\(\int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx\) [32]

Optimal result

Integrand size = 18, antiderivative size = 120 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\frac {b f (d e-c f) x}{d^2}+\frac {b f^2 (c+d x)^2}{6 d^3}+\frac {(e+f x)^3 (a+b \text {arctanh}(c+d x))}{3 f}+\frac {b (d e+f-c f)^3 \log (1-c-d x)}{6 d^3 f}-\frac {b (d e-(1+c) f)^3 \log (1+c+d x)}{6 d^3 f} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.45 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\frac {2 d \left (3 a d^2 e^2+b f (3 d e-2 c f)\right ) x+d^2 f (6 a d e+b f) x^2+2 a d^3 f^2 x^3+2 b d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \text {arctanh}(c+d x)-b (-1+c) \left (3 d^2 e^2-3 (-1+c) d e f+(-1+c)^2 f^2\right ) \log (1-c-d x)+b (1+c) \left (3 d^2 e^2-3 (1+c) d e f+(1+c)^2 f^2\right ) \log (1+c+d x)}{6 d^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.41 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {6661, 27, 6478, 477, 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.

\(\displaystyle \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx\)

\(\Big \downarrow \) 6661

\(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 (a+b \text {arctanh}(c+d x))}{d^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 (a+b \text {arctanh}(c+d x))d(c+d x)}{d^3}\)

\(\Big \downarrow \) 6478

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \text {arctanh}(c+d x))}{3 f}-\frac {b \int \frac {(d e-c f+f (c+d x))^3}{1-(c+d x)^2}d(c+d x)}{3 f}}{d^3}\)

\(\Big \downarrow \) 477

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \text {arctanh}(c+d x))}{3 f}-\frac {b \int \left (-\left ((c+d x) f^3\right )-3 (d e-c f) f^2+\frac {(d e-c f+f)^3}{2 (-c-d x+1)}+\frac {(d e-(c+1) f)^3}{2 (c+d x+1)}\right )d(c+d x)}{3 f}}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 (a+b \text {arctanh}(c+d x))}{3 f}-\frac {b \left (-3 f^2 (c+d x) (d e-c f)-\frac {1}{2} (-c f+d e+f)^3 \log (-c-d x+1)+\frac {1}{2} (d e-(c+1) f)^3 \log (c+d x+1)-\frac {1}{2} f^3 (c+d x)^2\right )}{3 f}}{d^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 477

Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ 
a^p   Int[ExpandIntegrand[(c + d*x)^n*(1 - Rt[-b/a, 2]*x)^p*(1 + Rt[-b/a, 2 
]*x)^p, x], x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[p, 0] && IntegerQ[n] & 
& NiceSqrtQ[-b/a] &&  !FractionalPowerFactorQ[Rt[-b/a, 2]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6478

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol 
] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])/(e*(q + 1))), x] - Simp[b 
*(c/(e*(q + 1)))   Int[(d + e*x)^(q + 1)/(1 - c^2*x^2), x], x] /; FreeQ[{a, 
 b, c, d, e, q}, x] && NeQ[q, -1]
 

rule 6661

Int[((a_.) + ArcTanh[(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* 
ArcTanh[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IG 
tQ[p, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(350\) vs. \(2(112)=224\).

