∫ (e + fx)2(a + b coth−1(c + dx))2 dx [109]

Optimal. Leaf size=374 \[ \frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d^3} \]

1/3*b^2*f^2*x/d^2+2*a*b*f*(-c*f+d*e)*x/d^2+2*b^2*f*(-c*f+d*e)*(d*x+c)*arccoth(d*x+c)/d^3+1/3*b*f^2*(d*x+c)^2*(

a+b*arccoth(d*x+c))/d^3-1/3*(-c*f+d*e)*(d^2*e^2-2*c*d*e*f+(c^2+3)*f^2)*(a+b*arccoth(d*x+c))^2/d^3/f+1/3*(3*d^2

*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))^2/d^3+1/3*(f*x+e)^3*(a+b*arccoth(d*x+c))^2/f-1/3*b^2*f^2*ar

ctanh(d*x+c)/d^3-2/3*b*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1))/d^3+b^2*f*(-c

*f+d*e)*ln(1-(d*x+c)^2)/d^3-1/3*b^2*(3*d^2*e^2-6*c*d*e*f+(3*c^2+1)*f^2)*polylog(2,(-d*x-c-1)/(-d*x-c+1))/d^3

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Rubi [A]
time = 0.42, antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {6247, 6066, 6022, 266, 6038, 327, 212, 6196, 6096, 6132, 6056, 2449, 2352} \begin {gather*} -\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \end {gather*}

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

(b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCoth[c + d*x])/d^3 + (b*

f^2*(c + d*x)^2*(a + b*ArcCoth[c + d*x]))/(3*d^3) - ((d*e - c*f)*(d^2*e^2 - 2*c*d*e*f + (3 + c^2)*f^2)*(a + b*

ArcCoth[c + d*x])^2)/(3*d^3*f) + ((3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])^2)/(3*d^3

) + ((e + f*x)^3*(a + b*ArcCoth[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTanh[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*

c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCoth[c + d*x])*Log[2/(1 - c - d*x)])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 -

(c + d*x)^2])/d^3 - (b^2*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))]

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]

/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

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

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,

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

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]

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x^n])^p, x] - Dist[b

*c*n*p, Int[x^n*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*

x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))

), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x])^p)

*(Log[2/(1 + e*(x/d))]/e), x] + Dist[b*c*(p/e), Int[(a + b*ArcCoth[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]

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((

a + b*ArcCoth[c*x])^p/(e*(q + 1))), x] - Dist[b*c*(p/(e*(q + 1))), Int[ExpandIntegrand[(a + b*ArcCoth[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] &

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[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]

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcCoth[c

*x])^(p + 1)/(b*e*(p + 1)), x] + Dist[1/(c*d), Int[(a + b*ArcCoth[c*x])^p/(1 - c*x), x], x] /; FreeQ[{a, b, c,

d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :>

Int[ExpandIntegrand[(a + b*ArcCoth[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]

Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[

Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcCoth[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] &

