Integrand size = 18, antiderivative size = 953 \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\frac {3 b c (a+b \text {arctanh}(c x))^2}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {(a+b \text {arctanh}(c x))^3}{2 e (d+e x)^2}-\frac {3 b^2 c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right )}{2 (c d-e) (c d+e)^2}+\frac {3 b c^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{4 e (c d+e)^2}-\frac {3 b^2 c^2 e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)}-\frac {3 b c^2 (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}+\frac {3 b c^3 d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^2 e (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b c^3 d (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 (c d-e) (c d+e)^2}+\frac {3 b^2 c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right )}{4 e (c d+e)^2}+\frac {3 b^3 c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 (c d+e)}+\frac {3 b^2 c^2 (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{4 (c d-e)^2 e}-\frac {3 b^2 c^3 d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^2 c^3 d (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{(c d-e)^2 (c d+e)^2}-\frac {3 b^3 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right )}{8 e (c d+e)^2}+\frac {3 b^3 c^2 \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{8 (c d-e)^2 e}-\frac {3 b^3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2}{1+c x}\right )}{2 (c d-e)^2 (c d+e)^2}+\frac {3 b^3 c^3 d \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (1+c x)}\right )}{2 (c d-e)^2 (c d+e)^2} \]
3/2*b*c*(a+b*arctanh(c*x))^2/(c^2*d^2-e^2)/(e*x+d)-1/2*(a+b*arctanh(c*x))^ 3/e/(e*x+d)^2-3/2*b^2*c^2*(a+b*arctanh(c*x))*ln(2/(-c*x+1))/(c*d-e)/(c*d+e )^2+3/4*b*c^2*(a+b*arctanh(c*x))^2*ln(2/(-c*x+1))/e/(c*d+e)^2+3/2*b^2*c^2* (a+b*arctanh(c*x))*ln(2/(c*x+1))/(c*d-e)^2/(c*d+e)-3*b^2*c^2*e*(a+b*arctan h(c*x))*ln(2/(c*x+1))/(c^2*d^2-e^2)^2-3/4*b*c^2*(a+b*arctanh(c*x))^2*ln(2/ (c*x+1))/(c*d-e)^2/e+3*b*c^3*d*(a+b*arctanh(c*x))^2*ln(2/(c*x+1))/(c^2*d^2 -e^2)^2+3*b^2*c^2*e*(a+b*arctanh(c*x))*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^ 2*d^2-e^2)^2-3*b*c^3*d*(a+b*arctanh(c*x))^2*ln(2*c*(e*x+d)/(c*d+e)/(c*x+1) )/(c^2*d^2-e^2)^2-3/4*b^3*c^2*polylog(2,1-2/(-c*x+1))/(c*d-e)/(c*d+e)^2+3/ 4*b^2*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(-c*x+1))/e/(c*d+e)^2-3/4*b^3*c ^2*polylog(2,1-2/(c*x+1))/(c*d-e)^2/(c*d+e)+3/2*b^3*c^2*e*polylog(2,1-2/(c *x+1))/(c^2*d^2-e^2)^2+3/4*b^2*c^2*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1 ))/(c*d-e)^2/e-3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog(2,1-2/(c*x+1))/(c^2* d^2-e^2)^2-3/2*b^3*c^2*e*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c*x+1))/(c^2*d^2 -e^2)^2+3*b^2*c^3*d*(a+b*arctanh(c*x))*polylog(2,1-2*c*(e*x+d)/(c*d+e)/(c* x+1))/(c^2*d^2-e^2)^2-3/8*b^3*c^2*polylog(3,1-2/(-c*x+1))/e/(c*d+e)^2+3/8* b^3*c^2*polylog(3,1-2/(c*x+1))/(c*d-e)^2/e-3/2*b^3*c^3*d*polylog(3,1-2/(c* x+1))/(c^2*d^2-e^2)^2+3/2*b^3*c^3*d*polylog(3,1-2*c*(e*x+d)/(c*d+e)/(c*x+1 ))/(c^2*d^2-e^2)^2
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx \]
Time = 1.43 (sec) , antiderivative size = 896, normalized size of antiderivative = 0.94, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6480, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 6480 |
