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

Optimal. Leaf size=374 \[ -\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {Li}_2\left (-\frac {c+d x+1}{-c-d x+1}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]

[Out]

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 = 0.64, 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 = {6112, 5929, 5911, 260, 5917, 321, 206, 6049, 5949, 5985, 5919, 2402, 2315} \[ -\frac {b^2 \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{3 d^3}-\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {2 b \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {2 a b f x (d e-c f)}{d^2}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}+\frac {2 b^2 f (c+d x) (d e-c f) \coth ^{-1}(c+d x)}{d^3}-\frac {b^2 f^2 \tanh ^{-1}(c+d x)}{3 d^3}+\frac {b^2 f^2 x}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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))]
)/(3*d^3)

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 260

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 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 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]

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 5911

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

Rule 5917

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcC
oth[c*x])^p)/(d*(m + 1)), x] - Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCoth[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 5919

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]

Rule 5929

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] &
& NeQ[q, -1]

Rule 5949

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]

Rule 5985

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]

Rule 6049

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]

Rule 6112

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] &
& IGtQ[p, 0]

Rubi steps

\begin {align*} \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx &=\frac {\operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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)) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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]  time = 7.36, size = 1054, normalized size = 2.82 \[ \frac {1}{3} a^2 f^2 x^3+a^2 e f x^2+a^2 e^2 x+\frac {1}{3} a b \left (2 x \left (3 e^2+3 f x e+f^2 x^2\right ) \coth ^{-1}(c+d x)+\frac {d f x (6 d e-4 c f+d f x)-(c-1) \left (3 d^2 e^2-3 (c-1) d f e+(c-1)^2 f^2\right ) \log (-c-d x+1)+(c+1) \left (3 d^2 e^2-3 (c+1) d f e+(c+1)^2 f^2\right ) \log (c+d x+1)}{d^3}\right )+\frac {b^2 e^2 \left (1-(c+d x)^2\right ) \left (\coth ^{-1}(c+d x) \left (-\left ((c+d x) \coth ^{-1}(c+d x)\right )+\coth ^{-1}(c+d x)+2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )-\text {Li}_2\left (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) \left (c \coth ^{-1}(c+d x)-1\right ) \coth ^{-1}(c+d x)+4 c \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) \coth ^{-1}(c+d x)-2 \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-2 c \text {Li}_2\left (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 {9 \coth ^{-1}(c+d x)^2 c^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+3 \coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right ) c^2+\frac {18 \coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right ) c^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-3 \coth ^{-1}(c+d x)^2 \sinh \left (3 \coth ^{-1}(c+d x)\right ) c^2-6 \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 ) c^2-\frac {12 \coth ^{-1}(c+d x)^2 c}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-6 \coth ^{-1}(c+d x) \cosh \left (3 \coth ^{-1}(c+d x)\right ) c-\frac {18 \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right ) c}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+6 \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 ) c+\frac {3 \coth ^{-1}(c+d x)^2}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {4 \coth ^{-1}(c+d x)}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}+\frac {-3 c^2 \coth ^{-1}(c+d x)^2+3 \coth ^{-1}(c+d x)^2+6 c \coth ^{-1}(c+d x)-1}{\sqrt {1-\frac {1}{(c+d x)^2}}}+\coth ^{-1}(c+d x)^2 \cosh \left (3 \coth ^{-1}(c+d x)\right )+\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 {4 \left (3 c^2+1\right ) \text {Li}_2\left (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 )-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 )\right )}{12 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} + {\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname {arcoth}\left (d x + c\right )^{2} + 2 \, {\left (a b f^{2} x^{2} + 2 \, a b e f x + a b e^{2}\right )} \operatorname {arcoth}\left (d x + c\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.08, size = 2694, normalized size = 7.20 \[ \text {Expression too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.62, size = 791, normalized size = 2.11 \[ \frac {1}{3} \, a^{2} f^{2} x^{3} + a^{2} e f x^{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 e f + \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} + a^{2} e^{2} x + \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 (3 \, d^{2} e^{2} - 6 \, c d e f + 3 \, c^{2} f^{2} + f^{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 )} b^{2}}{3 \, d^{3}} - \frac {{\left (5 \, c^{2} f^{2} - 6 \, d e f - 6 \, {\left (d e f - f^{2}\right )} c + f^{2}\right )} b^{2} \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} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} + 3 \, d^{2} e^{2} - 3 \, {\left (d e f - f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} - 2 \, d e f + f^{2}\right )} c + 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} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} - 3 \, d^{2} e^{2} - 3 \, {\left (d e f + f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} + 2 \, d e f + f^{2}\right )} c - 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 (3 \, d^{2} e f - 2 \, c d f^{2}\right )} b^{2} x - {\left (b^{2} d^{3} f^{2} x^{3} + 3 \, b^{2} d^{3} e f x^{2} + 3 \, b^{2} d^{3} e^{2} x + {\left (c^{3} f^{2} - 3 \, d^{2} e^{2} - 3 \, {\left (d e f + f^{2}\right )} c^{2} - 3 \, d e f + 3 \, {\left (d^{2} e^{2} + 2 \, d e f + f^{2}\right )} c - 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} + 2 \, {\left (3 \, d^{2} e f - 2 \, c d f^{2}\right )} b^{2} x - {\left (5 \, c^{2} f^{2} + 6 \, d e f - 6 \, {\left (d e f + f^{2}\right )} c + f^{2}\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{12 \, d^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*f^2*x^3 + a^2*e*f*x^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*e*f + 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 + a^2*e^2*x +
(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e^2/d - 1/3*(3*d^2*e^2 - 6*c*d*e*f + 3*c^2*f^2 + f^
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))*b^2/d^3 - 1/6*(5*c^2*f^2 - 6*
d*e*f - 6*(d*e*f - f^2)*c + 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
*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 + 3*d^2*e^2 - 3*(d*e*f - f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 - 2*d*e*f + f^2
)*c + f^2)*b^2)*log(d*x + c + 1)^2 + (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f^2 - 3*d^2
*e^2 - 3*(d*e*f + f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^2)*log(d*x + c - 1)^2 + 2*(b^2*d
^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x - (b^2*d^3*f^2*x^3 + 3*b^2*d^3*e*f*x^2 + 3*b^2*d^3*e^2*x + (c^3*f
^2 - 3*d^2*e^2 - 3*(d*e*f + f^2)*c^2 - 3*d*e*f + 3*(d^2*e^2 + 2*d*e*f + f^2)*c - f^2)*b^2)*log(d*x + c - 1))*l
og(d*x + c + 1) - 2*(b^2*d^2*f^2*x^2 + 2*(3*d^2*e*f - 2*c*d*f^2)*b^2*x - (5*c^2*f^2 + 6*d*e*f - 6*(d*e*f + f^2
)*c + f^2)*b^2)*log(d*x + c - 1))/d^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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