\begin {align*} \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right )^2 \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \left (-\frac {3 f^2 (d e-c f) \left (a+b \coth ^{-1}(x)\right )}{d^3}-\frac {f^3 x \left (a+b \coth ^{-1}(x)\right )}{d^3}+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{d^3 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{3 f}\\ &=\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \frac {\left ((d e-c f) \left (d^2 e^2-2 c d e f+3 f^2+c^2 f^2\right )+f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}+\frac {\left (2 b f^2\right ) \text {Subst}\left (\int x \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{3 d^3}+\frac {(2 b f (d e-c f)) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {2 a b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {(2 b) \text {Subst}\left (\int \left (\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2}+\frac {f \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{3 d^3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}+\frac {\left (2 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {\left (b^2 f^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}-\frac {\left (2 b^2 f (d e-c f)\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^3}-\frac {\left (2 b (d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x^2} \, dx,x,c+d x\right )}{3 d^3 f}-\frac {\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {\left (2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {\left (2 b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{3 d^3}\\ &=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \text {Li}_2\left (1-\frac {2}{1-c-d x}\right )}{3 d^3}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1054\) vs. \(2(374)=748\).
time = 6.82, size = 1054, normalized size = 2.82 \begin {gather*} a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {1}{3} a b \left (2 x \left (3 e^2+3 e f x+f^2 x^2\right ) \coth ^{-1}(c+d x)+\frac {d f x (6 d e-4 c f+d f x)-(-1+c) \left (3 d^2 e^2-3 (-1+c) d e f+(-1+c)^2 f^2\right ) \log (1-c-d x)+(1+c) \left (3 d^2 e^2-3 (1+c) d e f+(1+c)^2 f^2\right ) \log (1+c+d x)}{d^3}\right )+\frac {b^2 e^2 \left (1-(c+d x)^2\right ) \left (\coth ^{-1}(c+d x) \left (\coth ^{-1}(c+d x)-(c+d x) \coth ^{-1}(c+d x)+2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )-\text {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d (c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right )}-\frac {b^2 e f \left (1-(c+d x)^2\right ) \left (2 c \coth ^{-1}(c+d x)^2+(c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right ) \coth ^{-1}(c+d x)^2-2 (c+d x) \coth ^{-1}(c+d x) \left (-1+c \coth ^{-1}(c+d x)\right )+4 c \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )-2 \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-2 c \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d^2 (c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right )}-\frac {b^2 f^2 (c+d x) \sqrt {1-\frac {1}{(c+d x)^2}} \left (1-(c+d x)^2\right ) \left (\frac {4 \coth ^{-1}(c+d x)}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {3 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {12 c \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {9 c^2 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {-1+6 c \coth ^{-1}(c+d x)+3 \coth ^{-1}(c+d x)^2-3 c^2 \coth ^{-1}(c+d x)^2}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\cosh \left (3 \coth ^{-1}(c+d x)\right )-6 c \coth ^{-1}(c+d x) \cosh \left (3 \coth ^{-1}(c+d x)\right )+\coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+3 c^2 \coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+\frac {6 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {18 c^2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {18 c \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {4 \left (1+3 c^2\right ) \text {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{(c+d x)^3 \left (1-\frac {1}{(c+d x)^2}\right )^{3/2}}-\coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right )-3 c^2 \coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right )-2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )-6 c^2 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )+6 c \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )\right )}{12 d^3} \end {gather*}

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

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(2*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCoth[c + d*x] + (d*f*x*(6

*d*e - 4*c*f + d*f*x) - (-1 + c)*(3*d^2*e^2 - 3*(-1 + c)*d*e*f + (-1 + c)^2*f^2)*Log[1 - c - d*x] + (1 + c)*(3

*d^2*e^2 - 3*(1 + c)*d*e*f + (1 + c)^2*f^2)*Log[1 + c + d*x])/d^3))/3 + (b^2*e^2*(1 - (c + d*x)^2)*(ArcCoth[c

+ d*x]*(ArcCoth[c + d*x] - (c + d*x)*ArcCoth[c + d*x] + 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) - PolyLog[2, E^(-2

*ArcCoth[c + d*x])]))/(d*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^2*e*f*(1 - (c + d*x)^2)*(2*c*ArcCoth[c + d*x]^

2 + (c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d*x]^2 - 2*(c + d*x)*ArcCoth[c + d*x]*(-1 + c*ArcCoth[c + d*x

]) + 4*c*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - 2*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 2

*c*PolyLog[2, E^(-2*ArcCoth[c + d*x])]))/(d^2*(c + d*x)^2*(1 - (c + d*x)^(-2))) - (b^2*f^2*(c + d*x)*Sqrt[1 -

(c + d*x)^(-2)]*(1 - (c + d*x)^2)*((4*ArcCoth[c + d*x])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (3*ArcCoth[c +

d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (12*c*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])

+ (9*c^2*ArcCoth[c + d*x]^2)/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (-1 + 6*c*ArcCoth[c + d*x] + 3*ArcCoth[c +

d*x]^2 - 3*c^2*ArcCoth[c + d*x]^2)/Sqrt[1 - (c + d*x)^(-2)] + Cosh[3*ArcCoth[c + d*x]] - 6*c*ArcCoth[c + d*x]

