\(\int (e+f x) (a+b \coth ^{-1}(c+d x))^2 \, dx\) [110]

Optimal result

Integrand size = 18, antiderivative size = 221 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{d^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6247, 6066, 6022, 266, 6196, 6096, 6132, 6056, 2449, 2352} \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {\left (-\frac {\left (c^2+1\right ) f}{d}+2 c e-\frac {d e^2}{f}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {2 b (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {a b f x}{d}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2} \]

[In]

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

[Out]

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

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 6022

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
, 0] && (EqQ[n, 1] || EqQ[p, 1])

Rule 6056

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 6066

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 6096

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 6132

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 6196

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 6247

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*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {d e-c f}{d}+\frac {f x}{d}\right ) \left (a+b \coth ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (-\frac {f^2 \left (a+b \coth ^{-1}(x)\right )}{d^2}+\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )}{d^2 \left (1-x^2\right )}\right ) \, dx,x,c+d x\right )}{f} \\ & = \frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2+2 f (d e-c f) x\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2 f}+\frac {(b f) \text {Subst}\left (\int \left (a+b \coth ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {b \text {Subst}\left (\int \left (\frac {d^2 e^2 \left (1+\frac {f \left (-2 c d e+f+c^2 f\right )}{d^2 e^2}\right ) \left (a+b \coth ^{-1}(x)\right )}{1-x^2}-\frac {2 f (-d e+c f) x \left (a+b \coth ^{-1}(x)\right )}{1-x^2}\right ) \, dx,x,c+d x\right )}{d^2 f}+\frac {\left (b^2 f\right ) \text {Subst}\left (\int \coth ^{-1}(x) \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {\left (b^2 f\right ) \text {Subst}\left (\int \frac {x}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {x \left (a+b \coth ^{-1}(x)\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2}-\frac {\left (b \left (d^2 e^2-2 c d e f+\left (1+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 )}{d^2 f} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {(2 b (d e-c f)) \text {Subst}\left (\int \frac {a+b \coth ^{-1}(x)}{1-x} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}+\frac {\left (2 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {\left (2 b^2 (d e-c f)\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-c-d x}\right )}{d^2} \\ & = \frac {a b f x}{d}+\frac {b^2 f (c+d x) \coth ^{-1}(c+d x)}{d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 f}-\frac {2 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}+\frac {b^2 f \log \left (1-(c+d x)^2\right )}{2 d^2}-\frac {b^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.33 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {2 a^2 c d e+2 a b c f-a^2 c^2 f+2 a^2 d^2 e x+2 a b d f x+a^2 d^2 f x^2+b^2 (-1+c+d x) (2 d e+f-c f+d f x) \coth ^{-1}(c+d x)^2+2 b \coth ^{-1}(c+d x) \left (-((c+d x) (-b f+a c f-a d (2 e+f x)))-2 b (d e-c f) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+a b f \log (1-c-d x)-a b f \log (1+c+d x)-4 a b d e \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )-2 b^2 f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+4 a b c f \log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )+2 b^2 (d e-c f) \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )}{2 d^2} \]

[In]

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

[Out]

(2*a^2*c*d*e + 2*a*b*c*f - a^2*c^2*f + 2*a^2*d^2*e*x + 2*a*b*d*f*x + a^2*d^2*f*x^2 + b^2*(-1 + c + d*x)*(2*d*e
+ f - c*f + d*f*x)*ArcCoth[c + d*x]^2 + 2*b*ArcCoth[c + d*x]*(-((c + d*x)*(-(b*f) + a*c*f - a*d*(2*e + f*x)))
- 2*b*(d*e - c*f)*Log[1 - E^(-2*ArcCoth[c + d*x])]) + a*b*f*Log[1 - c - d*x] - a*b*f*Log[1 + c + d*x] - 4*a*b
*d*e*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] - 2*b^2*f*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] + 4*a*b
*c*f*Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])] + 2*b^2*(d*e - c*f)*PolyLog[2, E^(-2*ArcCoth[c + d*x])])/(2*d
^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(449\) vs. \(2(217)=434\).

