Integrand size = 18, antiderivative size = 326 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {3 b f \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {3 b f (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2}{2 d^2}+\frac {(d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d^2}-\frac {\left (d^2 e^2-2 c d e f+\left (1+c^2\right ) f^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 d^2 f}+\frac {(e+f x)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b^2 f \left (a+b \coth ^{-1}(c+d x)\right ) \log \left (\frac {2}{1-c-d x}\right )}{d^2}-\frac {3 b (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^2 \log \left (\frac {2}{1-c-d x}\right )}{d^2}-\frac {3 b^3 f \operatorname {PolyLog}\left (2,-\frac {1+c+d x}{1-c-d x}\right )}{2 d^2}-\frac {3 b^2 (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c-d x}\right )}{d^2}+\frac {3 b^3 (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{1-c-d x}\right )}{2 d^2} \]
3/2*b*f*(a+b*arccoth(d*x+c))^2/d^2+3/2*b*f*(d*x+c)*(a+b*arccoth(d*x+c))^2/ d^2+(-c*f+d*e)*(a+b*arccoth(d*x+c))^3/d^2-1/2*(d^2*e^2-2*c*d*e*f+(c^2+1)*f ^2)*(a+b*arccoth(d*x+c))^3/d^2/f+1/2*(f*x+e)^2*(a+b*arccoth(d*x+c))^3/f-3* b^2*f*(a+b*arccoth(d*x+c))*ln(2/(-d*x-c+1))/d^2-3*b*(-c*f+d*e)*(a+b*arccot h(d*x+c))^2*ln(2/(-d*x-c+1))/d^2-3/2*b^3*f*polylog(2,(-d*x-c-1)/(-d*x-c+1) )/d^2-3*b^2*(-c*f+d*e)*(a+b*arccoth(d*x+c))*polylog(2,1-2/(-d*x-c+1))/d^2+ 3/2*b^3*(-c*f+d*e)*polylog(3,1-2/(-d*x-c+1))/d^2
Result contains complex when optimal does not.
Time = 3.69 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.84 \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\frac {2 a^2 (2 a d e+3 b f-2 a c f) (c+d x)+2 a^3 f (c+d x)^2-6 a^2 b (c+d x) (c f-d (2 e+f x)) \coth ^{-1}(c+d x)+3 a^2 b (2 d e+f-2 c f) \log (1-c-d x)+3 a^2 b (2 d e-(1+2 c) f) \log (1+c+d x)+12 a b^2 f \left ((c+d x) \coth ^{-1}(c+d x)+\frac {1}{2} \left (-1+(c+d x)^2\right ) \coth ^{-1}(c+d x)^2-\log \left (\frac {1}{(c+d x) \sqrt {1-\frac {1}{(c+d x)^2}}}\right )\right )+12 a b^2 d e \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )-12 a b^2 c f \left (\coth ^{-1}(c+d x) \left ((-1+c+d x) \coth ^{-1}(c+d x)-2 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+2 b^3 f \left (\coth ^{-1}(c+d x) \left (3 (-1+c+d x) \coth ^{-1}(c+d x)+\left (-1+c^2+2 c d x+d^2 x^2\right ) \coth ^{-1}(c+d x)^2-6 \log \left (1-e^{-2 \coth ^{-1}(c+d x)}\right )\right )+3 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c+d x)}\right )\right )+4 b^3 d e \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )-4 b^3 c f \left (-\frac {i \pi ^3}{8}+\coth ^{-1}(c+d x)^3+(c+d x) \coth ^{-1}(c+d x)^3-3 \coth ^{-1}(c+d x)^2 \log \left (1-e^{2 \coth ^{-1}(c+d x)}\right )-3 \coth ^{-1}(c+d x) \operatorname {PolyLog}\left (2,e^{2 \coth ^{-1}(c+d x)}\right )+\frac {3}{2} \operatorname {PolyLog}\left (3,e^{2 \coth ^{-1}(c+d x)}\right )\right )}{4 d^2} \]
