Integrand size = 20, antiderivative size = 621 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\frac {b d (a+b \text {arctanh}(c+d x))}{(d e+f-c f) (d e-(1+c) f) (e+f x)}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}+\frac {b d^2 (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1-c-d x}\right )}{2 f (d e+f-c f)^2}+\frac {b^2 d^2 \log (1-c-d x)}{2 (d e+f-c f)^2 (d e-(1+c) f)}-\frac {b d^2 (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{2 f (d e-f-c f)^2}+\frac {2 b d^2 (d e-c f) (a+b \text {arctanh}(c+d x)) \log \left (\frac {2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac {b^2 d^2 \log (1+c+d x)}{2 (d e+f-c f) (d e-(1+c) f)^2}+\frac {b^2 d^2 f \log (e+f x)}{(d e+f-c f)^2 (d e-(1+c) f)^2}-\frac {2 b d^2 (d e-c f) (a+b \text {arctanh}(c+d x)) \log \left (\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac {b^2 d^2 \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{4 f (d e+f-c f)^2}+\frac {b^2 d^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{4 f (d e-f-c f)^2}-\frac {b^2 d^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c+d x}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2}+\frac {b^2 d^2 (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2 d (e+f x)}{(d e+f-c f) (1+c+d x)}\right )}{(d e+f-c f)^2 (d e-(1+c) f)^2} \] Output:
b*d*(a+b*arctanh(d*x+c))/(-c*f+d*e+f)/(d*e-(1+c)*f)/(f*x+e)-1/2*(a+b*arcta nh(d*x+c))^2/f/(f*x+e)^2+1/2*b*d^2*(a+b*arctanh(d*x+c))*ln(2/(-d*x-c+1))/f /(-c*f+d*e+f)^2+1/2*b^2*d^2*ln(-d*x-c+1)/(-c*f+d*e+f)^2/(d*e-(1+c)*f)-1/2* b*d^2*(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/f/(-c*f+d*e-f)^2+2*b*d^2*(-c*f+ d*e)*(a+b*arctanh(d*x+c))*ln(2/(d*x+c+1))/(-c*f+d*e+f)^2/(d*e-(1+c)*f)^2-1 /2*b^2*d^2*ln(d*x+c+1)/(-c*f+d*e+f)/(d*e-(1+c)*f)^2+b^2*d^2*f*ln(f*x+e)/(- c*f+d*e+f)^2/(d*e-(1+c)*f)^2-2*b*d^2*(-c*f+d*e)*(a+b*arctanh(d*x+c))*ln(2* d*(f*x+e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e+f)^2/(d*e-(1+c)*f)^2+1/4*b^2*d ^2*polylog(2,-(d*x+c+1)/(-d*x-c+1))/f/(-c*f+d*e+f)^2+1/4*b^2*d^2*polylog(2 ,1-2/(d*x+c+1))/f/(-c*f+d*e-f)^2-b^2*d^2*(-c*f+d*e)*polylog(2,1-2/(d*x+c+1 ))/(-c*f+d*e+f)^2/(d*e-(1+c)*f)^2+b^2*d^2*(-c*f+d*e)*polylog(2,1-2*d*(f*x+ e)/(-c*f+d*e+f)/(d*x+c+1))/(-c*f+d*e+f)^2/(d*e-(1+c)*f)^2
Result contains complex when optimal does not.