Time = 0.53 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.92

method	result	size
parallelrisch	\(\frac {b \,f^{2}-12 a c \,d^{2} e^{2}+6 a e f d +6 \,\operatorname {arctanh}\left (d x +c \right ) b c \,d^{2} e^{2}-6 \,\operatorname {arctanh}\left (d x +c \right ) b d e f -4 x b c d \,f^{2}+6 x b \,d^{2} e f +6 x^{2} a \,d^{3} e f +2 x^{3} \operatorname {arctanh}\left (d x +c \right ) b \,d^{3} f^{2}+6 x \,\operatorname {arctanh}\left (d x +c \right ) b \,d^{3} e^{2}+2 \,\operatorname {arctanh}\left (d x +c \right ) b \,f^{2}+2 \ln \left (d x +c -1\right ) b \,f^{2}-6 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{2} d e f -12 \ln \left (d x +c -1\right ) b c d e f +6 x^{2} \operatorname {arctanh}\left (d x +c \right ) b \,d^{3} e f -12 \,\operatorname {arctanh}\left (d x +c \right ) b c d e f +x^{2} b \,d^{2} f^{2}+6 \ln \left (d x +c -1\right ) b \,c^{2} f^{2}+6 \ln \left (d x +c -1\right ) b \,d^{2} e^{2}+2 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{3} f^{2}+6 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{2} f^{2}+6 \,\operatorname {arctanh}\left (d x +c \right ) b \,d^{2} e^{2}+6 \,\operatorname {arctanh}\left (d x +c \right ) b c \,f^{2}+6 x a \,d^{3} e^{2}+2 x^{3} a \,d^{3} f^{2}-6 a \,c^{2} e f d +7 b \,c^{2} f^{2}-12 b c d e f}{6 d^{3}}\)	\(351\)
parts	\(\frac {a \left (f x +e \right )^{3}}{3 f}+\frac {b \left (\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{3}}{3 d^{2}}-\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} c}{d^{2}}+\frac {f \,\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} e}{d}+\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c^{2}}{d^{2}}-\frac {2 f \,\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c e}{d}+\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) e^{2}-\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3}}{3 d^{2}}+\frac {f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} e}{d}-\operatorname {arctanh}\left (d x +c \right ) c \,e^{2}+\frac {d \,\operatorname {arctanh}\left (d x +c \right ) e^{3}}{3 f}-\frac {-\frac {f^{3} \left (d x +c \right )^{2}}{2}+3 f^{3} c \left (d x +c \right )-3 f^{2} d e \left (d x +c \right )-\frac {\left (-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,e^{2} f \,d^{2}+d^{3} e^{3}+3 c^{2} f^{3}-6 c e \,f^{2} d +3 f \,e^{2} d^{2}-3 c \,f^{3}+3 e \,f^{2} d +f^{3}\right ) \ln \left (d x +c -1\right )}{2}+\frac {\left (-c^{3} f^{3}+3 c^{2} d e \,f^{2}-3 c \,e^{2} f \,d^{2}+d^{3} e^{3}-3 c^{2} f^{3}+6 c e \,f^{2} d -3 f \,e^{2} d^{2}-3 c \,f^{3}+3 e \,f^{2} d -f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 d^{2} f}\right )}{d}\)	\(415\)
derivativedivides	\(\frac {-\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{3}}{3 d^{2} f}+\frac {b \left (-\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3}}{3}+f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} d e +f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{2} \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) c \,d^{2} e^{2}-2 f \,\operatorname {arctanh}\left (d x +c \right ) c d e \left (d x +c \right )-f^{2} \operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )^{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) d^{3} e^{3}}{3 f}+\operatorname {arctanh}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )+f \,\operatorname {arctanh}\left (d x +c \right ) d e \left (d x +c \right )^{2}+\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{3}}{3}+\frac {-3 f^{3} c \left (d x +c \right )+3 f^{2} d e \left (d x +c \right )+\frac {f^{3} \left (d x +c \right )^{2}}{2}-\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,e^{2} f \,d^{2}-d^{3} e^{3}-3 c^{2} f^{3}+6 c e \,f^{2} d -3 f \,e^{2} d^{2}+3 c \,f^{3}-3 e \,f^{2} d -f^{3}\right ) \ln \left (d x +c -1\right )}{2}+\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,e^{2} f \,d^{2}-d^{3} e^{3}+3 c^{2} f^{3}-6 c e \,f^{2} d +3 f \,e^{2} d^{2}+3 c \,f^{3}-3 e \,f^{2} d +f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 f}\right )}{d^{2}}}{d}\)	\(420\)
default	\(\frac {-\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{3}}{3 d^{2} f}+\frac {b \left (-\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3}}{3}+f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} d e +f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{2} \left (d x +c \right )-\operatorname {arctanh}\left (d x +c \right ) c \,d^{2} e^{2}-2 f \,\operatorname {arctanh}\left (d x +c \right ) c d e \left (d x +c \right )-f^{2} \operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )^{2}+\frac {\operatorname {arctanh}\left (d x +c \right ) d^{3} e^{3}}{3 f}+\operatorname {arctanh}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )+f \,\operatorname {arctanh}\left (d x +c \right ) d e \left (d x +c \right )^{2}+\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{3}}{3}+\frac {-3 f^{3} c \left (d x +c \right )+3 f^{2} d e \left (d x +c \right )+\frac {f^{3} \left (d x +c \right )^{2}}{2}-\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,e^{2} f \,d^{2}-d^{3} e^{3}-3 c^{2} f^{3}+6 c e \,f^{2} d -3 f \,e^{2} d^{2}+3 c \,f^{3}-3 e \,f^{2} d -f^{3}\right ) \ln \left (d x +c -1\right )}{2}+\frac {\left (c^{3} f^{3}-3 c^{2} d e \,f^{2}+3 c \,e^{2} f \,d^{2}-d^{3} e^{3}+3 c^{2} f^{3}-6 c e \,f^{2} d +3 f \,e^{2} d^{2}+3 c \,f^{3}-3 e \,f^{2} d +f^{3}\right ) \ln \left (d x +c +1\right )}{2}}{3 f}\right )}{d^{2}}}{d}\)	\(420\)
risch	\(-\frac {f \ln \left (-d x -c -1\right ) b \,c^{2} e}{2 d^{2}}+\frac {f \ln \left (d x +c -1\right ) b \,c^{2} e}{2 d^{2}}-\frac {f \ln \left (-d x -c -1\right ) b c e}{d^{2}}-\frac {f \ln \left (d x +c -1\right ) b c e}{d^{2}}+f a e \,x^{2}+a \,e^{2} x -\frac {2 f^{2} b c x}{3 d^{2}}+\frac {f b e x}{d}-\frac {f b e \,x^{2} \ln \left (-d x -c +1\right )}{2}+\frac {\ln \left (-d x -c -1\right ) b c \,e^{2}}{2 d}-\frac {\ln \left (d x +c -1\right ) b c \,e^{2}}{2 d}+\frac {f^{2} \ln \left (-d x -c -1\right ) b \,c^{3}}{6 d^{3}}-\frac {f^{2} \ln \left (d x +c -1\right ) b \,c^{3}}{6 d^{3}}+\frac {f^{2} \ln \left (-d x -c -1\right ) b \,c^{2}}{2 d^{3}}+\frac {f^{2} \ln \left (d x +c -1\right ) b \,c^{2}}{2 d^{3}}+\frac {f^{2} \ln \left (-d x -c -1\right ) b c}{2 d^{3}}-\frac {f \ln \left (-d x -c -1\right ) b e}{2 d^{2}}-\frac {f^{2} \ln \left (d x +c -1\right ) b c}{2 d^{3}}+\frac {f \ln \left (d x +c -1\right ) b e}{2 d^{2}}+\frac {f^{2} a \,x^{3}}{3}+\frac {f^{2} b \,x^{2}}{6 d}-\frac {b \,e^{2} x \ln \left (-d x -c +1\right )}{2}-\frac {f^{2} b \,x^{3} \ln \left (-d x -c +1\right )}{6}-\frac {\ln \left (-d x -c -1\right ) b \,e^{3}}{6 f}-\frac {b \,e^{3} \ln \left (-d x -c +1\right )}{6 f}+\frac {\ln \left (d x +c -1\right ) b \,e^{3}}{6 f}+\frac {\ln \left (-d x -c -1\right ) b \,e^{2}}{2 d}+\frac {\ln \left (d x +c -1\right ) b \,e^{2}}{2 d}+\frac {f^{2} \ln \left (-d x -c -1\right ) b}{6 d^{3}}+\frac {f^{2} \ln \left (d x +c -1\right ) b}{6 d^{3}}+\frac {\left (f x +e \right )^{3} b \ln \left (d x +c +1\right )}{6 f}\)	\(516\)