(2*a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + 2*a^3*f*(c + d*x)^2 - 6*a^2 *b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcCoth[c + d*x] + 3*a^2*b*(2*d*e + f - 2*c*f)*Log[1 - c - d*x] + 3*a^2*b*(2*d*e - (1 + 2*c)*f)*Log[1 + c + d*x] + 12*a*b^2*f*((c + d*x)*ArcCoth[c + d*x] + ((-1 + (c + d*x)^2)*ArcCoth[c + d*x]^2)/2 - Log[1/((c + d*x)*Sqrt[1 - (c + d*x)^(-2)])]) + 12*a*b^2*d*e*( ArcCoth[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*ArcCot h[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) - 12*a*b^2*c*f*(ArcCo th[c + d*x]*((-1 + c + d*x)*ArcCoth[c + d*x] - 2*Log[1 - E^(-2*ArcCoth[c + d*x])]) + PolyLog[2, E^(-2*ArcCoth[c + d*x])]) + 2*b^3*f*(ArcCoth[c + d*x ]*(3*(-1 + c + d*x)*ArcCoth[c + d*x] + (-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcC oth[c + d*x]^2 - 6*Log[1 - E^(-2*ArcCoth[c + d*x])]) + 3*PolyLog[2, E^(-2* ArcCoth[c + d*x])]) + 4*b^3*d*e*((-1/8*I)*Pi^3 + ArcCoth[c + d*x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*ArcCoth[c + d *x])] - 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog [3, E^(2*ArcCoth[c + d*x])])/2) - 4*b^3*c*f*((-1/8*I)*Pi^3 + ArcCoth[c + d *x]^3 + (c + d*x)*ArcCoth[c + d*x]^3 - 3*ArcCoth[c + d*x]^2*Log[1 - E^(2*A rcCoth[c + d*x])] - 3*ArcCoth[c + d*x]*PolyLog[2, E^(2*ArcCoth[c + d*x])] + (3*PolyLog[3, E^(2*ArcCoth[c + d*x])])/2))/(4*d^2)
Time = 0.90 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, 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.
| \(\displaystyle \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx\) |
| \(\Big \downarrow \) 6662 |
| \(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{d}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int (d e-c f+f (c+d x)) \left (a+b \coth ^{-1}(c+d x)\right )^3d(c+d x)}{d^2}\) |
| \(\Big \downarrow \) 6481 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b \int \left (\frac {\left (d^2 e^2-2 c d f e+\left (c^2+1\right ) f^2+2 f (d e-c f) (c+d x)\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2}{1-(c+d x)^2}-f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2\right )d(c+d x)}{2 f}}{d^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^2 \left (a+b \coth ^{-1}(c+d x)\right )^3}{2 f}-\frac {3 b \left (\frac {\left (\left (c^2+1\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+2 b f (d e-c f) \operatorname {PolyLog}\left (2,1-\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )-\frac {2 f (d e-c f) \left (a+b \coth ^{-1}(c+d x)\right )^3}{3 b}+2 f (d e-c f) \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )^2-f^2 \left (a+b \coth ^{-1}(c+d x)\right )^2-f^2 (c+d x) \left (a+b \coth ^{-1}(c+d x)\right )^2+2 b f^2 \log \left (\frac {2}{-c-d x+1}\right ) \left (a+b \coth ^{-1}(c+d x)\right )-b^2 f (d e-c f) \operatorname {PolyLog}\left (3,1-\frac {2}{-c-d x+1}\right )+b^2 f^2 \operatorname {PolyLog}\left (2,-\frac {c+d x+1}{-c-d x+1}\right )\right )}{2 f}}{d^2}\) |