*Cosh[3*ArcCoth[c + d*x]] + ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcC

oth[c + d*x]] + (6*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (

18*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) - (18*c*Log[1/(

(c + d*x)*Sqrt[1 - (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)]) + (4*(1 + 3*c^2)*PolyLog[2, E^(-2*A

rcCoth[c + d*x])])/((c + d*x)^3*(1 - (c + d*x)^(-2))^(3/2)) - ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 3*

c^2*ArcCoth[c + d*x]^2*Sinh[3*ArcCoth[c + d*x]] - 2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*A

rcCoth[c + d*x]] - 6*c^2*ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])]*Sinh[3*ArcCoth[c + d*x]] + 6*c*Log[

1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]*Sinh[3*ArcCoth[c + d*x]]))/(12*d^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2889\) vs. \(2(360)=720\).
time = 0.41, size = 2890, normalized size = 7.73


method	result	size

risch	\(\text {Expression too large to display}\)	\(2100\)
derivativedivides	\(\text {Expression too large to display}\)	\(2890\)
default	\(\text {Expression too large to display}\)	\(2890\)

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

1/d*(-a*b/d*f*ln(d*x+c+1)*e-4*a*b/d*f*arccoth(d*x+c)*c*e*(d*x+c)-1/4*b^2*ln(d*x+c+1)^2*e^2+1/4*b^2*ln(d*x+c-1)

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

)+1/12*b^2/d^2*f^2*ln(d*x+c-1)^2+1/6*b^2/d^2*f^2*ln(d*x+c-1)-1/6*b^2/d^2*f^2*ln(d*x+c+1)-1/4*b^2*ln(d*x+c-1)^2

*c*e^2+b^2*arccoth(d*x+c)*ln(d*x+c+1)*e^2-b^2*arccoth(d*x+c)^2*c*e^2+b^2*arccoth(d*x+c)^2*e^2*(d*x+c)-1/3*(c*f

-d*e-f*(d*x+c))^3*a^2/d^2/f+a*b*ln(d*x+c-1)*e^2+a*b*ln(d*x+c+1)*e^2-1/4*b^2*ln(d*x+c+1)^2*c*e^2+b^2*arccoth(d*

x+c)*ln(d*x+c-1)*e^2-1/2*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e^2+1/2*b^2*ln(-1/2*d*x-1/2*c+1/2)*l

n(d*x+c+1)*e^2-1/2*b^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*e^2+1/3*b^2/d^2*f^2*(d*x+c)-1/4*b^2/d^2*f^2*ln(d*x+c+

1)^2*c^2-1/4*b^2/d^2*f^2*ln(d*x+c+1)^2*c+1/6*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)-1/6*b^2/d^2*f^2*ln

(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)-b^2/d^2*f^2*dilog(1/2*d*x+1/2*c+1/2)*c^2-1/6*b^2/d^2*f^2*ln(d*x+c-1

)*ln(1/2*d*x+1/2*c+1/2)-1/12*b^2/d^2*f^2*ln(d*x+c-1)^2*c^3+1/4*b^2/d^2*f^2*ln(d*x+c-1)^2*c^2+1/4*b^2/d*f*ln(d*

x+c+1)^2*e+1/4*b^2/d*f*ln(d*x+c-1)^2*e+b^2/d*f*ln(d*x+c+1)*e+b^2/d*f*ln(d*x+c-1)*e+1/12*b^2*d/f*ln(d*x+c-1)^2*

e^3+1/12*b^2*d/f*ln(d*x+c+1)^2*e^3+1/3*a*b/d^2*f^2*(d*x+c)^2-2*a*b*arccoth(d*x+c)*c*e^2+2*a*b*arccoth(d*x+c)*e

^2*(d*x+c)-a*b*ln(d*x+c-1)*c*e^2+a*b*ln(d*x+c+1)*c*e^2+1/3*a*b/d^2*f^2*ln(d*x+c+1)+1/3*a*b/d^2*f^2*ln(d*x+c-1)