| \(\displaystyle \frac {3 b c \int \left (\frac {c^2 (a+b \text {arctanh}(c x))^2}{2 (c d+e)^2 (1-c x)}+\frac {c^2 (a+b \text {arctanh}(c x))^2}{2 (c d-e)^2 (c x+1)}-\frac {2 c^2 d e^2 (a+b \text {arctanh}(c x))^2}{(c d-e)^2 (c d+e)^2 (d+e x)}-\frac {e^2 (a+b \text {arctanh}(c x))^2}{(c d-e) (c d+e) (d+e x)^2}\right )dx}{2 e}-\frac {(a+b \text {arctanh}(c x))^3}{2 e (d+e x)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b c \left (-\frac {c e \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) b^2}{2 (c d-e) (c d+e)^2}-\frac {c e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^2}{2 (c d-e)^2 (c d+e)}+\frac {c e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}-\frac {c e^2 \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}-\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{1-c x}\right ) b^2}{4 (c d+e)^2}+\frac {c \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) b^2}{4 (c d-e)^2}-\frac {c^2 d e \operatorname {PolyLog}\left (3,1-\frac {2}{c x+1}\right ) b^2}{(c d-e)^2 (c d+e)^2}+\frac {c^2 d e \operatorname {PolyLog}\left (3,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b^2}{(c d-e)^2 (c d+e)^2}-\frac {c e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{1-c x}\right ) b}{(c d-e) (c d+e)^2}+\frac {c e (a+b \text {arctanh}(c x)) \log \left (\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)}-\frac {2 c e^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}+\frac {2 c e^2 (a+b \text {arctanh}(c x)) \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}+\frac {c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{1-c x}\right ) b}{2 (c d+e)^2}+\frac {c (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b}{2 (c d-e)^2}-\frac {2 c^2 d e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right ) b}{(c d-e)^2 (c d+e)^2}+\frac {2 c^2 d e (a+b \text {arctanh}(c x)) \operatorname {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right ) b}{(c d-e)^2 (c d+e)^2}+\frac {e (a+b \text {arctanh}(c x))^2}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{1-c x}\right )}{2 (c d+e)^2}-\frac {c (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{c x+1}\right )}{2 (c d-e)^2}+\frac {2 c^2 d e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2}{c x+1}\right )}{(c d-e)^2 (c d+e)^2}-\frac {2 c^2 d e (a+b \text {arctanh}(c x))^2 \log \left (\frac {2 c (d+e x)}{(c d+e) (c x+1)}\right )}{(c d-e)^2 (c d+e)^2}\right )}{2 e}-\frac {(a+b \text {arctanh}(c x))^3}{2 e (d+e x)^2}\) |
-1/2*(a + b*ArcTanh[c*x])^3/(e*(d + e*x)^2) + (3*b*c*((e*(a + b*ArcTanh[c* x])^2)/((c^2*d^2 - e^2)*(d + e*x)) - (b*c*e*(a + b*ArcTanh[c*x])*Log[2/(1 - c*x)])/((c*d - e)*(c*d + e)^2) + (c*(a + b*ArcTanh[c*x])^2*Log[2/(1 - c* x)])/(2*(c*d + e)^2) - (2*b*c*e^2*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/( (c*d - e)^2*(c*d + e)^2) + (b*c*e*(a + b*ArcTanh[c*x])*Log[2/(1 + c*x)])/( (c*d - e)^2*(c*d + e)) - (c*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/(2*(c *d - e)^2) + (2*c^2*d*e*(a + b*ArcTanh[c*x])^2*Log[2/(1 + c*x)])/((c*d - e )^2*(c*d + e)^2) + (2*b*c*e^2*(a + b*ArcTanh[c*x])*Log[(2*c*(d + e*x))/((c *d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) - (2*c^2*d*e*(a + b*ArcTanh [c*x])^2*Log[(2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e )^2) - (b^2*c*e*PolyLog[2, 1 - 2/(1 - c*x)])/(2*(c*d - e)*(c*d + e)^2) + ( b*c*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 - c*x)])/(2*(c*d + e)^2) + (b ^2*c*e^2*PolyLog[2, 1 - 2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) - (b^2*c*e *PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2*(c*d + e)) + (b*c*(a + b*ArcT anh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/(2*(c*d - e)^2) - (2*b*c^2*d*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - 2/(1 + c*x)])/((c*d - e)^2*(c*d + e)^2) - (b^2*c*e^2*PolyLog[2, 1 - (2*c*(d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) + (2*b*c^2*d*e*(a + b*ArcTanh[c*x])*PolyLog[2, 1 - (2*c* (d + e*x))/((c*d + e)*(1 + c*x))])/((c*d - e)^2*(c*d + e)^2) - (b^2*c*Poly Log[3, 1 - 2/(1 - c*x)])/(4*(c*d + e)^2) + (b^2*c*PolyLog[3, 1 - 2/(1 +...
