Integrand size = 23, antiderivative size = 166 \[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {3 b (a+b \text {arctanh}(c+d x))^2}{2 d e^3}-\frac {3 b (a+b \text {arctanh}(c+d x))^2}{2 d e^3 (c+d x)}+\frac {(a+b \text {arctanh}(c+d x))^3}{2 d e^3}-\frac {(a+b \text {arctanh}(c+d x))^3}{2 d e^3 (c+d x)^2}+\frac {3 b^2 (a+b \text {arctanh}(c+d x)) \log \left (2-\frac {2}{1+c+d x}\right )}{d e^3}-\frac {3 b^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c+d x}\right )}{2 d e^3} \]
3/2*b*(a+b*arctanh(d*x+c))^2/d/e^3-3/2*b*(a+b*arctanh(d*x+c))^2/d/e^3/(d*x +c)+1/2*(a+b*arctanh(d*x+c))^3/d/e^3-1/2*(a+b*arctanh(d*x+c))^3/d/e^3/(d*x +c)^2+3*b^2*(a+b*arctanh(d*x+c))*ln(2-2/(d*x+c+1))/d/e^3-3/2*b^3*polylog(2 ,-1+2/(d*x+c+1))/d/e^3
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.02 \[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {-4 a^3-12 a^2 b c+i b^3 c^3 \pi ^3-12 a^2 b d x+2 i b^3 c^2 d \pi ^3 x+i b^3 c d^2 \pi ^3 x^2+12 b^2 (-1+c+d x) (b (c+d x)+a (1+c+d x)) \text {arctanh}(c+d x)^2+4 b^3 \left (-1+c^2+2 c d x+d^2 x^2\right ) \text {arctanh}(c+d x)^3+12 b \text {arctanh}(c+d x) \left (a \left (-2 b (c+d x)+a \left (-1+c^2+2 c d x+d^2 x^2\right )\right )+2 b^2 (c+d x)^2 \log \left (1-e^{-2 \text {arctanh}(c+d x)}\right )\right )+24 a b^2 c^2 \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )+48 a b^2 c d x \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )+24 a b^2 d^2 x^2 \log \left (\frac {c+d x}{\sqrt {1-(c+d x)^2}}\right )-12 b^3 (c+d x)^2 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c+d x)}\right )}{8 d e^3 (c+d x)^2} \]
(-4*a^3 - 12*a^2*b*c + I*b^3*c^3*Pi^3 - 12*a^2*b*d*x + (2*I)*b^3*c^2*d*Pi^ 3*x + I*b^3*c*d^2*Pi^3*x^2 + 12*b^2*(-1 + c + d*x)*(b*(c + d*x) + a*(1 + c + d*x))*ArcTanh[c + d*x]^2 + 4*b^3*(-1 + c^2 + 2*c*d*x + d^2*x^2)*ArcTanh [c + d*x]^3 + 12*b*ArcTanh[c + d*x]*(a*(-2*b*(c + d*x) + a*(-1 + c^2 + 2*c *d*x + d^2*x^2)) + 2*b^2*(c + d*x)^2*Log[1 - E^(-2*ArcTanh[c + d*x])]) + 2 4*a*b^2*c^2*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] + 48*a*b^2*c*d*x*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] + 24*a*b^2*d^2*x^2*Log[(c + d*x)/Sqrt[1 - (c + d*x)^2]] - 12*b^3*(c + d*x)^2*PolyLog[2, E^(-2*ArcTanh[c + d*x])])/(8*d* e^3*(c + d*x)^2)
Time = 1.02 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {6657, 27, 6452, 6544, 6452, 6510, 6550, 6494, 2897}
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.