(((d*e - c*f + f*(c + d*x))^2*(a + b*ArcCoth[c + d*x])^3)/(2*f) - (3*b*(-( f^2*(a + b*ArcCoth[c + d*x])^2) - f^2*(c + d*x)*(a + b*ArcCoth[c + d*x])^2 - (2*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^3)/(3*b) + ((d^2*e^2 - 2*c*d* e*f + (1 + c^2)*f^2)*(a + b*ArcCoth[c + d*x])^3)/(3*b) + 2*b*f^2*(a + b*Ar cCoth[c + d*x])*Log[2/(1 - c - d*x)] + 2*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])^2*Log[2/(1 - c - d*x)] + b^2*f^2*PolyLog[2, -((1 + c + d*x)/(1 - c - d*x))] + 2*b*f*(d*e - c*f)*(a + b*ArcCoth[c + d*x])*PolyLog[2, 1 - 2/(1 - c - d*x)] - b^2*f*(d*e - c*f)*PolyLog[3, 1 - 2/(1 - c - d*x)]))/(2*f))/d^ 2
3.2.15.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.56 (sec) , antiderivative size = 7528, normalized size of antiderivative = 23.09
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(7528\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(7638\) |
| default | \(\text {Expression too large to display}\) | \(7638\) |
\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
integral(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*arccoth(d*x + c)^3 + 3*(a*b^2 *f*x + a*b^2*e)*arccoth(d*x + c)^2 + 3*(a^2*b*f*x + a^2*b*e)*arccoth(d*x + c), x)
\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (a + b \operatorname {acoth}{\left (c + d x \right )}\right )^{3} \left (e + f x\right )\, dx \]
\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
1/2*a^3*f*x^2 + 3/4*(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^2*b*f + a ^3*e*x + 3/2*(2*(d*x + c)*arccoth(d*x + c) + log(-(d*x + c)^2 + 1))*a^2*b* e/d + 1/16*((b^3*d^2*f*x^2 + 2*b^3*d^2*e*x - (c^2*f - 2*(d*e - f)*c - 2*d* e + f)*b^3)*log(d*x + c + 1)^3 + 3*(2*a*b^2*d^2*f*x^2 + 2*(2*a*b^2*d^2*e + b^3*d*f)*x - (b^3*d^2*f*x^2 + 2*b^3*d^2*e*x - (c^2*f - 2*(d*e + f)*c + 2* d*e + f)*b^3)*log(d*x + c - 1))*log(d*x + c + 1)^2)/d^2 + integrate(-1/8*( (b^3*d^2*f*x^2 + (d^2*e + c*d*f + d*f)*b^3*x + (c*d*e + d*e)*b^3)*log(d*x + c - 1)^3 - 6*(a*b^2*d^2*f*x^2 + (d^2*e + c*d*f + d*f)*a*b^2*x + (c*d*e + d*e)*a*b^2)*log(d*x + c - 1)^2 + 3*(2*a*b^2*d^2*f*x^2 - (b^3*d^2*f*x^2 + (d^2*e + c*d*f + d*f)*b^3*x + (c*d*e + d*e)*b^3)*log(d*x + c - 1)^2 + 2*(2 *a*b^2*d^2*e + b^3*d*f)*x + (4*(c*d*e + d*e)*a*b^2 + (c^2*f - 2*(d*e + f)* c + 2*d*e + f)*b^3 + (4*a*b^2*d^2*f - b^3*d^2*f)*x^2 - 2*(b^3*d^2*e - 2*(d ^2*e + c*d*f + d*f)*a*b^2)*x)*log(d*x + c - 1))*log(d*x + c + 1))/(d^2*x + c*d + d), x)
\[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int { {\left (f x + e\right )} {\left (b \operatorname {arcoth}\left (d x + c\right ) + a\right )}^{3} \,d x } \]
Timed out. \[ \int (e+f x) \left (a+b \coth ^{-1}(c+d x)\right )^3 \, dx=\int \left (e+f\,x\right )\,{\left (a+b\,\mathrm {acoth}\left (c+d\,x\right )\right )}^3 \,d x \]