Time = 11.30 (sec) , antiderivative size = 1318, normalized size of antiderivative = 2.12 \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*ArcTanh[c + d*x])^2/(e + f*x)^3,x]
Output:
-1/2*a^2/(f*(e + f*x)^2) + (a*b*(d*e - c*f + f*(c + d*x))^3*((f*(2 + ((d*e + f - c*f)*(d*e - (1 + c)*f))/((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2])^2)*ArcTanh[c + d*x])/((d*e + f - c*f)^2*(-( d*e) + f + c*f)^2) - ((c + d*x)*(f - 2*d*e*ArcTanh[c + d*x] + 2*c*f*ArcTan h[c + d*x]))/((d*e - c*f)*(d*e + f - c*f)*(d*e - (1 + c)*f)*Sqrt[1 - (c + d*x)^2]*((d*e - c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d *x)^2])) - (2*(d*e - c*f)*Log[(d*e)/Sqrt[1 - (c + d*x)^2] - (c*f)/Sqrt[1 - (c + d*x)^2] + (f*(c + d*x))/Sqrt[1 - (c + d*x)^2]])/(d^2*e^2 - 2*c*d*e*f + (-1 + c^2)*f^2)^2))/(d*(e + f*x)^3) + (b^2*(d*e - c*f + f*(c + d*x))^3* ((d^2*(-(f^2*ArcTanh[c + d*x]) - I*d^2*e^2*Pi*ArcTanh[c + d*x] + (2*I)*c*d *e*f*Pi*ArcTanh[c + d*x] - I*c^2*f^2*Pi*ArcTanh[c + d*x] + d*e*E^ArcTanh[c - (d*e)/f]*Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]*f*ArcTanh[c + d*x] ^2 - c*E^ArcTanh[c - (d*e)/f]*Sqrt[1 - c^2 - (d^2*e^2)/f^2 + (2*c*d*e)/f]* f^2*ArcTanh[c + d*x]^2 - 2*d^2*e^2*ArcTanh[c + d*x]*Log[1 - E^(2*ArcTanh[c - (d*e)/f] - 2*ArcTanh[c + d*x])] + 4*c*d*e*f*ArcTanh[c + d*x]*Log[1 - E^ (2*ArcTanh[c - (d*e)/f] - 2*ArcTanh[c + d*x])] - 2*c^2*f^2*ArcTanh[c + d*x ]*Log[1 - E^(2*ArcTanh[c - (d*e)/f] - 2*ArcTanh[c + d*x])] + I*d^2*e^2*Pi* Log[1 + E^(2*ArcTanh[c + d*x])] - (2*I)*c*d*e*f*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] + I*c^2*f^2*Pi*Log[1 + E^(2*ArcTanh[c + d*x])] - I*d^2*e^2*Pi*Log [1/Sqrt[1 - (c + d*x)^2]] + (2*I)*c*d*e*f*Pi*Log[1/Sqrt[1 - (c + d*x)^2...
Time = 2.19 (sec) , antiderivative size = 739, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6659, 7292, 6671, 27, 7276, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx\) |
| \(\Big \downarrow \) 6659 |
| \(\displaystyle \frac {b d \int \frac {a+b \text {arctanh}(c+d x)}{(e+f x)^2 \left (1-(c+d x)^2\right )}dx}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \frac {b d \int \frac {a+b \text {arctanh}(c+d x)}{(e+f x)^2 \left (-c^2-2 d x c-d^2 x^2+1\right )}dx}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
| \(\Big \downarrow \) 6671 |
| \(\displaystyle \frac {b \int \frac {d^2 (a+b \text {arctanh}(c+d x))}{\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b d^2 \int \frac {a+b \text {arctanh}(c+d x)}{(d e-c f+f (c+d x))^2 \left (1-(c+d x)^2\right )}d(c+d x)}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {b d^2 \int \left (-\frac {a}{(d e-c f+f (c+d x))^2 \left ((c+d x)^2-1\right )}-\frac {b \text {arctanh}(c+d x)}{(d e-c f+f (c+d x))^2 \left ((c+d x)^2-1\right )}\right )d(c+d x)}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b d^2 \left (\frac {a f}{(-c f+d e+f) (d e-(c+1) f) (f (c+d x)-c f+d e)}-\frac {2 a f (d e-c f) \log (f (c+d x)-c f+d e)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac {a \log (-c-d x+1)}{2 (-c f+d e+f)^2}+\frac {a \log (c+d x+1)}{2 (d e-(c+1) f)^2}+\frac {b f \text {arctanh}(c+d x)}{(-c f+d e+f) (d e-(c+1) f) (f (c+d x)-c f+d e)}+\frac {2 b f \text {arctanh}(c+d x) (d e-c f) \log \left (\frac {2}{c+d x+1}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac {2 b f \text {arctanh}(c+d x) (d e-c f) \log \left (\frac {2 (f (c+d x)-c f+d e)}{(c+d x+1) (-c f+d