Input:

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

Output:

1/6*(b*f^2-12*a*c*d^2*e^2+6*a*e*f*d+6*arctanh(d*x+c)*b*c*d^2*e^2-6*arctanh 
(d*x+c)*b*d*e*f-4*x*b*c*d*f^2+6*x*b*d^2*e*f+6*x^2*a*d^3*e*f+2*x^3*arctanh( 
d*x+c)*b*d^3*f^2+6*x*arctanh(d*x+c)*b*d^3*e^2+2*arctanh(d*x+c)*b*f^2+2*ln( 
d*x+c-1)*b*f^2-6*arctanh(d*x+c)*b*c^2*d*e*f-12*ln(d*x+c-1)*b*c*d*e*f+6*x^2 
*arctanh(d*x+c)*b*d^3*e*f-12*arctanh(d*x+c)*b*c*d*e*f+x^2*b*d^2*f^2+6*ln(d 
*x+c-1)*b*c^2*f^2+6*ln(d*x+c-1)*b*d^2*e^2+2*arctanh(d*x+c)*b*c^3*f^2+6*arc 
tanh(d*x+c)*b*c^2*f^2+6*arctanh(d*x+c)*b*d^2*e^2+6*arctanh(d*x+c)*b*c*f^2+ 
6*x*a*d^3*e^2+2*x^3*a*d^3*f^2-6*a*c^2*e*f*d+7*b*c^2*f^2-12*b*c*d*e*f)/d^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (112) = 224\).