Time = 0.36 (sec) , antiderivative size = 450, normalized size of antiderivative = 2.04

method	result	size
parts	\(a^{2} \left (\frac {1}{2} f \,x^{2}+e x \right )+\frac {b^{2} \left (\frac {\operatorname {arccoth}\left (d x +c \right )^{2} \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} c f \left (d x +c \right )}{d}+\operatorname {arccoth}\left (d x +c \right )^{2} e \left (d x +c \right )+\frac {\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}-\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\ln \left (d x +c -1\right ) f}{2}+\frac {\ln \left (d x +c +1\right ) f}{2}+\frac {\left (-2 c f +2 d e +f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (-2 c f +2 d e -f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}}{d}\right )}{d}+\frac {2 a b \left (\frac {\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right )^{2} f}{2 d}-\frac {\operatorname {arccoth}\left (d x +c \right ) c f \left (d x +c \right )}{d}+\operatorname {arccoth}\left (d x +c \right ) e \left (d x +c \right )+\frac {f \left (d x +c \right )+\frac {\left (-2 c f +2 d e +f \right ) \ln \left (d x +c -1\right )}{2}-\frac {\left (2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )}{2}}{2 d}\right )}{d}\)	\(450\)
derivativedivides	\(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\)	\(462\)
default	\(\frac {-\frac {a^{2} \left (f c \left (d x +c \right )-e d \left (d x +c \right )-\frac {f \left (d x +c \right )^{2}}{2}\right )}{d}-\frac {b^{2} \left (\operatorname {arccoth}\left (d x +c \right )^{2} f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right )^{2} e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right )^{2} f \left (d x +c \right )^{2}}{2}-\operatorname {arccoth}\left (d x +c \right ) \left (d x +c \right ) f +\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) d e -\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c -1\right ) f}{2}+\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) c f -\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) d e +\frac {\operatorname {arccoth}\left (d x +c \right ) \ln \left (d x +c +1\right ) f}{2}+\frac {\left (2 c f -2 d e -f \right ) \left (\frac {\ln \left (d x +c -1\right )^{2}}{4}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}+\frac {\left (2 c f -2 d e +f \right ) \left (-\frac {\ln \left (d x +c +1\right )^{2}}{4}+\frac {\left (\ln \left (d x +c +1\right )-\ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {d x}{2}-\frac {c}{2}+\frac {1}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right )}{2}\right )}{2}-\frac {\ln \left (d x +c -1\right ) f}{2}-\frac {\ln \left (d x +c +1\right ) f}{2}\right )}{d}-\frac {2 a b \left (\operatorname {arccoth}\left (d x +c \right ) f c \left (d x +c \right )-\operatorname {arccoth}\left (d x +c \right ) e d \left (d x +c \right )-\frac {\operatorname {arccoth}\left (d x +c \right ) f \left (d x +c \right )^{2}}{2}-\frac {f \left (d x +c \right )}{2}+\frac {\left (2 c f -2 d e -f \right ) \ln \left (d x +c -1\right )}{4}-\frac {\left (-2 c f +2 d e -f \right ) \ln \left (d x +c +1\right )}{4}\right )}{d}}{d}\)	\(462\)
risch	\(\frac {a^{2} c e}{d}-\frac {b a f}{d^{2}}+\frac {a^{2} c f}{d^{2}}-\frac {a^{2} f \,c^{2}}{2 d^{2}}+\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) c f}{d^{2}}-\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c -1\right ) e}{d}+\frac {b \ln \left (d x +c +1\right ) a e}{d}+\frac {b^{2} \ln \left (d x +c +1\right ) c f}{2 d^{2}}-\frac {b \ln \left (d x +c +1\right ) a f}{2 d^{2}}-\frac {b^{2} f \ln \left (d x +c -1\right )^{2} c^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (d x +c -1\right )^{2} c}{4 d^{2}}-\frac {b^{2} f \ln \left (d x +c -1\right ) x}{2 d}-\frac {b^{2} f \ln \left (d x +c -1\right ) c}{2 d^{2}}+\frac {\ln \left (d x +c -1\right ) a b e}{d}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} c e}{4 d}+\frac {b a f \ln \left (d x +c -1\right )}{2 d^{2}}-\ln \left (d x +c -1\right ) a b e x -\frac {b a f \ln \left (d x +c -1\right ) x^{2}}{2}-\frac {b^{2} \left (-d^{2} f \,x^{2}-2 d^{2} e x +c^{2} f -2 c d e +2 c f -2 d e +f \right ) \ln \left (d x +c +1\right )^{2}}{8 d^{2}}-\frac {a^{2} e}{d}-\frac {a^{2} f}{2 d^{2}}-\frac {b a f \ln \left (d x +c -1\right ) c}{d^{2}}-\frac {\ln \left (d x +c -1\right ) a b c e}{d}+\frac {b a f \ln \left (d x +c -1\right ) c^{2}}{2 d^{2}}-\frac {b \ln \left (d x +c +1\right ) a \,c^{2} f}{2 d^{2}}+\frac {b \ln \left (d x +c +1\right ) a c e}{d}-\frac {b \ln \left (d x +c +1\right ) a c f}{d^{2}}+\frac {b a c f}{d^{2}}+\frac {b^{2} \ln \left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) \ln \left (d x +c -1\right ) c f}{d^{2}}+\left (-\frac {b^{2} x \left (f x +2 e \right ) \ln \left (d x +c -1\right )}{4}-\frac {b \left (-2 a \,d^{2} f \,x^{2}-4 a \,d^{2} e x -\ln \left (d x +c -1\right ) b \,c^{2} f +2 \ln \left (d x +c -1\right ) b c d e +2 \ln \left (d x +c -1\right ) b c f -2 \ln \left (d x +c -1\right ) b d e -2 b d f x -\ln \left (d x +c -1\right ) b f \right )}{4 d^{2}}\right ) \ln \left (d x +c +1\right )-\frac {b^{2} \operatorname {dilog}\left (\frac {d x}{2}+\frac {c}{2}+\frac {1}{2}\right ) e}{d}+\frac {b^{2} \ln \left (d x +c +1\right ) f}{2 d^{2}}+\frac {a^{2} f \,x^{2}}{2}+a^{2} e x +\frac {b^{2} f \ln \left (d x +c -1\right )^{2} x^{2}}{8}+\frac {\ln \left (d x +c -1\right )^{2} b^{2} e x}{4}-\frac {\ln \left (d x +c -1\right )^{2} b^{2} e}{4 d}-\frac {b^{2} f \ln \left (d x +c -1\right )^{2}}{8 d^{2}}+\frac {b^{2} f \ln \left (d x +c -1\right )}{2 d^{2}}+\frac {a b f x}{d}\)	\(751\)

