Integrand size = 14, antiderivative size = 130 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {b^2 c^2}{3 x}+\frac {1}{3} b^2 c^3 \text {arctanh}(c x)-\frac {b c (a+b \text {arctanh}(c x))}{3 x^2}+\frac {1}{3} c^3 (a+b \text {arctanh}(c x))^2-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}+\frac {2}{3} b c^3 (a+b \text {arctanh}(c x)) \log \left (2-\frac {2}{1+c x}\right )-\frac {1}{3} b^2 c^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1+c x}\right ) \]
-1/3*b^2*c^2/x+1/3*b^2*c^3*arctanh(c*x)-1/3*b*c*(a+b*arctanh(c*x))/x^2+1/3 *c^3*(a+b*arctanh(c*x))^2-1/3*(a+b*arctanh(c*x))^2/x^3+2/3*b*c^3*(a+b*arct anh(c*x))*ln(2-2/(c*x+1))-1/3*b^2*c^3*polylog(2,-1+2/(c*x+1))
Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=-\frac {a^2+a b c x+b^2 c^2 x^2+b^2 \left (1-c^3 x^3\right ) \text {arctanh}(c x)^2+b \text {arctanh}(c x) \left (2 a+b c x-b c^3 x^3-2 b c^3 x^3 \log \left (1-e^{-2 \text {arctanh}(c x)}\right )\right )-2 a b c^3 x^3 \log (c x)+a b c^3 x^3 \log \left (1-c^2 x^2\right )+b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{-2 \text {arctanh}(c x)}\right )}{3 x^3} \]
-1/3*(a^2 + a*b*c*x + b^2*c^2*x^2 + b^2*(1 - c^3*x^3)*ArcTanh[c*x]^2 + b*A rcTanh[c*x]*(2*a + b*c*x - b*c^3*x^3 - 2*b*c^3*x^3*Log[1 - E^(-2*ArcTanh[c *x])]) - 2*a*b*c^3*x^3*Log[c*x] + a*b*c^3*x^3*Log[1 - c^2*x^2] + b^2*c^3*x ^3*PolyLog[2, E^(-2*ArcTanh[c*x])])/x^3
Time = 0.79 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6452, 6544, 6452, 264, 219, 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.
| \(\displaystyle \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {2}{3} b c \int \frac {a+b \text {arctanh}(c x)}{x^3 \left (1-c^2 x^2\right )}dx-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6544 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\int \frac {a+b \text {arctanh}(c x)}{x^3}dx\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6452 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx+\frac {1}{2} b c \left (c^2 \int \frac {1}{1-c^2 x^2}dx-\frac {1}{x}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \int \frac {a+b \text {arctanh}(c x)}{x \left (1-c^2 x^2\right )}dx-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6550 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \left (\int \frac {a+b \text {arctanh}(c x)}{x (c x+1)}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 6494 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \left (-b c \int \frac {\log \left (2-\frac {2}{c x+1}\right )}{1-c^2 x^2}dx+\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
| \(\Big \downarrow \) 2897 |
| \(\displaystyle \frac {2}{3} b c \left (c^2 \left (\frac {(a+b \text {arctanh}(c x))^2}{2 b}+\log \left (2-\frac {2}{c x+1}\right ) (a+b \text {arctanh}(c x))-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {2}{c x+1}-1\right )\right )-\frac {a+b \text {arctanh}(c x)}{2 x^2}+\frac {1}{2} b c \left (c \text {arctanh}(c x)-\frac {1}{x}\right )\right )-\frac {(a+b \text {arctanh}(c x))^2}{3 x^3}\) |
-1/3*(a + b*ArcTanh[c*x])^2/x^3 + (2*b*c*(-1/2*(a + b*ArcTanh[c*x])/x^2 + (b*c*(-x^(-1) + c*ArcTanh[c*x]))/2 + c^2*((a + b*ArcTanh[c*x])^2/(2*b) + ( a + b*ArcTanh[c*x])*Log[2 - 2/(1 + c*x)] - (b*PolyLog[2, -1 + 2/(1 + c*x)] )/2)))/3
3.1.22.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, 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_.)*((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]
Leaf count of result is larger than twice the leaf count of optimal. \(246\) vs. \(2(116)=232\).