Time = 0.10 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.02 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\frac {2 \, a d^{3} f^{2} x^{3} + {\left (6 \, a d^{3} e f + b d^{2} f^{2}\right )} x^{2} + 2 \, {\left (3 \, a d^{3} e^{2} + 3 \, b d^{2} e f - 2 \, b c d f^{2}\right )} x + {\left (3 \, {\left (b c + b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} + 2 \, b c + b\right )} d e f + {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} f^{2}\right )} \log \left (d x + c + 1\right ) - {\left (3 \, {\left (b c - b\right )} d^{2} e^{2} - 3 \, {\left (b c^{2} - 2 \, b c + b\right )} d e f + {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} f^{2}\right )} \log \left (d x + c - 1\right ) + {\left (b d^{3} f^{2} x^{3} + 3 \, b d^{3} e f x^{2} + 3 \, b d^{3} e^{2} x\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{6 \, d^{3}} \] Input:

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

Output:

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (105) = 210\).

Time = 1.84 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.08 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\begin {cases} a e^{2} x + a e f x^{2} + \frac {a f^{2} x^{3}}{3} + \frac {b c^{3} f^{2} \operatorname {atanh}{\left (c + d x \right )}}{3 d^{3}} - \frac {b c^{2} e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} + \frac {b c^{2} f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{3}} - \frac {b c^{2} f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + \frac {b c e^{2} \operatorname {atanh}{\left (c + d x \right )}}{d} - \frac {2 b c e f \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d^{2}} + \frac {2 b c e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} - \frac {2 b c f^{2} x}{3 d^{2}} + \frac {b c f^{2} \operatorname {atanh}{\left (c + d x \right )}}{d^{3}} + b e^{2} x \operatorname {atanh}{\left (c + d x \right )} + b e f x^{2} \operatorname {atanh}{\left (c + d x \right )} + \frac {b f^{2} x^{3} \operatorname {atanh}{\left (c + d x \right )}}{3} + \frac {b e^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{d} - \frac {b e^{2} \operatorname {atanh}{\left (c + d x \right )}}{d} + \frac {b e f x}{d} + \frac {b f^{2} x^{2}}{6 d} - \frac {b e f \operatorname {atanh}{\left (c + d x \right )}}{d^{2}} + \frac {b f^{2} \log {\left (\frac {c}{d} + x + \frac {1}{d} \right )}}{3 d^{3}} - \frac {b f^{2} \operatorname {atanh}{\left (c + d x \right )}}{3 d^{3}} & \text {for}\: d \neq 0 \\\left (a + b \operatorname {atanh}{\left (c \right )}\right ) \left (e^{2} x + e f x^{2} + \frac {f^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a*e**2*x + a*e*f*x**2 + a*f**2*x**3/3 + b*c**3*f**2*atanh(c + d 
*x)/(3*d**3) - b*c**2*e*f*atanh(c + d*x)/d**2 + b*c**2*f**2*log(c/d + x + 
1/d)/d**3 - b*c**2*f**2*atanh(c + d*x)/d**3 + b*c*e**2*atanh(c + d*x)/d - 
2*b*c*e*f*log(c/d + x + 1/d)/d**2 + 2*b*c*e*f*atanh(c + d*x)/d**2 - 2*b*c* 
f**2*x/(3*d**2) + b*c*f**2*atanh(c + d*x)/d**3 + b*e**2*x*atanh(c + d*x) + 
 b*e*f*x**2*atanh(c + d*x) + b*f**2*x**3*atanh(c + d*x)/3 + b*e**2*log(c/d 
 + x + 1/d)/d - b*e**2*atanh(c + d*x)/d + b*e*f*x/d + b*f**2*x**2/(6*d) - 
b*e*f*atanh(c + d*x)/d**2 + b*f**2*log(c/d + x + 1/d)/(3*d**3) - b*f**2*at 
anh(c + d*x)/(3*d**3), Ne(d, 0)), ((a + b*atanh(c))*(e**2*x + e*f*x**2 + f 
**2*x**3/3), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.72 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\frac {1}{3} \, a f^{2} x^{3} + a e f x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b e f + \frac {1}{6} \, {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b f^{2} + a e^{2} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{2}}{2 \, d} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (112) = 224\).