-1/3*b^2/d^2*f^2*arccoth(d*x+c)^2*c^3+b^2*arccoth(d*x+c)*ln(d*x+c+1)*c*e^2-b^2*arccoth(d*x+c)*ln(d*x+c-1)*c*e^

2-1/2*b^2/d^2*f^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*c^2+1/2*b^2/d^2*f^2*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*c+2*

a*b/d*f*e*(d*x+c)+a*b/d*f*ln(d*x+c-1)*e-a*b/d^2*f^2*ln(d*x+c-1)*c+1/3*a*b*d/f*ln(d*x+c-1)*e^3-1/3*a*b*d/f*ln(d

*x+c+1)*e^3+2/3*a*b*d/f*arccoth(d*x+c)*e^3-1/3*a*b/d^2*f^2*ln(d*x+c-1)*c^3+a*b/d^2*f^2*ln(d*x+c-1)*c^2+1/3*a*b

/d^2*f^2*ln(d*x+c+1)*c^3+a*b/d^2*f^2*ln(d*x+c+1)*c^2-2/3*a*b/d^2*f^2*arccoth(d*x+c)*c^3+2/3*a*b/d^2*f^2*arccot

h(d*x+c)*(d*x+c)^3+a*b/d^2*f^2*ln(d*x+c+1)*c+1/3*b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+c+1)*c^3+1/2*b^2/d*f*ln(d*x

+c+1)^2*c*e-1/2*b^2/d*f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*e+1/3*b^2*d/f*arccoth(d*x+c)*ln(d*x+c-1)*e^3-1/3*b^

2*d/f*arccoth(d*x+c)*ln(d*x+c+1)*e^3+2*b^2/d*f*dilog(1/2*d*x+1/2*c+1/2)*c*e-1/2*b^2/d*f*ln(d*x+c-1)*ln(1/2*d*x

+1/2*c+1/2)*e+1/4*b^2/d*f*ln(d*x+c+1)^2*c^2*e-2*a*b/d^2*f^2*c*(d*x+c)+b^2/d*f*arccoth(d*x+c)^2*c^2*e+b^2/d*f*a

rccoth(d*x+c)^2*e*(d*x+c)^2+b^2/d^2*f^2*arccoth(d*x+c)^2*c^2*(d*x+c)-b^2/d^2*f^2*arccoth(d*x+c)^2*c*(d*x+c)^2-

1/6*b^2*d/f*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*e^3+2*b^2/d*f*arccoth(d*x+c)*e*(d*x+c)+b^2/d*f*arccoth(d*x+c)*ln

(d*x+c-1)*e-b^2/d*f*arccoth(d*x+c)*ln(d*x+c+1)*e+1/4*b^2/d*f*ln(d*x+c-1)^2*c^2*e-1/2*b^2/d*f*ln(d*x+c-1)^2*c*e

-1/6*b^2*d/f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*e^3+1/6*b^2*d/f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e

^3+1/2*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c^2+1/2*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c

-1/6*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c^3-1/2*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/

2*d*x+1/2*c+1/2)*c^2-1/2*b^2/d^2*f^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c+1/6*b^2/d^2*f^2*ln(-1/2*d*

x-1/2*c+1/2)*ln(d*x+c+1)*c^3-2*b^2/d^2*f^2*arccoth(d*x+c)*c*(d*x+c)+1/6*b^2/d^2*f^2*ln(d*x+c-1)*ln(1/2*d*x+1/2

*c+1/2)*c^3+b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+c+1)*c^2+b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+c+1)*c-1/3*b^2/d^2*f^

2*arccoth(d*x+c)*ln(d*x+c-1)*c^3+b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+c-1)*c^2-b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+

c-1)*c+1/2*b^2/d*f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*e+1/3*b^2/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)^3+1

/3*b^2*d/f*arccoth(d*x+c)^2*e^3-1/2*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c*e^2+1/2*b^2*ln(d*x+c-1)

