Integrand size = 18, antiderivative size = 168 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\frac {b f \left (6 d^2 e^2-12 c d e f+\left (1+6 c^2\right ) f^2\right ) x}{4 d^3}+\frac {b f^2 (d e-c f) (c+d x)^2}{2 d^4}+\frac {b f^3 (c+d x)^3}{12 d^4}+\frac {(e+f x)^4 (a+b \text {arctanh}(c+d x))}{4 f}+\frac {b (d e+f-c f)^4 \log (1-c-d x)}{8 d^4 f}-\frac {b (d e-f-c f)^4 \log (1+c+d x)}{8 d^4 f} \] Output:
1/4*b*f*(6*d^2*e^2-12*c*d*e*f+(6*c^2+1)*f^2)*x/d^3+1/2*b*f^2*(-c*f+d*e)*(d *x+c)^2/d^4+1/12*b*f^3*(d*x+c)^3/d^4+1/4*(f*x+e)^4*(a+b*arctanh(d*x+c))/f+ 1/8*b*(-c*f+d*e+f)^4*ln(-d*x-c+1)/d^4/f-1/8*b*(-c*f+d*e-f)^4*ln(d*x+c+1)/d ^4/f
Time = 0.14 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.61 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\frac {6 d \left (4 a d^3 e^3+b f \left (6 d^2 e^2-8 c d e f+\left (1+3 c^2\right ) f^2\right )\right ) x+6 d^2 f \left (6 a d^2 e^2+b f (2 d e-c f)\right ) x^2+2 d^3 f^2 (12 a d e+b f) x^3+6 a d^4 f^3 x^4+6 b d^4 x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right ) \text {arctanh}(c+d x)-3 b (-1+c) \left (4 d^3 e^3-6 (-1+c) d^2 e^2 f+4 (-1+c)^2 d e f^2-(-1+c)^3 f^3\right ) \log (1-c-d x)-3 b (1+c) \left (-4 d^3 e^3+6 (1+c) d^2 e^2 f-4 (1+c)^2 d e f^2+(1+c)^3 f^3\right ) \log (1+c+d x)}{24 d^4} \] Input:
Integrate[(e + f*x)^3*(a + b*ArcTanh[c + d*x]),x]
Output:
(6*d*(4*a*d^3*e^3 + b*f*(6*d^2*e^2 - 8*c*d*e*f + (1 + 3*c^2)*f^2))*x + 6*d ^2*f*(6*a*d^2*e^2 + b*f*(2*d*e - c*f))*x^2 + 2*d^3*f^2*(12*a*d*e + b*f)*x^ 3 + 6*a*d^4*f^3*x^4 + 6*b*d^4*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x^2 + f^3*x^3 )*ArcTanh[c + d*x] - 3*b*(-1 + c)*(4*d^3*e^3 - 6*(-1 + c)*d^2*e^2*f + 4*(- 1 + c)^2*d*e*f^2 - (-1 + c)^3*f^3)*Log[1 - c - d*x] - 3*b*(1 + c)*(-4*d^3* e^3 + 6*(1 + c)*d^2*e^2*f - 4*(1 + c)^2*d*e*f^2 + (1 + c)^3*f^3)*Log[1 + c + d*x])/(24*d^4)
Time = 0.49 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99, 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.
| \(\displaystyle \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx\) |
| \(\Big \downarrow \) 6661 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^3 (a+b \text {arctanh}(c+d x))}{d^3}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x))^3 (a+b \text {arctanh}(c+d x))d(c+d x)}{d^4}\) |
| \(\Big \downarrow \) 6478 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^4 (a+b \text {arctanh}(c+d x))}{4 f}-\frac {b \int \frac {(d e-c f+f (c+d x))^4}{1-(c+d x)^2}d(c+d x)}{4 f}}{d^4}\) |
| \(\Big \downarrow \) 477 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^4 (a+b \text {arctanh}(c+d x))}{4 f}-\frac {b \int \left (-(c+d x)^2 f^4-4 (d e-c f) (c+d x) f^3-\left (6 d^2 e^2-12 c d f e+\left (6 c^2+1\right ) f^2\right ) f^2+\frac {(d e-c f+f)^4}{2 (-c-d x+1)}+\frac {(d e-c f-f)^4}{2 (c+d x+1)}\right )d(c+d x)}{4 f}}{d^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^4 (a+b \text {arctanh}(c+d x))}{4 f}-\frac {b \left (-f^2 (c+d x) \left (\left (6 c^2+1\right ) f^2-12 c d e f+6 d^2 e^2\right )-2 f^3 (c+d x)^2 (d e-c f)-\frac {1}{2} (-c f+d e+f)^4 \log (-c-d x+1)+\frac {1}{2} (-c f+d e-f)^4 \log (c+d x+1)-\frac {1}{3} f^4 (c+d x)^3\right )}{4 f}}{d^4}\) |
Input:
Int[(e + f*x)^3*(a + b*ArcTanh[c + d*x]),x]
Output:
(((d*e - c*f + f*(c + d*x))^4*(a + b*ArcTanh[c + d*x]))/(4*f) - (b*(-(f^2* (6*d^2*e^2 - 12*c*d*e*f + (1 + 6*c^2)*f^2)*(c + d*x)) - 2*f^3*(d*e - c*f)* (c + d*x)^2 - (f^4*(c + d*x)^3)/3 - ((d*e + f - c*f)^4*Log[1 - c - d*x])/2 + ((d*e - f - c*f)^4*Log[1 + c + d*x])/2))/(4*f))/d^4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(598\) vs. \(2(156)=312\).