e+f)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b \text {arctanh}(c+d x) \log \left (\frac {2}{-c-d x+1}\right )}{2 (-c f+d e+f)^2}-\frac {b \text {arctanh}(c+d x) \log \left (\frac {2}{c+d x+1}\right )}{2 (d e-(c+1) f)^2}+\frac {b f^2 \log (f (c+d x)-c f+d e)}{(-c f+d e+f)^2 (d e-(c+1) f)^2}-\frac {b f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2 (d e-c f+f (c+d x))}{(d e-c f+f) (c+d x+1)}\right )}{(-c f+d e+f)^2 (d e-(c+1) f)^2}+\frac {b \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )}{4 (-c f+d e+f)^2}+\frac {b \operatorname {PolyLog}\left (2,1-\frac {2}{c+d x+1}\right )}{4 (d e-(c+1) f)^2}+\frac {b f \log (-c-d x+1)}{2 (-c f+d e+f)^2 (d e-(c+1) f)}-\frac {b f \log (c+d x+1)}{2 (-c f+d e+f) (d e-(c+1) f)^2}\right )}{f}-\frac {(a+b \text {arctanh}(c+d x))^2}{2 f (e+f x)^2}\) |
Input:
Int[(a + b*ArcTanh[c + d*x])^2/(e + f*x)^3,x]
Output:
-1/2*(a + b*ArcTanh[c + d*x])^2/(f*(e + f*x)^2) + (b*d^2*((a*f)/((d*e + f - c*f)*(d*e - (1 + c)*f)*(d*e - c*f + f*(c + d*x))) + (b*f*ArcTanh[c + d*x ])/((d*e + f - c*f)*(d*e - (1 + c)*f)*(d*e - c*f + f*(c + d*x))) + (b*ArcT anh[c + d*x]*Log[2/(1 - c - d*x)])/(2*(d*e + f - c*f)^2) - (a*Log[1 - c - d*x])/(2*(d*e + f - c*f)^2) + (b*f*Log[1 - c - d*x])/(2*(d*e + f - c*f)^2* (d*e - (1 + c)*f)) - (b*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/(2*(d*e - ( 1 + c)*f)^2) + (2*b*f*(d*e - c*f)*ArcTanh[c + d*x]*Log[2/(1 + c + d*x)])/( (d*e + f - c*f)^2*(d*e - (1 + c)*f)^2) + (a*Log[1 + c + d*x])/(2*(d*e - (1 + c)*f)^2) - (b*f*Log[1 + c + d*x])/(2*(d*e + f - c*f)*(d*e - (1 + c)*f)^ 2) + (b*f^2*Log[d*e - c*f + f*(c + d*x)])/((d*e + f - c*f)^2*(d*e - (1 + c )*f)^2) - (2*a*f*(d*e - c*f)*Log[d*e - c*f + f*(c + d*x)])/((d*e + f - c*f )^2*(d*e - (1 + c)*f)^2) - (2*b*f*(d*e - c*f)*ArcTanh[c + d*x]*Log[(2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^ 2*(d*e - (1 + c)*f)^2) + (b*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))])/(4 *(d*e + f - c*f)^2) + (b*PolyLog[2, 1 - 2/(1 + c + d*x)])/(4*(d*e - (1 + c )*f)^2) - (b*f*(d*e - c*f)*PolyLog[2, 1 - 2/(1 + c + d*x)])/((d*e + f - c* f)^2*(d*e - (1 + c)*f)^2) + (b*f*(d*e - c*f)*PolyLog[2, 1 - (2*(d*e - c*f + f*(c + d*x)))/((d*e + f - c*f)*(1 + c + d*x))])/((d*e + f - c*f)^2*(d*e - (1 + c)*f)^2)))/f
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_), x_Symbol] :> Simp[(e + f*x)^(m + 1)*((a + b*ArcTanh[c + d*x])^p/(f*(m + 1))), x] - Simp[b*d*(p/(f*(m + 1))) Int[(e + f*x)^(m + 1)*((a + b*ArcTa nh[c + d*x])^(p - 1)/(1 - (c + d*x)^2)), x], x] /; FreeQ[{a, b, c, d, e, f} , x] && IGtQ[p, 0] && ILtQ[m, -1]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/d Sub st[Int[((d*e - c*f)/d + f*(x/d))^m*(-C/d^2 + (C/d^2)*x^2)^q*(a + b*ArcTanh[ x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, p, q}, x ] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 2.01 (sec) , antiderivative size = 870, normalized size of antiderivative = 1.40
| method | result | size |