\(\displaystyle \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx\) |
\(\Big \downarrow \) 6657 |
\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^3}{e^3 (c+d x)^3}d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {(a+b \text {arctanh}(c+d x))^3}{(c+d x)^3}d(c+d x)}{d e^3}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {\frac {3}{2} b \int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^2 \left (1-(c+d x)^2\right )}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6544 |
\(\displaystyle \frac {\frac {3}{2} b \left (\int \frac {(a+b \text {arctanh}(c+d x))^2}{(c+d x)^2}d(c+d x)+\int \frac {(a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6452 |
\(\displaystyle \frac {\frac {3}{2} b \left (2 b \int \frac {a+b \text {arctanh}(c+d x)}{(c+d x) \left (1-(c+d x)^2\right )}d(c+d x)+\int \frac {(a+b \text {arctanh}(c+d x))^2}{1-(c+d x)^2}d(c+d x)-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6510 |
\(\displaystyle \frac {\frac {3}{2} b \left (2 b \int \frac {a+b \text {arctanh}(c+d x)}{(c+d x) \left (1-(c+d x)^2\right )}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6550 |
\(\displaystyle \frac {\frac {3}{2} b \left (2 b \left (\int \frac {a+b \text {arctanh}(c+d x)}{(c+d x) (c+d x+1)}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 6494 |
\(\displaystyle \frac {\frac {3}{2} b \left (2 b \left (-b \int \frac {\log \left (2-\frac {2}{c+d x+1}\right )}{1-(c+d x)^2}d(c+d x)+\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (2-\frac {2}{c+d x+1}\right ) (a+b \text {arctanh}(c+d x))\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
\(\Big \downarrow \) 2897 |
\(\displaystyle \frac {\frac {3}{2} b \left (2 b \left (\frac {(a+b \text {arctanh}(c+d x))^2}{2 b}+\log \left (2-\frac {2}{c+d x+1}\right ) (a+b \text {arctanh}(c+d x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c+d x+1}-1\right )\right )+\frac {(a+b \text {arctanh}(c+d x))^3}{3 b}-\frac {(a+b \text {arctanh}(c+d x))^2}{c+d x}\right )-\frac {(a+b \text {arctanh}(c+d x))^3}{2 (c+d x)^2}}{d e^3}\) |
(-1/2*(a + b*ArcTanh[c + d*x])^3/(c + d*x)^2 + (3*b*(-((a + b*ArcTanh[c + d*x])^2/(c + d*x)) + (a + b*ArcTanh[c + d*x])^3/(3*b) + 2*b*((a + b*ArcTan h[c + d*x])^2/(2*b) + (a + b*ArcTanh[c + d*x])*Log[2 - 2/(1 + c + d*x)] - (b*PolyLog[2, -1 + 2/(1 + c + d*x)])/2)))/2)/(d*e^3)
3.1.27.3.1 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[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, x][[2]], Expon[Pq, x]]
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] : > Simp[x^(m + 1)*((a + b*ArcTanh[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTanh[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))), x ], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1 ] && IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x _Symbol] :> Simp[(a + b*ArcTanh[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Simp[b*c*(p/d) Int[(a + b*ArcTanh[c*x])^(p - 1)*(Log[2 - 2/(1 + e*(x/d))] /(1 - c^2*x^2)), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[c ^2*d^2 - e^2, 0]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b , c, d, e, p}, x] && EqQ[c^2*d + e, 0] && NeQ[p, -1]
Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + ( e_.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTanh[c*x])^p, x ], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTanh[c*x])^p/(d + e*x ^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[(a + b*ArcTanh[c*x])^(p + 1)/(b*d*(p + 1)), x] + Simp[1/ d Int[(a + b*ArcTanh[c*x])^p/(x*(1 + c*x)), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[p, 0]
Int[((a_.) + ArcTanh[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[(f*(x/d))^m*(a + b*ArcTanh[x])^p, x ], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[d*e - c*f, 0] && IGtQ[p, 0]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.94 (sec) , antiderivative size = 4949, normalized size of antiderivative = 29.81
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(4949\) |
default | \(\text {Expression too large to display}\) | \(4949\) |
parts | \(\text {Expression too large to display}\) | \(4957\) |
1/d*(-1/2*a^3/e^3/(d*x+c)^2+b^3/e^3*(3/4*I*Pi*arctanh(d*x+c)^2-3/8*I*Pi*cs gn(I*(d*x+c+1)^2/((d*x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-(d*x+c +1)^2/((d*x+c)^2-1)))^2*arctanh(d*x+c)*ln(1-(d*x+c+1)/(1-(d*x+c)^2)^(1/2)) -3/2/(d*x+c)*arctanh(d*x+c)^2-3/8*I*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1) ))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-( d*x+c+1)^2/((d*x+c)^2-1)))*arctanh(d*x+c)*ln(1-(d*x+c+1)/(1-(d*x+c)^2)^(1/ 2))-3/8*I*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*csgn(I*(d*x+c+1)^2/((d* x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1))) *polylog(2,(d*x+c+1)/(1-(d*x+c)^2)^(1/2))-3/8*I*Pi*csgn(I/(1-(d*x+c+1)^2/( (d*x+c)^2-1)))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+ c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*polylog(2,-(d*x+c+1)/(1-(d*x+c)^2)^ (1/2))-3/8*I*Pi*csgn(I/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*csgn(I*(d*x+c+1)^2/( (d*x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1 )))*dilog((d*x+c+1)/(1-(d*x+c)^2)^(1/2))+3/8*I*Pi*csgn(I/(1-(d*x+c+1)^2/(( d*x+c)^2-1)))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1))*csgn(I*(d*x+c+1)^2/((d*x+c )^2-1)/(1-(d*x+c+1)^2/((d*x+c)^2-1)))*dilog(1+(d*x+c+1)/(1-(d*x+c)^2)^(1/2 ))+3/8*I*Pi*csgn(I*(d*x+c+1)/(1-(d*x+c)^2)^(1/2))^2*csgn(I*(d*x+c+1)^2/((d *x+c)^2-1))*arctanh(d*x+c)*ln(1-(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+3/4*I*Pi*cs gn(I*(d*x+c+1)/(1-(d*x+c)^2)^(1/2))*csgn(I*(d*x+c+1)^2/((d*x+c)^2-1))^2*ar ctanh(d*x+c)*ln(1-(d*x+c+1)/(1-(d*x+c)^2)^(1/2))+3/8*I*Pi*csgn(I/(1-(d*...