Time = 0.76 (sec) , antiderivative size = 599, normalized size of antiderivative = 3.57
| method | result | size |
| parallelrisch | \(-\frac {24 a c \,d^{3} e^{3}-18 f \,e^{2} a \,d^{2}-6 b e \,f^{2} d +9 b c \,f^{3}+18 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{2} f^{3}+12 \,\operatorname {arctanh}\left (d x +c \right ) b c \,f^{3}+12 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{3} f^{3}+12 \ln \left (d x +c -1\right ) b \,c^{3} f^{3}-12 \ln \left (d x +c -1\right ) b \,d^{3} e^{3}+12 \ln \left (d x +c -1\right ) b c \,f^{3}-3 x^{4} a \,d^{4} f^{3}-12 x a \,d^{4} e^{3}-3 x b d \,f^{3}-x^{3} b \,d^{3} f^{3}+3 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{4} f^{3}-12 \,\operatorname {arctanh}\left (d x +c \right ) b \,d^{3} e^{3}-3 x^{4} \operatorname {arctanh}\left (d x +c \right ) b \,d^{4} f^{3}-18 x^{2} a \,d^{4} e^{2} f -12 \ln \left (d x +c -1\right ) b d e \,f^{2}-9 x b \,c^{2} d \,f^{3}-18 x b \,d^{3} e^{2} f -12 x^{3} a \,d^{4} e \,f^{2}-12 x \,\operatorname {arctanh}\left (d x +c \right ) b \,d^{4} e^{3}-12 \,\operatorname {arctanh}\left (d x +c \right ) b c \,d^{3} e^{3}+18 \,\operatorname {arctanh}\left (d x +c \right ) b \,d^{2} e^{2} f -12 \,\operatorname {arctanh}\left (d x +c \right ) b d e \,f^{2}+3 x^{2} b c \,d^{2} f^{3}-6 x^{2} b \,d^{3} e \,f^{2}+3 \,\operatorname {arctanh}\left (d x +c \right ) b \,f^{3}+18 a \,c^{2} d^{2} e^{2} f +18 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{2} d^{2} e^{2} f -36 \ln \left (d x +c -1\right ) b \,c^{2} d e \,f^{2}+36 \ln \left (d x +c -1\right ) b c \,d^{2} e^{2} f -12 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{3} d e \,f^{2}-36 \,\operatorname {arctanh}\left (d x +c \right ) b \,c^{2} d e \,f^{2}+36 \,\operatorname {arctanh}\left (d x +c \right ) b c \,d^{2} e^{2} f -36 \,\operatorname {arctanh}\left (d x +c \right ) b c d e \,f^{2}-12 x^{3} \operatorname {arctanh}\left (d x +c \right ) b \,d^{4} e \,f^{2}+24 x b c \,d^{2} e \,f^{2}-18 x^{2} \operatorname {arctanh}\left (d x +c \right ) b \,d^{4} e^{2} f -42 b \,c^{2} d e \,f^{2}+36 b c \,d^{2} e^{2} f +15 b \,c^{3} f^{3}}{12 d^{4}}\) | \(599\) |
| derivativedivides | \(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arctanh}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arctanh}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arctanh}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arctanh}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arctanh}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arctanh}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}+\frac {12 c d e \,f^{3} \left (d x +c \right )-6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 d e \,f^{3} \left (d x +c \right )^{2}+2 c \,f^{4} \left (d x +c \right )^{2}+\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}-\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-f^{4} \left (d x +c \right )-\frac {f^{4} \left (d x +c \right )^{3}}{3}-6 c^{2} f^{4} \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\) | \(694\) |