| parts | \(-\frac {a^{2}}{2 \left (f x +e \right )^{2} f}+\frac {b^{2} \left (-\frac {d^{3} \operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (f \left (d x +c \right )-c f +d e \right )^{2} f}+\frac {d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right ) f}{\left (c f -d e +f \right ) \left (c f -d e -f \right ) \left (f \left (d x +c \right )-c f +d e \right )}+\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f^{2} \ln \left (f \left (d x +c \right )-c f +d e \right ) c}{\left (c f -d e +f \right )^{2} \left (c f -d e -f \right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f \ln \left (f \left (d x +c \right )-c f +d e \right ) d e}{\left (c f -d e +f \right )^{2} \left (c f -d e -f \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}-\frac {-\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}+\frac {\ln \left (d x +c -1\right )^{2}}{4}}{2 \left (c f -d e -f \right )^{2}}+\frac {-\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}}{2 \left (c f -d e +f \right )^{2}}+\frac {f \left (\frac {f \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c f -d e +f \right ) \left (c f -d e -f \right )}-\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}+\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}\right )}{\left (c f -d e +f \right ) \left (c f -d e -f \right )}+\frac {2 \left (c f -d e \right ) \left (\frac {f \left (\operatorname {dilog}\left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )+\ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )-f}{c f -d e -f}\right )\right )}{2}-\frac {f \left (\operatorname {dilog}\left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )+\ln \left (f \left (d x +c \right )-c f +d e \right ) \ln \left (\frac {f \left (d x +c \right )+f}{c f -d e +f}\right )\right )}{2}\right )}{\left (c f -d e +f \right )^{2} \left (c f -d e -f \right )^{2}}\right )}{f}\right )}{d}+\frac {2 b a \left (-\frac {d^{3} \operatorname {arctanh}\left (d x +c \right )}{2 \left (f \left (d x +c \right )-c f +d e \right )^{2} f}+\frac {d^{3} \left (\frac {f}{\left (c f -d e +f \right ) \left (c f -d e -f \right ) \left (f \left (d x +c \right )-c f +d e \right )}+\frac {2 f \left (c f -d e \right ) \ln \left (f \left (d x +c \right )-c f +d e \right )}{\left (c f -d e +f \right )^{2} \left (c f -d e -f \right )^{2}}-\frac {\ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}\right )}{2 f}\right )}{d}\) | \(870\) |
| derivativedivides | \(\frac {-\frac {a^{2} d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b^{2} d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) f}{\left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f^{2} \ln \left (c f -d e -f \left (d x +c \right )\right ) c}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right ) e d}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {-\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}+\frac {\ln \left (d x +c -1\right )^{2}}{4}}{2 \left (c f -d e -f \right )^{2}}+\frac {-\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}}{2 \left (c f -d e +f \right )^{2}}-\frac {f \left (\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {2 \left (c f -d e \right ) \left (\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}\right )}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}}{f}\right )-2 b a \,d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {-\frac {\ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}-\frac {f}{\left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}}{2 f}\right )}{d}\) | \(896\) |