\[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
integral((b^3*arctanh(d*x + c)^3 + 3*a*b^2*arctanh(d*x + c)^2 + 3*a^2*b*ar ctanh(d*x + c) + a^3)/(d^3*e^3*x^3 + 3*c*d^2*e^3*x^2 + 3*c^2*d*e^3*x + c^3 *e^3), x)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\frac {\int \frac {a^{3}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {b^{3} \operatorname {atanh}^{3}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a b^{2} \operatorname {atanh}^{2}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx + \int \frac {3 a^{2} b \operatorname {atanh}{\left (c + d x \right )}}{c^{3} + 3 c^{2} d x + 3 c d^{2} x^{2} + d^{3} x^{3}}\, dx}{e^{3}} \]
(Integral(a**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x) + Integ ral(b**3*atanh(c + d*x)**3/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3) , x) + Integral(3*a*b**2*atanh(c + d*x)**2/(c**3 + 3*c**2*d*x + 3*c*d**2*x **2 + d**3*x**3), x) + Integral(3*a**2*b*atanh(c + d*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3), x))/e**3
\[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
-3/4*(d*(2/(d^3*e^3*x + c*d^2*e^3) - log(d*x + c + 1)/(d^2*e^3) + log(d*x + c - 1)/(d^2*e^3)) + 2*arctanh(d*x + c)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^ 2*d*e^3))*a^2*b - 3/8*(d^2*((log(d*x + c + 1)^2 - 2*log(d*x + c + 1)*log(d *x + c - 1) + log(d*x + c - 1)^2 + 4*log(d*x + c - 1))/(d^3*e^3) + 4*log(d *x + c + 1)/(d^3*e^3) - 8*log(d*x + c)/(d^3*e^3)) + 4*d*(2/(d^3*e^3*x + c* d^2*e^3) - log(d*x + c + 1)/(d^2*e^3) + log(d*x + c - 1)/(d^2*e^3))*arctan h(d*x + c))*a*b^2 - 1/16*b^3*(((d^2*x^2 + 2*c*d*x + c^2 - 1)*log(-d*x - c + 1)^3 + 3*(2*d*x - (d^2*x^2 + 2*c*d*x + c^2 - 1)*log(d*x + c + 1) + 2*c)* log(-d*x - c + 1)^2)/(d^3*e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) + 2*integra te(-((d*x + c - 1)*log(d*x + c + 1)^3 + 3*(2*d^2*x^2 + 4*c*d*x - (d*x + c - 1)*log(d*x + c + 1)^2 + 2*c^2 - (d^3*x^3 + 3*c*d^2*x^2 + c^3 + (3*c^2*d - d)*x - c)*log(d*x + c + 1))*log(-d*x - c + 1))/(d^4*e^3*x^4 + c^4*e^3 - c^3*e^3 + (4*c*d^3*e^3 - d^3*e^3)*x^3 + 3*(2*c^2*d^2*e^3 - c*d^2*e^3)*x^2 + (4*c^3*d*e^3 - 3*c^2*d*e^3)*x), x)) - 3/2*a*b^2*arctanh(d*x + c)^2/(d^3* e^3*x^2 + 2*c*d^2*e^3*x + c^2*d*e^3) - 1/2*a^3/(d^3*e^3*x^2 + 2*c*d^2*e^3* x + c^2*d*e^3)
\[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (d x + c\right ) + a\right )}^{3}}{{\left (d e x + c e\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c+d x))^3}{(c e+d e x)^3} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c+d\,x\right )\right )}^3}{{\left (c\,e+d\,e\,x\right )}^3} \,d x \]