| default | \(\frac {\frac {a \left (c f -d e -f \left (d x +c \right )\right )^{4}}{4 d^{3} f}-\frac {b \left (-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{4}}{4}+f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3} d e +f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} d^{2} e^{2}}{2}-3 f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{2} d e \left (d x +c \right )-\frac {3 f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{2} \left (d x +c \right )^{2}}{2}+\operatorname {arctanh}\left (d x +c \right ) c \,d^{3} e^{3}+3 f \,\operatorname {arctanh}\left (d x +c \right ) c \,d^{2} e^{2} \left (d x +c \right )+3 f^{2} \operatorname {arctanh}\left (d x +c \right ) c d e \left (d x +c \right )^{2}+f^{3} \operatorname {arctanh}\left (d x +c \right ) c \left (d x +c \right )^{3}-\frac {\operatorname {arctanh}\left (d x +c \right ) d^{4} e^{4}}{4 f}-\operatorname {arctanh}\left (d x +c \right ) d^{3} e^{3} \left (d x +c \right )-\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) d^{2} e^{2} \left (d x +c \right )^{2}}{2}-f^{2} \operatorname {arctanh}\left (d x +c \right ) d e \left (d x +c \right )^{3}-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{4}}{4}+\frac {12 c d e \,f^{3} \left (d x +c \right )-6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 d e \,f^{3} \left (d x +c \right )^{2}+2 c \,f^{4} \left (d x +c \right )^{2}+\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}-\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-f^{4} \left (d x +c \right )-\frac {f^{4} \left (d x +c \right )^{3}}{3}-6 c^{2} f^{4} \left (d x +c \right )}{4 f}\right )}{d^{3}}}{d}\) | \(694\) |
| parts | \(\frac {a \left (f x +e \right )^{4}}{4 f}+\frac {b \left (\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{4}}{4 d^{3}}-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{3} c}{d^{3}}+\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{3} e}{d^{2}}+\frac {3 f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} c^{2}}{2 d^{3}}-\frac {3 f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} c e}{d^{2}}+\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right )^{2} e^{2}}{2 d}-\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c^{3}}{d^{3}}+\frac {3 f^{2} \operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c^{2} e}{d^{2}}-\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) c \,e^{2}}{d}+\operatorname {arctanh}\left (d x +c \right ) \left (d x +c \right ) e^{3}+\frac {f^{3} \operatorname {arctanh}\left (d x +c \right ) c^{4}}{4 d^{3}}-\frac {f^{2} \operatorname {arctanh}\left (d x +c \right ) c^{3} e}{d^{2}}+\frac {3 f \,\operatorname {arctanh}\left (d x +c \right ) c^{2} e^{2}}{2 d}-\operatorname {arctanh}\left (d x +c \right ) c \,e^{3}+\frac {d \,\operatorname {arctanh}\left (d x +c \right ) e^{4}}{4 f}-\frac {12 c d e \,f^{3} \left (d x +c \right )-6 d^{2} e^{2} f^{2} \left (d x +c \right )-2 d e \,f^{3} \left (d x +c \right )^{2}+2 c \,f^{4} \left (d x +c \right )^{2}+\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}+4 c^{3} f^{4}-12 c^{2} d e \,f^{3}+12 c \,d^{2} e^{2} f^{2}-4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}+4 c \,f^{4}-4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c +1\right )}{2}-\frac {\left (c^{4} f^{4}-4 c^{3} d e \,f^{3}+6 c^{2} d^{2} e^{2} f^{2}-4 c \,d^{3} e^{3} f +d^{4} e^{4}-4 c^{3} f^{4}+12 c^{2} d e \,f^{3}-12 c \,d^{2} e^{2} f^{2}+4 d^{3} e^{3} f +6 c^{2} f^{4}-12 c d e \,f^{3}+6 d^{2} e^{2} f^{2}-4 c \,f^{4}+4 d e \,f^{3}+f^{4}\right ) \ln \left (d x +c -1\right )}{2}-f^{4} \left (d x +c \right )-\frac {f^{4} \left (d x +c \right )^{3}}{3}-6 c^{2} f^{4} \left (d x +c \right )}{4 d^{3} f}\right )}{d}\) | \(695\) |