| default | \(\frac {-\frac {a^{2} d^{3}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-b^{2} d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right )^{2}}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {-\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\operatorname {arctanh}\left (d x +c \right ) \ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}-\frac {\operatorname {arctanh}\left (d x +c \right ) f}{\left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f^{2} \ln \left (c f -d e -f \left (d x +c \right )\right ) c}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {2 \,\operatorname {arctanh}\left (d x +c \right ) f \ln \left (c f -d e -f \left (d x +c \right )\right ) e d}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}-\frac {-\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}+\frac {\ln \left (d x +c -1\right )^{2}}{4}}{2 \left (c f -d e -f \right )^{2}}+\frac {-\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}}{2 \left (c f -d e +f \right )^{2}}-\frac {f \left (\frac {\ln \left (d x +c -1\right )}{2 c f -2 d e -2 f}-\frac {f \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {\ln \left (d x +c +1\right )}{2 c f -2 d e +2 f}\right )}{\left (c f -d e -f \right ) \left (c f -d e +f \right )}-\frac {2 \left (c f -d e \right ) \left (\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )-f}{-c f +d e -f}\right )\right )}{2}-\frac {f \left (\operatorname {dilog}\left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )+\ln \left (c f -d e -f \left (d x +c \right )\right ) \ln \left (\frac {-f \left (d x +c \right )+f}{-c f +d e +f}\right )\right )}{2}\right )}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}}{f}\right )-2 b a \,d^{3} \left (\frac {\operatorname {arctanh}\left (d x +c \right )}{2 \left (c f -d e -f \left (d x +c \right )\right )^{2} f}-\frac {-\frac {\ln \left (d x +c -1\right )}{2 \left (c f -d e -f \right )^{2}}+\frac {\ln \left (d x +c +1\right )}{2 \left (c f -d e +f \right )^{2}}-\frac {f}{\left (c f -d e -f \right ) \left (c f -d e +f \right ) \left (c f -d e -f \left (d x +c \right )\right )}+\frac {2 f \left (c f -d e \right ) \ln \left (c f -d e -f \left (d x +c \right )\right )}{\left (c f -d e -f \right )^{2} \left (c f -d e +f \right )^{2}}}{2 f}\right )}{d}\) | \(896\) |
Input:
int((a+b*arctanh(d*x+c))^2/(f*x+e)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a^2/(f*x+e)^2/f+b^2/d*(-1/2*d^3/(f*(d*x+c)-c*f+d*e)^2/f*arctanh(d*x+c )^2+d^3/f*(arctanh(d*x+c)*f/(c*f-d*e+f)/(c*f-d*e-f)/(f*(d*x+c)-c*f+d*e)+2* arctanh(d*x+c)*f^2/(c*f-d*e+f)^2/(c*f-d*e-f)^2*ln(f*(d*x+c)-c*f+d*e)*c-2*a rctanh(d*x+c)*f/(c*f-d*e+f)^2/(c*f-d*e-f)^2*ln(f*(d*x+c)-c*f+d*e)*d*e-1/2* arctanh(d*x+c)/(c*f-d*e-f)^2*ln(d*x+c-1)+1/2*arctanh(d*x+c)/(c*f-d*e+f)^2* ln(d*x+c+1)-1/2/(c*f-d*e-f)^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)+1/4*ln(d*x+c-1)^2)+1/2/(c*f-d*e+f)^2*(-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))+f/(c*f-d*e+f)/(c*f-d*e-f)*(f/(c*f-d*e+f)/(c*f-d *e-f)*ln(f*(d*x+c)-c*f+d*e)-1/(2*c*f-2*d*e-2*f)*ln(d*x+c-1)+1/(2*c*f-2*d*e +2*f)*ln(d*x+c+1))+2*(c*f-d*e)/(c*f-d*e+f)^2/(c*f-d*e-f)^2*(1/2*f*(dilog(( f*(d*x+c)-f)/(c*f-d*e-f))+ln(f*(d*x+c)-c*f+d*e)*ln((f*(d*x+c)-f)/(c*f-d*e- f)))-1/2*f*(dilog((f*(d*x+c)+f)/(c*f-d*e+f))+ln(f*(d*x+c)-c*f+d*e)*ln((f*( d*x+c)+f)/(c*f-d*e+f))))))+2*b*a/d*(-1/2*d^3/(f*(d*x+c)-c*f+d*e)^2/f*arcta nh(d*x+c)+1/2*d^3/f*(f/(c*f-d*e+f)/(c*f-d*e-f)/(f*(d*x+c)-c*f+d*e)+2*f*(c* f-d*e)/(c*f-d*e+f)^2/(c*f-d*e-f)^2*ln(f*(d*x+c)-c*f+d*e)-1/2/(c*f-d*e-f)^2 *ln(d*x+c-1)+1/2/(c*f-d*e+f)^2*ln(d*x+c+1)))
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))^2/(f*x+e)^3,x, algorithm="fricas")
