Integrand size = 18, antiderivative size = 272 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=-\frac {(a+b \text {arctanh}(c x))^3 \log \left (\frac {2}{1+c x}\right )}{e}+\frac {(a+b \text {arctanh}(c x))^3 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{e}+\frac {3 b (a+b \text {arctanh}(c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b (a+b \text {arctanh}(c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 e}-\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 e}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{1+c x}\right )}{4 e}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{4 e} \] Output:
-(a+b*arctanh(c*x))^3*ln(2/(c*x+1))/e+(a+b*arctanh(c*x))^3*ln(2*c*(e*x+d)/ (c*d+e)/(c*x+1))/e+3/2*b*(a+b*arctanh(c*x))^2*polylog(2,1-2/(c*x+1))/e-3/2 *b*(a+b*arctanh(c*x))^2*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e+3/2*b^2 *(a+b*arctanh(c*x))*polylog(3,1-2/(c*x+1))/e-3/2*b^2*(a+b*arctanh(c*x))*po lylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e+3/4*b^3*polylog(4,1-2/(c*x+1))/e- 3/4*b^3*polylog(4,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/e
Result contains complex when optimal does not.
Time = 26.72 (sec) , antiderivative size = 2160, normalized size of antiderivative = 7.94 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\text {Result too large to show} \] Input:
Integrate[(a + b*ArcTanh[c*x])^3/(d + e*x),x]
Output:
(a^3*Log[d + e*x])/e - ((3*I)*a^2*b*(I*ArcTanh[c*x]*(-Log[1/Sqrt[1 - c^2*x ^2]] + Log[I*Sinh[ArcTanh[(c*d)/e] + ArcTanh[c*x]]]) + ((-I)*(I*ArcTanh[(c *d)/e] + I*ArcTanh[c*x])^2 - (I/4)*(Pi - (2*I)*ArcTanh[c*x])^2 + 2*(I*ArcT anh[(c*d)/e] + I*ArcTanh[c*x])*Log[1 - E^((2*I)*(I*ArcTanh[(c*d)/e] + I*Ar cTanh[c*x]))] + (Pi - (2*I)*ArcTanh[c*x])*Log[1 - E^(I*(Pi - (2*I)*ArcTanh [c*x]))] - (Pi - (2*I)*ArcTanh[c*x])*Log[2*Sin[(Pi - (2*I)*ArcTanh[c*x])/2 ]] - 2*(I*ArcTanh[(c*d)/e] + I*ArcTanh[c*x])*Log[(2*I)*Sinh[ArcTanh[(c*d)/ e] + ArcTanh[c*x]]] - I*PolyLog[2, E^((2*I)*(I*ArcTanh[(c*d)/e] + I*ArcTan h[c*x]))] - I*PolyLog[2, E^(I*(Pi - (2*I)*ArcTanh[c*x]))])/2))/e + (a*b^2* (-8*c*d*ArcTanh[c*x]^3 + 4*e*ArcTanh[c*x]^3 - (4*Sqrt[1 - (c^2*d^2)/e^2]*e *ArcTanh[c*x]^3)/E^ArcTanh[(c*d)/e] - 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(-2*A rcTanh[c*x])] - 6*c*d*ArcTanh[c*x]^2*Log[1 - (Sqrt[c*d + e]*E^ArcTanh[c*x] )/Sqrt[-(c*d) + e]] - 6*c*d*ArcTanh[c*x]^2*Log[1 + (Sqrt[c*d + e]*E^ArcTan h[c*x])/Sqrt[-(c*d) + e]] - (6*I)*c*d*Pi*ArcTanh[c*x]*Log[(1 + E^(2*ArcTan h[c*x]))/(2*E^ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(ArcTanh[(c* d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 + E^(ArcTanh[(c*d)/e] + ArcTanh[c*x])] + 6*c*d*ArcTanh[c*x]^2*Log[1 - E^(2*(ArcTanh[(c*d)/e] + A rcTanh[c*x]))] + 12*c*d*ArcTanh[(c*d)/e]*ArcTanh[c*x]*Log[(I/2)*E^(-ArcTan h[(c*d)/e] - ArcTanh[c*x])*(-1 + E^(2*(ArcTanh[(c*d)/e] + ArcTanh[c*x])))] + 6*c*d*ArcTanh[c*x]^2*Log[(e*(-1 + E^(2*ArcTanh[c*x])) + c*d*(1 + E^(...