*ln(1/2*d*x+1/2*c+1/2)*c*e^2+1/2*b^2*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c*e^2-1/4*b^2/d^2*f^2*ln(d*x+c-1)^2*c-

b^2/d^2*f^2*ln(d*x+c+1)*c+1/3*b^2/d^2*f^2*arccoth(d*x+c)*(d*x+c)^2-b^2/d^2*f^2*ln(d*x+c-1)*c+1/3*b^2/d^2*f^2*a

rccoth(d*x+c)*ln(d*x+c-1)+1/3*b^2/d^2*f^2*arccoth(d*x+c)*ln(d*x+c+1)-1/12*b^2/d^2*f^2*ln(d*x+c+1)^2*c^3-b^2/d*

f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c*e+b^2/d*f*arccoth(d*x+c)*ln(d*x+c-1)*c^2*e+a*b/d*f*ln(d*x+c-1)*c^2*e-2*

a*b/d*f*ln(d*x+c-1)*c*e-a*b/d*f*ln(d*x+c+1)*c^2*e-2*a*b/d*f*ln(d*x+c+1)*c*e+2*a*b/d^2*f^2*arccoth(d*x+c)*c^2*(

d*x+c)-2*a*b/d^2*f^2*arccoth(d*x+c)*c*(d*x+c)^2+2*a*b/d*f*arccoth(d*x+c)*c^2*e+2*a*b/d*f*arccoth(d*x+c)*e*(d*x

+c)^2-1/2*b^2/d*f*ln(-1/2*d*x-1/2*c+1/2)*ln(d*x+c+1)*c^2*e+b^2/d*f*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*c*e-2*b^2

/d*f*arccoth(d*x+c)*ln(d*x+c+1)*c*e+1/2*b^2/d*f*ln(-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c^2*e+b^2/d*f*ln(

-1/2*d*x-1/2*c+1/2)*ln(1/2*d*x+1/2*c+1/2)*c*e-1/2*b^2/d*f*ln(d*x+c-1)*ln(1/2*d*x+1/2*c+1/2)*c^2*e-2*b^2/d*f*ar

ccoth(d*x+c)*ln(d*x+c-1)*c*e-b^2/d*f*arccoth(d*x+c)*ln(d*x+c+1)*c^2*e-2*b^2/d*f*arccoth(d*x+c)^2*c*e*(d*x+c))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (361) = 722\).
time = 0.45, size = 843, normalized size = 2.25 \begin {gather*} \frac {1}{3} \, a^{2} f^{2} x^{3} + a^{2} f x^{2} e + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} a b f^{2} + {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f e + a^{2} x e^{2} + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e^{2}}{d} + \frac {{\left (6 \, b^{2} c d f e - 3 \, b^{2} d^{2} e^{2} - {\left (3 \, c^{2} f^{2} + f^{2}\right )} b^{2}\right )} {\left (\log \left (d x + c - 1\right ) \log \left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right )\right )}}{3 \, d^{3}} + \frac {{\left (6 \, {\left (c d f + d f\right )} b^{2} e - {\left (5 \, c^{2} f^{2} + 6 \, c f^{2} + f^{2}\right )} b^{2}\right )} \log \left (d x + c + 1\right )}{6 \, d^{3}} + \frac {4 \, b^{2} d f^{2} x + {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} f x^{2} e + 3 \, b^{2} d^{3} x e^{2} + 3 \, {\left (c d^{2} + d^{2}\right )} b^{2} e^{2} - 3 \, {\left (c^{2} d f + 2 \, c d f + d f\right )} b^{2} e + {\left (c^{3} f^{2} + 3 \, c^{2} f^{2} + 3 \, c f^{2} + f^{2}\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} f x^{2} e + 3 \, b^{2} d^{3} x e^{2} + 3 \, {\left (c d^{2} - d^{2}\right )} b^{2} e^{2} - 3 \, {\left (c^{2} d f - 2 \, c d f + d f\right )} b^{2} e + {\left (c^{3} f^{2} - 3 \, c^{2} f^{2} + 3 \, c f^{2} - f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (b^{2} d^{2} f^{2} x^{2} - 2 \, {\left (2 \, b^{2} c d f^{2} - 3 \, b^{2} d^{2} f e\right )} x - {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} f x^{2} e + 3 \, b^{2} d^{3} x e^{2} + 3 \, {\left (c d^{2} - d^{2}\right )} b^{2} e^{2} - 3 \, {\left (c^{2} d f - 2 \, c d f + d f\right )} b^{2} e + {\left (c^{3} f^{2} - 3 \, c^{2} f^{2} + 3 \, c f^{2} - f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) - 2 \, {\left (b^{2} d^{2} f^{2} x^{2} + 6 \, {\left (c d f - d f\right )} b^{2} e - {\left (5 \, c^{2} f^{2} - 6 \, c f^{2} + f^{2}\right )} b^{2} - 2 \, {\left (2 \, b^{2} c d f^{2} - 3 \, b^{2} d^{2} f e\right )} x\right )} \log \left (d x + c - 1\right )}{12 \, d^{3}} \end {gather*}