| risch | \(\frac {b \,e^{3} \ln \left (d x +c +1\right )}{2 d}+\frac {b \,e^{3} \ln \left (-d x -c +1\right )}{2 d}-\frac {2 f^{2} b c e x}{d^{2}}+\frac {f^{2} \ln \left (d x +c +1\right ) b \,c^{3} e}{2 d^{3}}-\frac {3 f \ln \left (d x +c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}-\frac {f^{2} \ln \left (-d x -c +1\right ) b \,c^{3} e}{2 d^{3}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,c^{2} e^{2}}{4 d^{2}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (d x +c +1\right ) b c \,e^{2}}{2 d^{2}}+\frac {3 f^{2} \ln \left (-d x -c +1\right ) b \,c^{2} e}{2 d^{3}}-\frac {3 f \ln \left (-d x -c +1\right ) b c \,e^{2}}{2 d^{2}}+\frac {3 f^{2} \ln \left (d x +c +1\right ) b c e}{2 d^{3}}-\frac {3 f^{2} \ln \left (-d x -c +1\right ) b c e}{2 d^{3}}+\frac {f^{2} \ln \left (d x +c +1\right ) b e}{2 d^{3}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b c}{2 d^{4}}+\frac {f^{2} \ln \left (-d x -c +1\right ) b e}{2 d^{3}}+f^{2} a e \,x^{3}+\frac {3 f a \,e^{2} x^{2}}{2}+a \,e^{3} x -\frac {f^{3} b c \,x^{2}}{4 d^{2}}+\frac {f^{2} b e \,x^{2}}{2 d}+\frac {3 f^{3} b \,c^{2} x}{4 d^{3}}+\frac {3 f b \,e^{2} x}{2 d}-\frac {f^{2} b e \,x^{3} \ln \left (-d x -c +1\right )}{2}-\frac {3 f b \,e^{2} x^{2} \ln \left (-d x -c +1\right )}{4}+\frac {\ln \left (d x +c +1\right ) b c \,e^{3}}{2 d}-\frac {\ln \left (-d x -c +1\right ) b c \,e^{3}}{2 d}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{4}}{8 d^{4}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{4}}{8 d^{4}}-\frac {f^{3} \ln \left (d x +c +1\right ) b \,c^{3}}{2 d^{4}}-\frac {f^{3} \ln \left (-d x -c +1\right ) b \,c^{3}}{2 d^{4}}-\frac {3 f^{3} \ln \left (d x +c +1\right ) b \,c^{2}}{4 d^{4}}-\frac {3 f \ln \left (d x +c +1\right ) b \,e^{2}}{4 d^{2}}+\frac {3 f^{3} \ln \left (-d x -c +1\right ) b \,c^{2}}{4 d^{4}}+\frac {3 f \ln \left (-d x -c +1\right ) b \,e^{2}}{4 d^{2}}-\frac {f^{3} \ln \left (d x +c +1\right ) b c}{2 d^{4}}-\frac {f^{3} \ln \left (d x +c +1\right ) b}{8 d^{4}}+\frac {f^{3} \ln \left (-d x -c +1\right ) b}{8 d^{4}}+\frac {f^{3} a \,x^{4}}{4}+\frac {f^{3} b \,x^{3}}{12 d}+\frac {f^{3} b x}{4 d^{3}}-\frac {b \,e^{3} x \ln \left (-d x -c +1\right )}{2}-\frac {f^{3} b \,x^{4} \ln \left (-d x -c +1\right )}{8}-\frac {\ln \left (d x +c +1\right ) b \,e^{4}}{8 f}+\frac {\left (f x +e \right )^{4} b \ln \left (d x +c +1\right )}{8 f}\) | \(780\) |
Input:
int((f*x+e)^3*(a+b*arctanh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/12*(24*a*c*d^3*e^3-18*f*e^2*a*d^2-6*b*e*f^2*d+9*b*c*f^3+18*arctanh(d*x+ c)*b*c^2*f^3+12*arctanh(d*x+c)*b*c*f^3+12*arctanh(d*x+c)*b*c^3*f^3+12*ln(d *x+c-1)*b*c^3*f^3-12*ln(d*x+c-1)*b*d^3*e^3+12*ln(d*x+c-1)*b*c*f^3-3*x^4*a* d^4*f^3-12*x*a*d^4*e^3-3*x*b*d*f^3-x^3*b*d^3*f^3+3*arctanh(d*x+c)*b*c^4*f^ 3-12*arctanh(d*x+c)*b*d^3*e^3-3*x^4*arctanh(d*x+c)*b*d^4*f^3-18*x^2*a*d^4* e^2*f-12*ln(d*x+c-1)*b*d*e*f^2-9*x*b*c^2*d*f^3-18*x*b*d^3*e^2*f-12*x^3*a*d ^4*e*f^2-12*x*arctanh(d*x+c)*b*d^4*e^3-12*arctanh(d*x+c)*b*c*d^3*e^3+18*ar ctanh(d*x+c)*b*d^2*e^2*f-12*arctanh(d*x+c)*b*d*e*f^2+3*x^2*b*c*d^2*f^3-6*x ^2*b*d^3*e*f^2+3*arctanh(d*x+c)*b*f^3+18*a*c^2*d^2*e^2*f+18*arctanh(d*x+c) *b*c^2*d^2*e^2*f-36*ln(d*x+c-1)*b*c^2*d*e*f^2+36*ln(d*x+c-1)*b*c*d^2*e^2*f -12*arctanh(d*x+c)*b*c^3*d*e*f^2-36*arctanh(d*x+c)*b*c^2*d*e*f^2+36*arctan h(d*x+c)*b*c*d^2*e^2*f-36*arctanh(d*x+c)*b*c*d*e*f^2-12*x^3*arctanh(d*x+c) *b*d^4*e*f^2+24*x*b*c*d^2*e*f^2-18*x^2*arctanh(d*x+c)*b*d^4*e^2*f-42*b*c^2 *d*e*f^2+36*b*c*d^2*e^2*f+15*b*c^3*f^3)/d^4
Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (156) = 312\).