Time = 0.15 (sec) , antiderivative size = 976, normalized size of antiderivative = 8.13 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\text {Too large to display} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 4.21 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.18 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=x^2\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{6\,d}-\frac {a\,c\,f^2}{d}\right )-\ln \left (1-d\,x-c\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )-x\,\left (\frac {2\,c\,\left (\frac {f\,\left (b\,f+6\,a\,c\,f+6\,a\,d\,e\right )}{3\,d}-\frac {2\,a\,c\,f^2}{d}\right )}{d}-\frac {3\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+3\,a\,d^2\,e^2+3\,b\,d\,e\,f-3\,a\,f^2}{3\,d^2}+\frac {a\,f^2\,\left (3\,c^2-3\right )}{3\,d^2}\right )+\ln \left (c+d\,x+1\right )\,\left (\frac {b\,e^2\,x}{2}+\frac {b\,e\,f\,x^2}{2}+\frac {b\,f^2\,x^3}{6}\right )+\frac {a\,f^2\,x^3}{3}+\frac {\ln \left (c+d\,x-1\right )\,\left (\frac {b\,f^2}{6}+d\,\left (\frac {b\,e\,f\,c^2}{2}-b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}-\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}-\frac {b\,c^3\,f^2}{6}-\frac {b\,c\,f^2}{2}\right )}{d^3}+\frac {\ln \left (c+d\,x+1\right )\,\left (\frac {b\,f^2}{6}-d\,\left (\frac {b\,e\,f\,c^2}{2}+b\,e\,f\,c+\frac {b\,e\,f}{2}\right )+d^2\,\left (\frac {b\,e^2}{2}+\frac {b\,c\,e^2}{2}\right )+\frac {b\,c^2\,f^2}{2}+\frac {b\,c^3\,f^2}{6}+\frac {b\,c\,f^2}{2}\right )}{d^3} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.53 \[ \int (e+f x)^2 (a+b \text {arctanh}(c+d x)) \, dx=\frac {2 \mathit {atanh} \left (d x +c \right ) b \,c^{3} f^{2}-6 \mathit {atanh} \left (d x +c \right ) b \,c^{2} d e f +6 \mathit {atanh} \left (d x +c \right ) b \,c^{2} f^{2}+6 \mathit {atanh} \left (d x +c \right ) b c \,d^{2} e^{2}-12 \mathit {atanh} \left (d x +c \right ) b c d e f +6 \mathit {atanh} \left (d x +c \right ) b c \,f^{2}+6 \mathit {atanh} \left (d x +c \right ) b \,d^{3} e^{2} x +6 \mathit {atanh} \left (d x +c \right ) b \,d^{3} e f \,x^{2}+2 \mathit {atanh} \left (d x +c \right ) b \,d^{3} f^{2} x^{3}+6 \mathit {atanh} \left (d x +c \right ) b \,d^{2} e^{2}-6 \mathit {atanh} \left (d x +c \right ) b d e f +2 \mathit {atanh} \left (d x +c \right ) b \,f^{2}+6 \,\mathrm {log}\left (d x +c -1\right ) b \,c^{2} f^{2}-12 \,\mathrm {log}\left (d x +c -1\right ) b c d e f +6 \,\mathrm {log}\left (d x +c -1\right ) b \,d^{2} e^{2}+2 \,\mathrm {log}\left (d x +c -1\right ) b \,f^{2}+6 a \,d^{3} e^{2} x +6 a \,d^{3} e f \,x^{2}+2 a \,d^{3} f^{2} x^{3}-4 b c d \,f^{2} x +6 b \,d^{2} e f x +b \,d^{2} f^{2} x^{2}}{6 d^{3}} \] Input:

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

Output:

(2*atanh(c + d*x)*b*c**3*f**2 - 6*atanh(c + d*x)*b*c**2*d*e*f + 6*atanh(c 
+ d*x)*b*c**2*f**2 + 6*atanh(c + d*x)*b*c*d**2*e**2 - 12*atanh(c + d*x)*b* 
c*d*e*f + 6*atanh(c + d*x)*b*c*f**2 + 6*atanh(c + d*x)*b*d**3*e**2*x + 6*a 
tanh(c + d*x)*b*d**3*e*f*x**2 + 2*atanh(c + d*x)*b*d**3*f**2*x**3 + 6*atan 
h(c + d*x)*b*d**2*e**2 - 6*atanh(c + d*x)*b*d*e*f + 2*atanh(c + d*x)*b*f** 
2 + 6*log(c + d*x - 1)*b*c**2*f**2 - 12*log(c + d*x - 1)*b*c*d*e*f + 6*log 
(c + d*x - 1)*b*d**2*e**2 + 2*log(c + d*x - 1)*b*f**2 + 6*a*d**3*e**2*x + 
6*a*d**3*e*f*x**2 + 2*a*d**3*f**2*x**3 - 4*b*c*d*f**2*x + 6*b*d**2*e*f*x + 
 b*d**2*f**2*x**2)/(6*d**3)