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

1/3*a^2*f^2*x^3 + a^2*f*x^2*e + 1/3*(2*x^3*arccoth(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*f^2 + (2*x^2*arccoth(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*f*e + a^2*x*e^2 +

(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e^2/d + 1/3*(6*b^2*c*d*f*e - 3*b^2*d^2*e^2 - (3*c^2

*f^2 + f^2)*b^2)*(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))/d^3 + 1/6*(6*(c

*d*f + d*f)*b^2*e - (5*c^2*f^2 + 6*c*f^2 + f^2)*b^2)*log(d*x + c + 1)/d^3 + 1/12*(4*b^2*d*f^2*x + (b^2*d^3*f^2

*x^3 + 3*b^2*d^3*f*x^2*e + 3*b^2*d^3*x*e^2 + 3*(c*d^2 + d^2)*b^2*e^2 - 3*(c^2*d*f + 2*c*d*f + d*f)*b^2*e + (c^

3*f^2 + 3*c^2*f^2 + 3*c*f^2 + f^2)*b^2)*log(d*x + c + 1)^2 + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*f*x^2*e + 3*b^2*d^3*

x*e^2 + 3*(c*d^2 - d^2)*b^2*e^2 - 3*(c^2*d*f - 2*c*d*f + d*f)*b^2*e + (c^3*f^2 - 3*c^2*f^2 + 3*c*f^2 - f^2)*b^

2)*log(d*x + c - 1)^2 + 2*(b^2*d^2*f^2*x^2 - 2*(2*b^2*c*d*f^2 - 3*b^2*d^2*f*e)*x - (b^2*d^3*f^2*x^3 + 3*b^2*d^

3*f*x^2*e + 3*b^2*d^3*x*e^2 + 3*(c*d^2 - d^2)*b^2*e^2 - 3*(c^2*d*f - 2*c*d*f + d*f)*b^2*e + (c^3*f^2 - 3*c^2*f

^2 + 3*c*f^2 - f^2)*b^2)*log(d*x + c - 1))*log(d*x + c + 1) - 2*(b^2*d^2*f^2*x^2 + 6*(c*d*f - d*f)*b^2*e - (5*

c^2*f^2 - 6*c*f^2 + f^2)*b^2 - 2*(2*b^2*c*d*f^2 - 3*b^2*d^2*f*e)*x)*log(d*x + c - 1))/d^3

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

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

integral(a^2*f^2*x^2 + 2*a^2*f*x*e + (b^2*f^2*x^2 + 2*b^2*f*x*e + b^2*e^2)*arccoth(d*x + c)^2 + a^2*e^2 + 2*(a

*b*f^2*x^2 + 2*a*b*f*x*e + a*b*e^2)*arccoth(d*x + c), x)

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

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

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

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

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

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

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

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

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

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3.2.9 \(\int (e+f x)^2 (a+b \coth ^{-1}(c+d x))^2 \, dx\) [109]