Time = 1.02 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.90
| method | result | size |
| parts | \(-\frac {a^{2}}{3 x^{3}}+b^{2} c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}\right )+2 a b \,c^{3} \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {1}{6 c^{2} x^{2}}+\frac {\ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}\right )\) | \(247\) |
| derivativedivides | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {1}{6 c^{2} x^{2}}+\frac {\ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}\right )\right )\) | \(248\) |
| default | \(c^{3} \left (-\frac {a^{2}}{3 c^{3} x^{3}}+b^{2} \left (-\frac {\operatorname {arctanh}\left (c x \right )^{2}}{3 c^{3} x^{3}}-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{2} x^{2}}+\frac {2 \ln \left (c x \right ) \operatorname {arctanh}\left (c x \right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {arctanh}\left (c x \right ) \ln \left (c x -1\right )}{3}-\frac {1}{3 c x}+\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}+\frac {\operatorname {dilog}\left (\frac {c x}{2}+\frac {1}{2}\right )}{3}+\frac {\ln \left (c x -1\right ) \ln \left (\frac {c x}{2}+\frac {1}{2}\right )}{6}-\frac {\ln \left (c x -1\right )^{2}}{12}-\frac {\left (\ln \left (c x +1\right )-\ln \left (\frac {c x}{2}+\frac {1}{2}\right )\right ) \ln \left (-\frac {c x}{2}+\frac {1}{2}\right )}{6}+\frac {\ln \left (c x +1\right )^{2}}{12}-\frac {\operatorname {dilog}\left (c x +1\right )}{3}-\frac {\ln \left (c x \right ) \ln \left (c x +1\right )}{3}-\frac {\operatorname {dilog}\left (c x \right )}{3}\right )+2 a b \left (-\frac {\operatorname {arctanh}\left (c x \right )}{3 c^{3} x^{3}}-\frac {1}{6 c^{2} x^{2}}+\frac {\ln \left (c x \right )}{3}-\frac {\ln \left (c x +1\right )}{6}-\frac {\ln \left (c x -1\right )}{6}\right )\right )\) | \(248\) |
-1/3*a^2/x^3+b^2*c^3*(-1/3/c^3/x^3*arctanh(c*x)^2-1/3/c^2/x^2*arctanh(c*x) +2/3*ln(c*x)*arctanh(c*x)-1/3*arctanh(c*x)*ln(c*x+1)-1/3*arctanh(c*x)*ln(c *x-1)-1/3/c/x+1/6*ln(c*x+1)-1/6*ln(c*x-1)+1/3*dilog(1/2*c*x+1/2)+1/6*ln(c* x-1)*ln(1/2*c*x+1/2)-1/12*ln(c*x-1)^2-1/6*(ln(c*x+1)-ln(1/2*c*x+1/2))*ln(- 1/2*c*x+1/2)+1/12*ln(c*x+1)^2-1/3*dilog(c*x+1)-1/3*ln(c*x)*ln(c*x+1)-1/3*d ilog(c*x))+2*a*b*c^3*(-1/3/c^3/x^3*arctanh(c*x)-1/6/c^2/x^2+1/3*ln(c*x)-1/ 6*ln(c*x+1)-1/6*ln(c*x-1))
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {\left (a + b \operatorname {atanh}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \]
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
-1/3*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3 )*a*b - 1/12*b^2*(log(-c*x + 1)^2/x^3 + 3*integrate(-1/3*(3*(c*x - 1)*log( c*x + 1)^2 + 2*(c*x - 3*(c*x - 1)*log(c*x + 1))*log(-c*x + 1))/(c*x^5 - x^ 4), x)) - 1/3*a^2/x^3
\[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int { \frac {{\left (b \operatorname {artanh}\left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]
Timed out. \[ \int \frac {(a+b \text {arctanh}(c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x\right )\right )}^2}{x^4} \,d x \]