))/(3*d)

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Maple [B]  time = 0.058, size = 583, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*d*arctanh(d*x+c)*x^2*a*b*c*e^2+arctanh(d*x+c)^2*x*b^2*c^2*e^2+2/3*arctanh(d*x+c)*x*b^2*c*e^2+2/3*x*a*b*c*e^2
+1/3*d*x^2*a*b*e^2+1/3*d*arctanh(d*x+c)*x^2*b^2*e^2+1/3/d*arctanh(d*x+c)^2*b^2*c^3*e^2+1/3/d*arctanh(d*x+c)*b^
2*c^2*e^2+1/6/d*e^2*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)+1/3/d*e^2*b^2*arctanh(d*x+c)*ln(d*x+c-1)+1/3/d*e^2*
b^2*arctanh(d*x+c)*ln(d*x+c+1)-1/6/d*e^2*b^2*ln(d*x+c-1)*ln(1/2+1/2*d*x+1/2*c)-1/6/d*e^2*b^2*ln(-1/2*d*x-1/2*c
+1/2)*ln(1/2+1/2*d*x+1/2*c)+1/3/d*e^2*a*b*ln(d*x+c-1)+1/3/d*e^2*a*b*ln(d*x+c+1)+d*x^2*a^2*c*e^2+1/3*b^2*e^2*x+
1/3*d^2*x^3*a^2*e^2+1/12/d*e^2*b^2*ln(d*x+c-1)^2-1/3/d*e^2*b^2*dilog(1/2+1/2*d*x+1/2*c)-1/12/d*e^2*b^2*ln(d*x+
c+1)^2-1/6/d*e^2*b^2*ln(d*x+c+1)+1/6/d*e^2*b^2*ln(d*x+c-1)+x*a^2*c^2*e^2+d*arctanh(d*x+c)^2*x^2*b^2*c*e^2+2/3/
d*arctanh(d*x+c)*a*b*c^3*e^2+1/3/d*a^2*c^3*e^2+1/3/d*b^2*c*e^2+1/3*d^2*arctanh(d*x+c)^2*x^3*b^2*e^2+1/3/d*a*b*
c^2*e^2+2*arctanh(d*x+c)*x*a*b*c^2*e^2+2/3*d^2*arctanh(d*x+c)*x^3*a*b*e^2

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Maxima [B]  time = 1.94724, size = 836, normalized size = 4.67 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*d^2*e^2*x^3 + a^2*c*d*e^2*x^2 + (2*x^2*arctanh(d*x + c) + d*(2*x/d^2 - (c^2 + 2*c + 1)*log(d*x + c + 1
)/d^3 + (c^2 - 2*c + 1)*log(d*x + c - 1)/d^3))*a*b*c*d*e^2 + 1/3*(2*x^3*arctanh(d*x + c) + d*((d*x^2 - 4*c*x)/
d^3 + (c^3 + 3*c^2 + 3*c + 1)*log(d*x + c + 1)/d^4 - (c^3 - 3*c^2 + 3*c - 1)*log(d*x + c - 1)/d^4))*a*b*d^2*e^
2 + a^2*c^2*e^2*x + (2*(d*x + c)*arctanh(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*c^2*e^2/d + 1/3*(log(d*x + c +
1)*log(-1/2*d*x - 1/2*c + 1/2) + dilog(1/2*d*x + 1/2*c + 1/2))*b^2*e^2/d + 1/6*(c^2*e^2 - e^2)*b^2*log(d*x + c
+ 1)/d - 1/6*(c^2*e^2 - e^2)*b^2*log(d*x + c - 1)/d + 1/12*(4*b^2*d*e^2*x + (b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^
2*x^2 + 3*b^2*c^2*d*e^2*x + (c^3*e^2 + e^2)*b^2)*log(d*x + c + 1)^2 + (b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^2*x^2 +
3*b^2*c^2*d*e^2*x + (c^3*e^2 - e^2)*b^2)*log(-d*x - c + 1)^2 + 2*(b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x)*log(d*x
+ c + 1) - 2*(b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + (b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^2*x^2 + 3*b^2*c^2*d*e^2*x +
(c^3*e^2 + e^2)*b^2)*log(d*x + c + 1))*log(-d*x - c + 1))/d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname{atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname{atanh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname{atanh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname{atanh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname{atanh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname{atanh}{\left (c + d x \right )}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**2*(Integral(a**2*c**2, x) + Integral(a**2*d**2*x**2, x) + Integral(b**2*c**2*atanh(c + d*x)**2, x) + Integr
al(2*a*b*c**2*atanh(c + d*x), x) + Integral(2*a**2*c*d*x, x) + Integral(b**2*d**2*x**2*atanh(c + d*x)**2, x) +
Integral(2*a*b*d**2*x**2*atanh(c + d*x), x) + Integral(2*b**2*c*d*x*atanh(c + d*x)**2, x) + Integral(4*a*b*c*
d*x*atanh(c + d*x), x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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