Integrand size = 20, antiderivative size = 374 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \coth ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f+\left (3+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {\left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 d^3}+\frac {(e+f x)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \text {arctanh}(c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1-(c+d x)^2\right )}{d^3}-\frac {b^2 \left (3 d^2 e^2-6 c d e f+\left (1+3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{3 d^3} \]
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)*arcc oth(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*arctanh(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)*polylo g(2,(-d*x-c-1)/(-d*x-c+1))/d^3
Leaf count is larger than twice the leaf count of optimal. \(1115\) vs. \(2(374)=748\).
Time = 7.06 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.98 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=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 {2 b^2 e f \left (1-(c+d x)^2\right ) \left (\frac {(c+d x) \coth ^{-1}(c+d x)}{d^2}-\frac {c (c+d x) \coth ^{-1}(c+d x)^2}{d^2}+\frac {(c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\right ) \coth ^{-1}(c+d x)^2}{2 d^2}-\frac {\log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )}{d^2}+\frac {2 c \left (\frac {1}{2} \coth ^{-1}(c+d x)^2+\coth ^{-1}(c+d x) \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )}{d^2}\right )}{(c+d x)^2 \left (1-\frac {1}{(c+d x)^2}\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 )-\operatorname {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 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}\right )}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}-\frac {18 c \log \left (\frac {1}{\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 ) \operatorname {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}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )+6 c \log \left (\frac {1}{\sqrt {1-\frac {1}{(c+d x)^2}}}\right ) \sinh \left (3 \coth ^{-1}(c+d x)\right )\right )}{12 d^3} \]
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 - (2*b ^2*e*f*(1 - (c + d*x)^2)*(((c + d*x)*ArcCoth[c + d*x])/d^2 - (c*(c + d*x)* ArcCoth[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 - (c + d*x)^(-2))*ArcCoth[c + d* x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]/d^2 + (2*c*(Ar cCoth[c + d*x]^2/2 + ArcCoth[c + d*x]*Log[1 - E^(-2*ArcCoth[c + d*x])] - P olyLog[2, E^(-2*ArcCoth[c + d*x])]/2))/d^2))/((c + d*x)^2*(1 - (c + d*x)^( -2))) + (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* 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*A rcCoth[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*A rcCoth[c + d*x]] + 3*c^2*ArcCoth[c + d*x]^2*Cosh[3*ArcCoth[c + d*x]] + ...
Time = 0.80 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6662, 27, 6481, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (a+b \coth ^{-1}(c+d x)\right )^2}{d^2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 \left (a+b \coth ^{-1}(c+d x)\right )^2d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 6481 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {2 b \int \left (-\left ((c+d x) \left (a+b \coth ^{-1}(c+d x)\right ) f^3\right )-3 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e+\left (c^2+3\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e+\left (3 c^2+1\right ) f^2\right ) (c+d x)\right ) \left (a+b \coth ^{-1}(c+d x)\right )}{1-(c+d x)^2}\right )d(c+d x)}{3 f}}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \coth ^{-1}(c+d x)\right )^2}{3 f}-\frac {2 b \left (-\frac {f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}+\frac {(d e-c f) \left (\left (c^2+3\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 b}+f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )-\frac {1}{2} f^3 (c+d x)^2 \left (a+b \coth ^{-1}(c+d x)\right )-3 a f^2 (c+d x) (d e-c f)+\frac {1}{2} b f^3 \text {arctanh}(c+d x)+\frac {1}{2} b f \left (\left (3 c^2+1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )-\frac {3}{2} b f^2 (d e-c f) \log \left (1-(c+d x)^2\right )-3 b f^2 (c+d x) (d e-c f) \coth ^{-1}(c+d x)-\frac {1}{2} b f^3 (c+d x)\right )}{3 f}}{d^3}\) |
(((d*e - c*f + f*(c + d*x))^3*(a + b*ArcCoth[c + d*x])^2)/(3*f) - (2*b*(-1 /2*(b*f^3*(c + d*x)) - 3*a*f^2*(d*e - c*f)*(c + d*x) - 3*b*f^2*(d*e - c*f) *(c + d*x)*ArcCoth[c + d*x] - (f^3*(c + d*x)^2*(a + b*ArcCoth[c + d*x]))/2 + ((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)/(2*b) - (f*(3*d^2*e^2 - 6*c*d*e*f + (1 + 3*c^2)*f^2)*(a + b*ArcCot h[c + d*x])^2)/(2*b) + (b*f^3*ArcTanh[c + d*x])/2 + f*(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*b *f^2*(d*e - c*f)*Log[1 - (c + d*x)^2])/2 + (b*f*(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))])/2))/(3*f))/d^3
3.2.9.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCoth[c*x])^p/(e*(q + 1))), x] - Simp[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]
Int[((a_.) + ArcCoth[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[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] && IG tQ[p, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1410\) vs. \(2(360)=720\).