Time = 0.10 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.30 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\frac {6 \, a d^{4} f^{3} x^{4} + 2 \, {\left (12 \, a d^{4} e f^{2} + b d^{3} f^{3}\right )} x^{3} + 6 \, {\left (6 \, a d^{4} e^{2} f + 2 \, b d^{3} e f^{2} - b c d^{2} f^{3}\right )} x^{2} + 6 \, {\left (4 \, a d^{4} e^{3} + 6 \, b d^{3} e^{2} f - 8 \, b c d^{2} e f^{2} + {\left (3 \, b c^{2} + b\right )} d f^{3}\right )} x + 3 \, {\left (4 \, {\left (b c + b\right )} d^{3} e^{3} - 6 \, {\left (b c^{2} + 2 \, b c + b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} + 3 \, b c^{2} + 3 \, b c + b\right )} d e f^{2} - {\left (b c^{4} + 4 \, b c^{3} + 6 \, b c^{2} + 4 \, b c + b\right )} f^{3}\right )} \log \left (d x + c + 1\right ) - 3 \, {\left (4 \, {\left (b c - b\right )} d^{3} e^{3} - 6 \, {\left (b c^{2} - 2 \, b c + b\right )} d^{2} e^{2} f + 4 \, {\left (b c^{3} - 3 \, b c^{2} + 3 \, b c - b\right )} d e f^{2} - {\left (b c^{4} - 4 \, b c^{3} + 6 \, b c^{2} - 4 \, b c + b\right )} f^{3}\right )} \log \left (d x + c - 1\right ) + 3 \, {\left (b d^{4} f^{3} x^{4} + 4 \, b d^{4} e f^{2} x^{3} + 6 \, b d^{4} e^{2} f x^{2} + 4 \, b d^{4} e^{3} x\right )} \log \left (-\frac {d x + c + 1}{d x + c - 1}\right )}{24 \, d^{4}} \] Input:
integrate((f*x+e)^3*(a+b*arctanh(d*x+c)),x, algorithm="fricas")
Output:
1/24*(6*a*d^4*f^3*x^4 + 2*(12*a*d^4*e*f^2 + b*d^3*f^3)*x^3 + 6*(6*a*d^4*e^ 2*f + 2*b*d^3*e*f^2 - b*c*d^2*f^3)*x^2 + 6*(4*a*d^4*e^3 + 6*b*d^3*e^2*f - 8*b*c*d^2*e*f^2 + (3*b*c^2 + b)*d*f^3)*x + 3*(4*(b*c + b)*d^3*e^3 - 6*(b*c ^2 + 2*b*c + b)*d^2*e^2*f + 4*(b*c^3 + 3*b*c^2 + 3*b*c + b)*d*e*f^2 - (b*c ^4 + 4*b*c^3 + 6*b*c^2 + 4*b*c + b)*f^3)*log(d*x + c + 1) - 3*(4*(b*c - b) *d^3*e^3 - 6*(b*c^2 - 2*b*c + b)*d^2*e^2*f + 4*(b*c^3 - 3*b*c^2 + 3*b*c - b)*d*e*f^2 - (b*c^4 - 4*b*c^3 + 6*b*c^2 - 4*b*c + b)*f^3)*log(d*x + c - 1) + 3*(b*d^4*f^3*x^4 + 4*b*d^4*e*f^2*x^3 + 6*b*d^4*e^2*f*x^2 + 4*b*d^4*e^3* x)*log(-(d*x + c + 1)/(d*x + c - 1)))/d^4
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (151) = 302\).
Time = 2.02 (sec) , antiderivative size = 644, normalized size of antiderivative = 3.83 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx =\text {Too large to display} \] Input:
integrate((f*x+e)**3*(a+b*atanh(d*x+c)),x)
Output:
Piecewise((a*e**3*x + 3*a*e**2*f*x**2/2 + a*e*f**2*x**3 + a*f**3*x**4/4 - b*c**4*f**3*atanh(c + d*x)/(4*d**4) + b*c**3*e*f**2*atanh(c + d*x)/d**3 - b*c**3*f**3*log(c/d + x + 1/d)/d**4 + b*c**3*f**3*atanh(c + d*x)/d**4 - 3* b*c**2*e**2*f*atanh(c + d*x)/(2*d**2) + 3*b*c**2*e*f**2*log(c/d + x + 1/d) /d**3 - 3*b*c**2*e*f**2*atanh(c + d*x)/d**3 + 3*b*c**2*f**3*x/(4*d**3) - 3 *b*c**2*f**3*atanh(c + d*x)/(2*d**4) + b*c*e**3*atanh(c + d*x)/d - 3*b*c*e **2*f*log(c/d + x + 1/d)/d**2 + 3*b*c*e**2*f*atanh(c + d*x)/d**2 - 2*b*c*e *f**2*x/d**2 - b*c*f**3*x**2/(4*d**2) + 3*b*c*e*f**2*atanh(c + d*x)/d**3 - b*c*f**3*log(c/d + x + 1/d)/d**4 + b*c*f**3*atanh(c + d*x)/d**4 + b*e**3* x*atanh(c + d*x) + 3*b*e**2*f*x**2*atanh(c + d*x)/2 + b*e*f**2*x**3*atanh( c + d*x) + b*f**3*x**4*atanh(c + d*x)/4 + b*e**3*log(c/d + x + 1/d)/d - b* e**3*atanh(c + d*x)/d + 3*b*e**2*f*x/(2*d) + b*e*f**2*x**2/(2*d) + b*f**3* x**3/(12*d) - 3*b*e**2*f*atanh(c + d*x)/(2*d**2) + b*e*f**2*log(c/d + x + 1/d)/d**3 - b*e*f**2*atanh(c + d*x)/d**3 + b*f**3*x/(4*d**3) - b*f**3*atan h(c + d*x)/(4*d**4), Ne(d, 0)), ((a + b*atanh(c))*(e**3*x + 3*e**2*f*x**2/ 2 + e*f**2*x**3 + f**3*x**4/4), True))
Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (156) = 312\).