Time = 0.38 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {6476}
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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx\) |
| \(\Big \downarrow \) 6476 |
| \(\displaystyle -\frac {3 b^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {3 b^2 \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))}{2 e}-\frac {3 b (a+b \text {arctanh}(c x))^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{2 e}+\frac {(a+b \text {arctanh}(c x))^3 \log \left (\frac {2 c (d+e x)}{(c x+1) (c d+e)}\right )}{e}+\frac {3 b \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^2}{2 e}-\frac {\log \left (\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))^3}{e}-\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{4 e}+\frac {3 b^3 \operatorname {PolyLog}\left (4,1-\frac {2}{c x+1}\right )}{4 e}\) |
Input:
Int[(a + b*ArcTanh[c*x])^3/(d + e*x),x]
Output:
-(((a + b*ArcTanh[c*x])^3*Log[2/(1 + c*x)])/e) + ((a + b*ArcTanh[c*x])^3*L og[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/e + (3*b*(a + b*ArcTanh[c*x])^2 *PolyLog[2, 1 - 2/(1 + c*x)])/(2*e) - (3*b*(a + b*ArcTanh[c*x])^2*PolyLog[ 2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e) + (3*b^2*(a + b*ArcTa nh[c*x])*PolyLog[3, 1 - 2/(1 + c*x)])/(2*e) - (3*b^2*(a + b*ArcTanh[c*x])* PolyLog[3, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/(2*e) + (3*b^3*Poly Log[4, 1 - 2/(1 + c*x)])/(4*e) - (3*b^3*PolyLog[4, 1 - (2*c*(d + e*x))/((c *d + e)*(1 + c*x))])/(4*e)
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^3/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x])^3)*(Log[2/(1 + c*x)]/e), x] + (Simp[(a + b*Arc Tanh[c*x])^3*(Log[2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/e), x] + Simp[3*b* (a + b*ArcTanh[c*x])^2*(PolyLog[2, 1 - 2/(1 + c*x)]/(2*e)), x] - Simp[3*b*( a + b*ArcTanh[c*x])^2*(PolyLog[2, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x))) ]/(2*e)), x] + Simp[3*b^2*(a + b*ArcTanh[c*x])*(PolyLog[3, 1 - 2/(1 + c*x)] /(2*e)), x] - Simp[3*b^2*(a + b*ArcTanh[c*x])*(PolyLog[3, 1 - 2*c*((d + e*x )/((c*d + e)*(1 + c*x)))]/(2*e)), x] + Simp[3*b^3*(PolyLog[4, 1 - 2/(1 + c* x)]/(4*e)), x] - Simp[3*b^3*(PolyLog[4, 1 - 2*c*((d + e*x)/((c*d + e)*(1 + c*x)))]/(4*e)), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.86 (sec) , antiderivative size = 2160, normalized size of antiderivative = 7.94
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2160\) |
| default | \(\text {Expression too large to display}\) | \(2160\) |
| parts | \(\text {Expression too large to display}\) | \(2164\) |
Input:
int((a+b*arctanh(c*x))^3/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/c*(a^3*c*ln(c*e*x+c*d)/e+b^3*c*(ln(c*e*x+c*d)/e*arctanh(c*x)^3-3/e*(1/3* arctanh(c*x)^3*ln(d*c*(1+(c*x+1)^2/(-c^2*x^2+1))+e*((c*x+1)^2/(-c^2*x^2+1) -1))-1/6*I*Pi*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2 -1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))*(csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2-1))+ e*(-(c*x+1)^2/(c^2*x^2-1)-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))-csgn(I*(d *c*(1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c ^2*x^2-1)))*csgn(I/(1-(c*x+1)^2/(c^2*x^2-1)))-csgn(I*(d*c*(1-(c*x+1)^2/(c^ 2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1)))*csgn(I*(d*c*(1-(c*x+1)^2/(c^2*x^2 -1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2*x^2-1)))+csgn(I*(d*c* (1-(c*x+1)^2/(c^2*x^2-1))+e*(-(c*x+1)^2/(c^2*x^2-1)-1))/(1-(c*x+1)^2/(c^2* x^2-1)))^2)*arctanh(c*x)^3+1/2*arctanh(c*x)^2*polylog(2,-(c*x+1)^2/(-c^2*x ^2+1))-1/2*arctanh(c*x)*polylog(3,-(c*x+1)^2/(-c^2*x^2+1))+1/4*polylog(4,- (c*x+1)^2/(-c^2*x^2+1))-1/3*e/(c*d+e)*arctanh(c*x)^3*ln(1-(c*d+e)*(c*x+1)^ 2/(-c^2*x^2+1)/(-c*d+e))-1/2*e/(c*d+e)*arctanh(c*x)^2*polylog(2,(c*d+e)*(c *x+1)^2/(-c^2*x^2+1)/(-c*d+e))+1/2*e/(c*d+e)*arctanh(c*x)*polylog(3,(c*d+e )*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/4*e/(c*d+e)*polylog(4,(c*d+e)*(c*x+1) ^2/(-c^2*x^2+1)/(-c*d+e))-1/3*d*c/(c*d+e)*arctanh(c*x)^3*ln(1-(c*d+e)*(c*x +1)^2/(-c^2*x^2+1)/(-c*d+e))-1/2*d*c/(c*d+e)*arctanh(c*x)^2*polylog(2,(c*d +e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))+1/2*d*c/(c*d+e)*arctanh(c*x)*polylog( 3,(c*d+e)*(c*x+1)^2/(-c^2*x^2+1)/(-c*d+e))-1/4*d*c/(c*d+e)*polylog(4,(c...
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^3/(e*x+d),x, algorithm="fricas")
Output:
integral((b^3*arctanh(c*x)^3 + 3*a*b^2*arctanh(c*x)^2 + 3*a^2*b*arctanh(c* x) + a^3)/(e*x + d), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{d + e x}\, dx \] Input:
integrate((a+b*atanh(c*x))**3/(e*x+d),x)
Output:
Integral((a + b*atanh(c*x))**3/(d + e*x), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^3/(e*x+d),x, algorithm="maxima")
Output:
a^3*log(e*x + d)/e + integrate(1/8*b^3*(log(c*x + 1) - log(-c*x + 1))^3/(e *x + d) + 3/4*a*b^2*(log(c*x + 1) - log(-c*x + 1))^2/(e*x + d) + 3/2*a^2*b *(log(c*x + 1) - log(-c*x + 1))/(e*x + d), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{e x + d} \,d x } \] Input:
integrate((a+b*arctanh(c*x))^3/(e*x+d),x, algorithm="giac")
Output:
integrate((b*arctanh(c*x) + a)^3/(e*x + d), x)
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{d+e\,x} \,d x \] Input:
int((a + b*atanh(c*x))^3/(d + e*x),x)
Output:
int((a + b*atanh(c*x))^3/(d + e*x), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{d+e x} \, dx=\frac {3 \left (\int \frac {\mathit {atanh} \left (c x \right )}{e x +d}d x \right ) a^{2} b e +\left (\int \frac {\mathit {atanh} \left (c x \right )^{3}}{e x +d}d x \right ) b^{3} e +3 \left (\int \frac {\mathit {atanh} \left (c x \right )^{2}}{e x +d}d x \right ) a \,b^{2} e +\mathrm {log}\left (e x +d \right ) a^{3}}{e} \] Input:
int((a+b*atanh(c*x))^3/(e*x+d),x)
Output:
(3*int(atanh(c*x)/(d + e*x),x)*a**2*b*e + int(atanh(c*x)**3/(d + e*x),x)*b **3*e + 3*int(atanh(c*x)**2/(d + e*x),x)*a*b**2*e + log(d + e*x)*a**3)/e