Time = 0.67 (sec) , antiderivative size = 1411, normalized size of antiderivative = 3.77
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1411\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1412\) |
| default | \(\text {Expression too large to display}\) | \(1412\) |
| risch | \(\text {Expression too large to display}\) | \(1687\) |
1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccoth(d*x+c)^2*(d*x+c)^3-1/d^2*f^ 2*arccoth(d*x+c)^2*(d*x+c)^2*c+1/d*f*arccoth(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^ 2*arccoth(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccoth(d*x+c)^2*(d*x+c)*c*e+arccoth( d*x+c)^2*(d*x+c)*e^2-1/3/d^2*f^2*arccoth(d*x+c)^2*c^3+1/d*f*arccoth(d*x+c) ^2*c^2*e-arccoth(d*x+c)^2*c*e^2+1/3*d/f*arccoth(d*x+c)^2*e^3+2/3/d^2/f*(1/ 2*arccoth(d*x+c)*f^3*(d*x+c)^2+1/2*arccoth(d*x+c)*ln(d*x+c-1)*f^3+1/2*arcc oth(d*x+c)*ln(d*x+c+1)*f^3-3/2*arccoth(d*x+c)*ln(d*x+c+1)*c^2*d*e*f^2+3/2* arccoth(d*x+c)*ln(d*x+c+1)*c*d^2*e^2*f-3*arccoth(d*x+c)*ln(d*x+c+1)*c*d*e* f^2+3/2*arccoth(d*x+c)*ln(d*x+c+1)*d^2*e^2*f-3/2*arccoth(d*x+c)*ln(d*x+c+1 )*d*e*f^2+3/2*arccoth(d*x+c)*ln(d*x+c-1)*c^2*d*e*f^2-3/2*arccoth(d*x+c)*ln (d*x+c-1)*c*d^2*e^2*f-3*arccoth(d*x+c)*ln(d*x+c-1)*c*d*e*f^2+3*arccoth(d*x +c)*d*e*f^2*(d*x+c)+3/2*arccoth(d*x+c)*ln(d*x+c-1)*d^2*e^2*f+3/2*arccoth(d *x+c)*ln(d*x+c-1)*d*e*f^2-1/2*arccoth(d*x+c)*ln(d*x+c-1)*c^3*f^3+1/2*arcco th(d*x+c)*ln(d*x+c-1)*d^3*e^3+3/2*arccoth(d*x+c)*ln(d*x+c-1)*c^2*f^3-3/2*a rccoth(d*x+c)*ln(d*x+c-1)*c*f^3+1/2*arccoth(d*x+c)*ln(d*x+c+1)*c^3*f^3-1/2 *arccoth(d*x+c)*ln(d*x+c+1)*d^3*e^3+3/2*arccoth(d*x+c)*ln(d*x+c+1)*c^2*f^3 +3/2*arccoth(d*x+c)*ln(d*x+c+1)*c*f^3-3*arccoth(d*x+c)*c*f^3*(d*x+c)+1/2*f ^2*(f*(d*x+c)+1/2*(-6*c*f+6*d*e+f)*ln(d*x+c-1)-1/2*(6*c*f-6*d*e+f)*ln(d*x+ c+1))+1/2*(c^3*f^3-3*c^2*d*e*f^2+3*c*d^2*e^2*f-d^3*e^3+3*c^2*f^3-6*c*d*e*f ^2+3*d^2*e^2*f+3*c*f^3-3*d*e*f^2+f^3)*(-1/4*ln(d*x+c+1)^2+1/2*(ln(d*x+c...
\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
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)*ar ccoth(d*x + c), x)
\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 791 vs. \(2 (350) = 700\).
Time = 0.42 (sec) , antiderivative size = 791, normalized size of antiderivative = 2.11 \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\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}} \]
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)*l og(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))*log(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
\[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^2 \,d x \]