Time = 0.03 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.98 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\frac {1}{4} \, a f^{3} x^{4} + a e f^{2} x^{3} + \frac {3}{2} \, a e^{2} f x^{2} + \frac {3}{4} \, {\left (2 \, x^{2} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, x}{d^{2}} - \frac {{\left (c^{2} + 2 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{3}} + \frac {{\left (c^{2} - 2 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{3}}\right )}\right )} b e^{2} f + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {d x^{2} - 4 \, c x}{d^{3}} + \frac {{\left (c^{3} + 3 \, c^{2} + 3 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{4}} - \frac {{\left (c^{3} - 3 \, c^{2} + 3 \, c - 1\right )} \log \left (d x + c - 1\right )}{d^{4}}\right )}\right )} b e f^{2} + \frac {1}{24} \, {\left (6 \, x^{4} \operatorname {artanh}\left (d x + c\right ) + d {\left (\frac {2 \, {\left (d^{2} x^{3} - 3 \, c d x^{2} + 3 \, {\left (3 \, c^{2} + 1\right )} x\right )}}{d^{4}} - \frac {3 \, {\left (c^{4} + 4 \, c^{3} + 6 \, c^{2} + 4 \, c + 1\right )} \log \left (d x + c + 1\right )}{d^{5}} + \frac {3 \, {\left (c^{4} - 4 \, c^{3} + 6 \, c^{2} - 4 \, c + 1\right )} \log \left (d x + c - 1\right )}{d^{5}}\right )}\right )} b f^{3} + a e^{3} x + \frac {{\left (2 \, {\left (d x + c\right )} \operatorname {artanh}\left (d x + c\right ) + \log \left (-{\left (d x + c\right )}^{2} + 1\right )\right )} b e^{3}}{2 \, d} \] Input:
integrate((f*x+e)^3*(a+b*arctanh(d*x+c)),x, algorithm="maxima")
Output:
1/4*a*f^3*x^4 + a*e*f^2*x^3 + 3/2*a*e^2*f*x^2 + 3/4*(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)*lo g(d*x + c - 1)/d^3))*b*e^2*f + 1/2*(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*e*f^2 + 1/24*(6*x^4*arctanh(d*x + c) + d *(2*(d^2*x^3 - 3*c*d*x^2 + 3*(3*c^2 + 1)*x)/d^4 - 3*(c^4 + 4*c^3 + 6*c^2 + 4*c + 1)*log(d*x + c + 1)/d^5 + 3*(c^4 - 4*c^3 + 6*c^2 - 4*c + 1)*log(d*x + c - 1)/d^5))*b*f^3 + a*e^3*x + 1/2*(2*(d*x + c)*arctanh(d*x + c) + log( -(d*x + c)^2 + 1))*b*e^3/d
Leaf count of result is larger than twice the leaf count of optimal. 2336 vs. \(2 (156) = 312\).
Time = 0.19 (sec) , antiderivative size = 2336, normalized size of antiderivative = 13.90 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\text {Too large to display} \] Input:
integrate((f*x+e)^3*(a+b*arctanh(d*x+c)),x, algorithm="giac")
Output:
1/6*((c + 1)*d - (c - 1)*d)*(3*((d*x + c + 1)^3*b*d^3*e^3/(d*x + c - 1)^3 - 3*(d*x + c + 1)^2*b*d^3*e^3/(d*x + c - 1)^2 + 3*(d*x + c + 1)*b*d^3*e^3/ (d*x + c - 1) - b*d^3*e^3 - 3*(d*x + c + 1)^3*b*c*d^2*e^2*f/(d*x + c - 1)^ 3 + 9*(d*x + c + 1)^2*b*c*d^2*e^2*f/(d*x + c - 1)^2 - 9*(d*x + c + 1)*b*c* d^2*e^2*f/(d*x + c - 1) + 3*b*c*d^2*e^2*f + 3*(d*x + c + 1)^3*b*c^2*d*e*f^ 2/(d*x + c - 1)^3 - 9*(d*x + c + 1)^2*b*c^2*d*e*f^2/(d*x + c - 1)^2 + 9*(d *x + c + 1)*b*c^2*d*e*f^2/(d*x + c - 1) - 3*b*c^2*d*e*f^2 - (d*x + c + 1)^ 3*b*c^3*f^3/(d*x + c - 1)^3 + 3*(d*x + c + 1)^2*b*c^3*f^3/(d*x + c - 1)^2 - 3*(d*x + c + 1)*b*c^3*f^3/(d*x + c - 1) + b*c^3*f^3 + 3*(d*x + c + 1)^3* b*d^2*e^2*f/(d*x + c - 1)^3 - 6*(d*x + c + 1)^2*b*d^2*e^2*f/(d*x + c - 1)^ 2 + 3*(d*x + c + 1)*b*d^2*e^2*f/(d*x + c - 1) - 6*(d*x + c + 1)^3*b*c*d*e* f^2/(d*x + c - 1)^3 + 12*(d*x + c + 1)^2*b*c*d*e*f^2/(d*x + c - 1)^2 - 6*( d*x + c + 1)*b*c*d*e*f^2/(d*x + c - 1) + 3*(d*x + c + 1)^3*b*c^2*f^3/(d*x + c - 1)^3 - 6*(d*x + c + 1)^2*b*c^2*f^3/(d*x + c - 1)^2 + 3*(d*x + c + 1) *b*c^2*f^3/(d*x + c - 1) + 3*(d*x + c + 1)^3*b*d*e*f^2/(d*x + c - 1)^3 - 3 *(d*x + c + 1)^2*b*d*e*f^2/(d*x + c - 1)^2 + (d*x + c + 1)*b*d*e*f^2/(d*x + c - 1) - b*d*e*f^2 - 3*(d*x + c + 1)^3*b*c*f^3/(d*x + c - 1)^3 + 3*(d*x + c + 1)^2*b*c*f^3/(d*x + c - 1)^2 - (d*x + c + 1)*b*c*f^3/(d*x + c - 1) + b*c*f^3 + (d*x + c + 1)^3*b*f^3/(d*x + c - 1)^3 + (d*x + c + 1)*b*f^3/(d* x + c - 1))*log(-(d*x + c + 1)/(d*x + c - 1))/((d*x + c + 1)^4*d^5/(d*x...