Output:
integral((b^2*arctanh(d*x + c)^2 + 2*a*b*arctanh(d*x + c) + a^2)/(f^3*x^3 + 3*e*f^2*x^2 + 3*e^2*f*x + e^3), x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c + d x \right )}\right )^{2}}{\left (e + f x\right )^{3}}\, dx \] Input:
integrate((a+b*atanh(d*x+c))**2/(f*x+e)**3,x)
Output:
Integral((a + b*atanh(c + d*x))**2/(e + f*x)**3, x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))^2/(f*x+e)^3,x, algorithm="maxima")
Output:
1/2*(d*(d*log(d*x + c + 1)/(d^2*e^2*f - 2*(c + 1)*d*e*f^2 + (c^2 + 2*c + 1 )*f^3) - d*log(d*x + c - 1)/(d^2*e^2*f - 2*(c - 1)*d*e*f^2 + (c^2 - 2*c + 1)*f^3) - 4*(d^2*e - c*d*f)*log(f*x + e)/(d^4*e^4 - 4*c*d^3*e^3*f + 2*(3*c ^2 - 1)*d^2*e^2*f^2 - 4*(c^3 - c)*d*e*f^3 + (c^4 - 2*c^2 + 1)*f^4) + 2/(d^ 2*e^3 - 2*c*d*e^2*f + (c^2 - 1)*e*f^2 + (d^2*e^2*f - 2*c*d*e*f^2 + (c^2 - 1)*f^3)*x)) - 2*arctanh(d*x + c)/(f^3*x^2 + 2*e*f^2*x + e^2*f))*a*b - 1/8* b^2*(log(-d*x - c + 1)^2/(f^3*x^2 + 2*e*f^2*x + e^2*f) + 2*integrate(-((d* f*x + c*f - f)*log(d*x + c + 1)^2 + (d*f*x + d*e - 2*(d*f*x + c*f - f)*log (d*x + c + 1))*log(-d*x - c + 1))/(d*f^4*x^4 + c*e^3*f - e^3*f + (3*d*e*f^ 3 + c*f^4 - f^4)*x^3 + 3*(d*e^2*f^2 + c*e*f^3 - e*f^3)*x^2 + (d*e^3*f + 3* c*e^2*f^2 - 3*e^2*f^2)*x), x)) - 1/2*a^2/(f^3*x^2 + 2*e*f^2*x + e^2*f)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{2}}{{\left (f x + e\right )}^{3}} \,d x } \] Input:
integrate((a+b*arctanh(d*x+c))^2/(f*x+e)^3,x, algorithm="giac")
Output:
integrate((b*arctanh(d*x + c) + a)^2/(f*x + e)^3, x)
Timed out. \[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^2}{{\left (e+f\,x\right )}^3} \,d x \] Input:
int((a + b*atanh(c + d*x))^2/(e + f*x)^3,x)
Output:
int((a + b*atanh(c + d*x))^2/(e + f*x)^3, x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^2}{(e+f x)^3} \, dx=\text {too large to display} \] Input:
int((a+b*atanh(d*x+c))^2/(f*x+e)^3,x)
Output:
( - 12*atanh(c + d*x)**2*b**2*c**9*e*f**9 + 90*atanh(c + d*x)**2*b**2*c**8 *d*e**2*f**8 - 288*atanh(c + d*x)**2*b**2*c**7*d**2*e**3*f**7 + 32*atanh(c + d*x)**2*b**2*c**7*e*f**9 + 510*atanh(c + d*x)**2*b**2*c**6*d**3*e**4*f* *6 + 12*atanh(c + d*x)**2*b**2*c**6*d**3*e**3*f**7*x + 6*atanh(c + d*x)**2 *b**2*c**6*d**3*e**2*f**8*x**2 - 200*atanh(c + d*x)**2*b**2*c**6*d*e**2*f* *8 - 540*atanh(c + d*x)**2*b**2*c**5*d**4*e**5*f**5 - 72*atanh(c + d*x)**2 *b**2*c**5*d**4*e**4*f**6*x - 36*atanh(c + d*x)**2*b**2*c**5*d**4*e**3*f** 7*x**2 + 528*atanh(c + d*x)**2*b**2*c**5*d**2*e**3*f**7 - 24*atanh(c + d*x )**2*b**2*c**5*e*f**9 + 342*atanh(c + d*x)**2*b**2*c**4*d**5*e**6*f**4 + 1 80*atanh(c + d*x)**2*b**2*c**4*d**5*e**5*f**5*x + 90*atanh(c + d*x)**2*b** 2*c**4*d**5*e**4*f**6*x**2 - 770*atanh(c + d*x)**2*b**2*c**4*d**3*e**4*f** 6 - 20*atanh(c + d*x)**2*b**2*c**4*d**3*e**3*f**7*x - 10*atanh(c + d*x)**2 *b**2*c**4*d**3*e**2*f**8*x**2 + 124*atanh(c + d*x)**2*b**2*c**4*d*e**2*f* *8 - 120*atanh(c + d*x)**2*b**2*c**3*d**6*e**7*f**3 - 240*atanh(c + d*x)** 2*b**2*c**3*d**6*e**6*f**4*x - 120*atanh(c + d*x)**2*b**2*c**3*d**6*e**5*f **5*x**2 + 680*atanh(c + d*x)**2*b**2*c**3*d**4*e**5*f**5 + 80*atanh(c + d *x)**2*b**2*c**3*d**4*e**4*f**6*x + 40*atanh(c + d*x)**2*b**2*c**3*d**4*e* *3*f**7*x**2 - 256*atanh(c + d*x)**2*b**2*c**3*d**2*e**3*f**7 + 18*atanh(c + d*x)**2*b**2*c**2*d**7*e**8*f**2 + 180*atanh(c + d*x)**2*b**2*c**2*d**7 *e**7*f**3*x + 90*atanh(c + d*x)**2*b**2*c**2*d**7*e**6*f**4*x**2 - 372...