3.1.20.3.1 Defintions of rubi rules used
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_S ymbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcTanh[c*x])^p/(e*(q + 1))), x] - Simp[b*c*(p/(e*(q + 1))) Int[ExpandIntegrand[(a + b*ArcTanh[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 14.39 (sec) , antiderivative size = 51695, normalized size of antiderivative = 54.24
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(51695\) |
| default | \(\text {Expression too large to display}\) | \(51695\) |
| parts | \(\text {Expression too large to display}\) | \(51703\) |
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
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^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3), x)
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{3}}\, dx \]
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
-3/4*((4*c^2*d*log(e*x + d)/(c^4*d^4 - 2*c^2*d^2*e^2 + e^4) - c*log(c*x + 1)/(c^2*d^2*e - 2*c*d*e^2 + e^3) + c*log(c*x - 1)/(c^2*d^2*e + 2*c*d*e^2 + e^3) - 2/(c^2*d^3 - d*e^2 + (c^2*d^2*e - e^3)*x))*c + 2*arctanh(c*x)/(e^3 *x^2 + 2*d*e^2*x + d^2*e))*a^2*b - 1/2*a^3/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/16*(((c^4*d^2*e^2 - 2*c^3*d*e^3 + c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e - 2*c ^3*d^2*e^2 + c^2*d*e^3)*b^3*x - (2*c^3*d^3*e - 3*c^2*d^2*e^2 + e^4)*b^3)*l og(-c*x + 1)^3 - 3*(2*(c^3*d^2*e^2 - c*e^4)*b^3*x - 2*(c^4*d^4 - 2*c^2*d^2 *e^2 + e^4)*a*b^2 + 2*(c^3*d^3*e - c*d*e^3)*b^3 + ((c^4*d^2*e^2 + 2*c^3*d* e^3 + c^2*e^4)*b^3*x^2 + 2*(c^4*d^3*e + 2*c^3*d^2*e^2 + c^2*d*e^3)*b^3*x + (2*c^3*d^3*e + 3*c^2*d^2*e^2 - e^4)*b^3)*log(c*x + 1))*log(-c*x + 1)^2)/( c^4*d^6*e - 2*c^2*d^4*e^3 + d^2*e^5 + (c^4*d^4*e^3 - 2*c^2*d^2*e^5 + e^7)* x^2 + 2*(c^4*d^5*e^2 - 2*c^2*d^3*e^4 + d*e^6)*x) - integrate(1/8*(((c^4*d^ 3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*b^3)*log(c*x + 1)^3 + 6*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d*e ^3 + c*e^4)*a*b^2*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*a*b^2)*log (c*x + 1)^2 - 3*(2*(c^3*d*e^3 - c^2*e^4)*b^3*x^2 - 2*(c^4*d^4 - c^3*d^3*e - c^2*d^2*e^2 + c*d*e^3)*a*b^2 + 2*(c^3*d^3*e - c^2*d^2*e^2)*b^3 + ((c^4*d ^3*e - c^3*d^2*e^2 - c^2*d*e^3 + c*e^4)*b^3*x - (c^3*d^3*e - c^2*d^2*e^2 - c*d*e^3 + e^4)*b^3)*log(c*x + 1)^2 - 2*((c^4*d^3*e - c^3*d^2*e^2 - c^2*d* e^3 + c*e^4)*a*b^2 - 2*(c^3*d^2*e^2 - c^2*d*e^3)*b^3)*x + ((c^4*d*e^3 +...
\[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^3}{(d+e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^3}{{\left (d+e\,x\right )}^3} \,d x \]