[In]

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

[Out]

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

Fricas [F]

\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F]

\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )\, dx \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.81 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {1}{2} \, a^{2} f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcoth}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} a b f + a^{2} e x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {arcoth}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} a b e}{d} - \frac {{\left (d e - c f\right )} {\left (\log \left (d x + c - 1\right ) \log \left (\frac {1}{2} \, d x + \frac {1}{2} \, c + \frac {1}{2}\right ) + {\rm Li}_2\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c + \frac {1}{2}\right )\right )} b^{2}}{d^{2}} + \frac {{\left (c f + f\right )} b^{2} \log \left (d x + c + 1\right )}{2 \, d^{2}} + \frac {{\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e - f\right )} c - 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c + 1\right )^{2} + {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e + f\right )} c + 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )^{2} + 2 \, {\left (2 \, b^{2} d f x - {\left (b^{2} d^{2} f x^{2} + 2 \, b^{2} d^{2} e x - {\left (c^{2} f - 2 \, {\left (d e + f\right )} c + 2 \, d e + f\right )} b^{2}\right )} \log \left (d x + c - 1\right )\right )} \log \left (d x + c + 1\right ) - 4 \, {\left (b^{2} d f x + {\left (c f - f\right )} b^{2}\right )} \log \left (d x + c - 1\right )}{8 \, d^{2}} \]

[In]

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

[Out]

1/2*a^2*f*x^2 + 1/2*(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 + a^2*e*x + (2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a*b*e/d -
(d*e - c*f)*(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^2 + 1/2*(c*f +
f)*b^2*log(d*x + c + 1)/d^2 + 1/8*((b^2*d^2*f*x^2 + 2*b^2*d^2*e*x - (c^2*f - 2*(d*e - f)*c - 2*d*e + f)*b^2)*
log(d*x + c + 1)^2 + (b^2*d^2*f*x^2 + 2*b^2*d^2*e*x - (c^2*f - 2*(d*e + f)*c + 2*d*e + f)*b^2)*log(d*x + c - 1
)^2 + 2*(2*b^2*d*f*x - (b^2*d^2*f*x^2 + 2*b^2*d^2*e*x - (c^2*f - 2*(d*e + f)*c + 2*d*e + f)*b^2)*log(d*x + c -
1))*log(d*x + c + 1) - 4*(b^2*d*f*x + (c*f - f)*b^2)*log(d*x + c - 1))/d^2

Giac [F]

\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]

[In]

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

[Out]

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