Time = 4.70 (sec) , antiderivative size = 737, normalized size of antiderivative = 4.39 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\ln \left (c+d\,x+1\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )-\ln \left (1-d\,x-c\right )\,\left (\frac {b\,e^3\,x}{2}+\frac {3\,b\,e^2\,f\,x^2}{4}+\frac {b\,e\,f^2\,x^3}{2}+\frac {b\,f^3\,x^4}{8}\right )+x\,\left (\frac {e\,\left (6\,a\,c^2\,f^2+12\,a\,c\,d\,e\,f+2\,a\,d^2\,e^2+3\,b\,d\,e\,f-6\,a\,f^2\right )}{2\,d^2}-\frac {\left (4\,c^2-4\right )\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{4\,d^2}+\frac {2\,c\,\left (\frac {2\,c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{4\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{4\,d^2}\right )}{d}\right )-x^2\,\left (\frac {c\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{4\,d}-\frac {2\,a\,c\,f^3}{d}\right )}{d}-\frac {4\,a\,c^2\,f^3+24\,a\,c\,d\,e\,f^2+12\,a\,d^2\,e^2\,f+4\,b\,d\,e\,f^2-4\,a\,f^3}{8\,d^2}+\frac {a\,f^3\,\left (4\,c^2-4\right )}{8\,d^2}\right )+x^3\,\left (\frac {f^2\,\left (b\,f+8\,a\,c\,f+12\,a\,d\,e\right )}{12\,d}-\frac {2\,a\,c\,f^3}{3\,d}\right )+\frac {a\,f^3\,x^4}{4}+\frac {\ln \left (c+d\,x-1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2-4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f+12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3-12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2-4\,b\,c\,f^3+4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f+4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4}-\frac {\ln \left (c+d\,x+1\right )\,\left (b\,c^4\,f^3-4\,b\,c^3\,d\,e\,f^2+4\,b\,c^3\,f^3+6\,b\,c^2\,d^2\,e^2\,f-12\,b\,c^2\,d\,e\,f^2+6\,b\,c^2\,f^3-4\,b\,c\,d^3\,e^3+12\,b\,c\,d^2\,e^2\,f-12\,b\,c\,d\,e\,f^2+4\,b\,c\,f^3-4\,b\,d^3\,e^3+6\,b\,d^2\,e^2\,f-4\,b\,d\,e\,f^2+b\,f^3\right )}{8\,d^4} \] Input:
int((e + f*x)^3*(a + b*atanh(c + d*x)),x)
Output:
log(c + d*x + 1)*((b*f^3*x^4)/8 + (b*e^3*x)/2 + (3*b*e^2*f*x^2)/4 + (b*e*f ^2*x^3)/2) - log(1 - d*x - c)*((b*f^3*x^4)/8 + (b*e^3*x)/2 + (3*b*e^2*f*x^ 2)/4 + (b*e*f^2*x^3)/2) + x*((e*(6*a*c^2*f^2 - 6*a*f^2 + 2*a*d^2*e^2 + 3*b *d*e*f + 12*a*c*d*e*f))/(2*d^2) - ((4*c^2 - 4)*((f^2*(b*f + 8*a*c*f + 12*a *d*e))/(4*d) - (2*a*c*f^3)/d))/(4*d^2) + (2*c*((2*c*((f^2*(b*f + 8*a*c*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d))/d - (4*a*c^2*f^3 - 4*a*f^3 + 4*b*d*e*f ^2 + 12*a*d^2*e^2*f + 24*a*c*d*e*f^2)/(4*d^2) + (a*f^3*(4*c^2 - 4))/(4*d^2 )))/d) - x^2*((c*((f^2*(b*f + 8*a*c*f + 12*a*d*e))/(4*d) - (2*a*c*f^3)/d)) /d - (4*a*c^2*f^3 - 4*a*f^3 + 4*b*d*e*f^2 + 12*a*d^2*e^2*f + 24*a*c*d*e*f^ 2)/(8*d^2) + (a*f^3*(4*c^2 - 4))/(8*d^2)) + x^3*((f^2*(b*f + 8*a*c*f + 12* a*d*e))/(12*d) - (2*a*c*f^3)/(3*d)) + (a*f^3*x^4)/4 + (log(c + d*x - 1)*(b *f^3 + 6*b*c^2*f^3 - 4*b*c^3*f^3 + 4*b*d^3*e^3 + b*c^4*f^3 - 4*b*c*f^3 + 4 *b*d*e*f^2 - 4*b*c*d^3*e^3 + 6*b*d^2*e^2*f - 12*b*c*d^2*e^2*f + 12*b*c^2*d *e*f^2 - 4*b*c^3*d*e*f^2 + 6*b*c^2*d^2*e^2*f - 12*b*c*d*e*f^2))/(8*d^4) - (log(c + d*x + 1)*(b*f^3 + 6*b*c^2*f^3 + 4*b*c^3*f^3 - 4*b*d^3*e^3 + b*c^4 *f^3 + 4*b*c*f^3 - 4*b*d*e*f^2 - 4*b*c*d^3*e^3 + 6*b*d^2*e^2*f + 12*b*c*d^ 2*e^2*f - 12*b*c^2*d*e*f^2 - 4*b*c^3*d*e*f^2 + 6*b*c^2*d^2*e^2*f - 12*b*c* d*e*f^2))/(8*d^4)
Time = 0.15 (sec) , antiderivative size = 518, normalized size of antiderivative = 3.08 \[ \int (e+f x)^3 (a+b \text {arctanh}(c+d x)) \, dx=\frac {12 \mathit {atanh} \left (d x +c \right ) b \,c^{3} d e \,f^{2}-18 \mathit {atanh} \left (d x +c \right ) b \,c^{2} d^{2} e^{2} f +36 \mathit {atanh} \left (d x +c \right ) b \,c^{2} d e \,f^{2}-36 \mathit {atanh} \left (d x +c \right ) b c \,d^{2} e^{2} f +36 \mathit {atanh} \left (d x +c \right ) b c d e \,f^{2}+18 \mathit {atanh} \left (d x +c \right ) b \,d^{4} e^{2} f \,x^{2}+12 \mathit {atanh} \left (d x +c \right ) b \,d^{4} e \,f^{2} x^{3}+36 \,\mathrm {log}\left (d x +c -1\right ) b \,c^{2} d e \,f^{2}-36 \,\mathrm {log}\left (d x +c -1\right ) b c \,d^{2} e^{2} f -24 b c \,d^{2} e \,f^{2} x +12 \mathit {atanh} \left (d x +c \right ) b c \,d^{3} e^{3}+12 \mathit {atanh} \left (d x +c \right ) b \,d^{4} e^{3} x +3 \mathit {atanh} \left (d x +c \right ) b \,d^{4} f^{3} x^{4}-18 \mathit {atanh} \left (d x +c \right ) b \,d^{2} e^{2} f +12 \mathit {atanh} \left (d x +c \right ) b d e \,f^{2}+12 \,\mathrm {log}\left (d x +c -1\right ) b d e \,f^{2}+18 a \,d^{4} e^{2} f \,x^{2}+12 a \,d^{4} e \,f^{2} x^{3}+9 b \,c^{2} d \,f^{3} x -3 \mathit {atanh} \left (d x +c \right ) b \,f^{3}-3 b c \,d^{2} f^{3} x^{2}+18 b \,d^{3} e^{2} f x +6 b \,d^{3} e \,f^{2} x^{2}+b \,d^{3} f^{3} x^{3}-3 \mathit {atanh} \left (d x +c \right ) b \,c^{4} f^{3}-12 \mathit {atanh} \left (d x +c \right ) b \,c^{3} f^{3}-18 \mathit {atanh} \left (d x +c \right ) b \,c^{2} f^{3}-12 \mathit {atanh} \left (d x +c \right ) b c \,f^{3}+12 \mathit {atanh} \left (d x +c \right ) b \,d^{3} e^{3}-12 \,\mathrm {log}\left (d x +c -1\right ) b \,c^{3} f^{3}-12 \,\mathrm {log}\left (d x +c -1\right ) b c \,f^{3}+12 \,\mathrm {log}\left (d x +c -1\right ) b \,d^{3} e^{3}+12 a \,d^{4} e^{3} x +3 a \,d^{4} f^{3} x^{4}+3 b d \,f^{3} x}{12 d^{4}} \] Input:
int((f*x+e)^3*(a+b*atanh(d*x+c)),x)
Output:
( - 3*atanh(c + d*x)*b*c**4*f**3 + 12*atanh(c + d*x)*b*c**3*d*e*f**2 - 12* atanh(c + d*x)*b*c**3*f**3 - 18*atanh(c + d*x)*b*c**2*d**2*e**2*f + 36*ata nh(c + d*x)*b*c**2*d*e*f**2 - 18*atanh(c + d*x)*b*c**2*f**3 + 12*atanh(c + d*x)*b*c*d**3*e**3 - 36*atanh(c + d*x)*b*c*d**2*e**2*f + 36*atanh(c + d*x )*b*c*d*e*f**2 - 12*atanh(c + d*x)*b*c*f**3 + 12*atanh(c + d*x)*b*d**4*e** 3*x + 18*atanh(c + d*x)*b*d**4*e**2*f*x**2 + 12*atanh(c + d*x)*b*d**4*e*f* *2*x**3 + 3*atanh(c + d*x)*b*d**4*f**3*x**4 + 12*atanh(c + d*x)*b*d**3*e** 3 - 18*atanh(c + d*x)*b*d**2*e**2*f + 12*atanh(c + d*x)*b*d*e*f**2 - 3*ata nh(c + d*x)*b*f**3 - 12*log(c + d*x - 1)*b*c**3*f**3 + 36*log(c + d*x - 1) *b*c**2*d*e*f**2 - 36*log(c + d*x - 1)*b*c*d**2*e**2*f - 12*log(c + d*x - 1)*b*c*f**3 + 12*log(c + d*x - 1)*b*d**3*e**3 + 12*log(c + d*x - 1)*b*d*e* f**2 + 12*a*d**4*e**3*x + 18*a*d**4*e**2*f*x**2 + 12*a*d**4*e*f**2*x**3 + 3*a*d**4*f**3*x**4 + 9*b*c**2*d*f**3*x - 24*b*c*d**2*e*f**2*x - 3*b*c*d**2 *f**3*x**2 + 18*b*d**3*e**2*f*x + 6*b*d**3*e*f**2*x**2 + b*d**3*f**3*x**3 + 3*b*d*f**3*x)/(12*d**4)