Integrand size = 16, antiderivative size = 832 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx =\text {Too large to display} \] Output:
1/2*(-b*x-a+1)*ln(-b*x-a+1)/b/c+1/2*(b*x+a+1)*ln(b*x+a+1)/b/c-1/6*d^(1/3)* ln(b*x+a+1)*ln(-b*(d^(1/3)+c^(1/3)*x)/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1 /6*d^(1/3)*ln(-b*x-a+1)*ln(b*(d^(1/3)+c^(1/3)*x)/((1-a)*c^(1/3)+b*d^(1/3)) )/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*ln(-b*x-a+1)*ln(-b*(d^(1/3)-(-1)^(1/3)*c^ (1/3)*x)/((-1)^(1/3)*(1-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(-1)^(2/3)*d^(1 /3)*ln(b*x+a+1)*ln(b*(d^(1/3)-(-1)^(1/3)*c^(1/3)*x)/((-1)^(1/3)*(1+a)*c^(1 /3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*ln(b*x+a+1)*ln(-b*(d^(1/3)+ (-1)^(2/3)*c^(1/3)*x)/((-1)^(2/3)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6*(- 1)^(1/3)*d^(1/3)*ln(-b*x-a+1)*ln(b*(d^(1/3)+(-1)^(2/3)*c^(1/3)*x)/((-1)^(2 /3)*(1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)+1/6*(-1)^(2/3)*d^(1/3)*polylog(2,(-1 )^(1/3)*c^(1/3)*(-b*x-a+1)/((-1)^(1/3)*(1-a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1 /6*d^(1/3)*polylog(2,c^(1/3)*(-b*x-a+1)/((1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3) -1/6*(-1)^(1/3)*d^(1/3)*polylog(2,(-1)^(2/3)*c^(1/3)*(-b*x-a+1)/((-1)^(2/3 )*(1-a)*c^(1/3)+b*d^(1/3)))/c^(4/3)-1/6*d^(1/3)*polylog(2,c^(1/3)*(b*x+a+1 )/((1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)+1/6*(-1)^(1/3)*d^(1/3)*polylog(2,(-1) ^(2/3)*c^(1/3)*(b*x+a+1)/((-1)^(2/3)*(1+a)*c^(1/3)-b*d^(1/3)))/c^(4/3)-1/6 *(-1)^(2/3)*d^(1/3)*polylog(2,(-1)^(1/3)*c^(1/3)*(b*x+a+1)/((-1)^(1/3)*(1+ a)*c^(1/3)+b*d^(1/3)))/c^(4/3)
Time = 0.67 (sec) , antiderivative size = 791, normalized size of antiderivative = 0.95 \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\frac {3 \sqrt [3]{c} \log (1-a-b x)-3 a \sqrt [3]{c} \log (1-a-b x)-3 b \sqrt [3]{c} x \log (1-a-b x)+3 \sqrt [3]{c} \log (1+a+b x)+3 a \sqrt [3]{c} \log (1+a+b x)+3 b \sqrt [3]{c} x \log (1+a+b x)+b \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{-\left ((-1+a) \sqrt [3]{c}\right )+b \sqrt [3]{d}}\right )-b \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}+\sqrt [3]{c} x\right )}{-\left ((1+a) \sqrt [3]{c}\right )+b \sqrt [3]{d}}\right )+(-1)^{2/3} b \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (-1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )-(-1)^{2/3} b \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )-\sqrt [3]{-1} b \sqrt [3]{d} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{-(-1)^{2/3} (-1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )+\sqrt [3]{-1} b \sqrt [3]{d} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{d}+(-1)^{2/3} \sqrt [3]{c} x\right )}{-(-1)^{2/3} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )+b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (-1+a+b x)}{(-1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )-\sqrt [3]{-1} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (-1+a+b x)}{(-1)^{2/3} (-1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )+(-1)^{2/3} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (-1+a+b x)}{\sqrt [3]{-1} (-1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )-b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (1+a+b x)}{(1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )+\sqrt [3]{-1} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (1+a+b x)}{(-1)^{2/3} (1+a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )-(-1)^{2/3} b \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (1+a+b x)}{\sqrt [3]{-1} (1+a) \sqrt [3]{c}+b \sqrt [3]{d}}\right )}{6 b c^{4/3}} \] Input:
Integrate[ArcTanh[a + b*x]/(c + d/x^3),x]
Output:
(3*c^(1/3)*Log[1 - a - b*x] - 3*a*c^(1/3)*Log[1 - a - b*x] - 3*b*c^(1/3)*x *Log[1 - a - b*x] + 3*c^(1/3)*Log[1 + a + b*x] + 3*a*c^(1/3)*Log[1 + a + b *x] + 3*b*c^(1/3)*x*Log[1 + a + b*x] + b*d^(1/3)*Log[1 - a - b*x]*Log[(b*( d^(1/3) + c^(1/3)*x))/(-((-1 + a)*c^(1/3)) + b*d^(1/3))] - b*d^(1/3)*Log[1 + a + b*x]*Log[(b*(d^(1/3) + c^(1/3)*x))/(-((1 + a)*c^(1/3)) + b*d^(1/3)) ] + (-1)^(2/3)*b*d^(1/3)*Log[1 - a - b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^( 1/3)*x))/((-1)^(1/3)*(-1 + a)*c^(1/3) + b*d^(1/3))] - (-1)^(2/3)*b*d^(1/3) *Log[1 + a + b*x]*Log[(b*(d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(1 + a)*c^(1/3) + b*d^(1/3))] - (-1)^(1/3)*b*d^(1/3)*Log[1 - a - b*x]*Log[(b* (d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/(-((-1)^(2/3)*(-1 + a)*c^(1/3)) + b*d^(1 /3))] + (-1)^(1/3)*b*d^(1/3)*Log[1 + a + b*x]*Log[(b*(d^(1/3) + (-1)^(2/3) *c^(1/3)*x))/(-((-1)^(2/3)*(1 + a)*c^(1/3)) + b*d^(1/3))] + b*d^(1/3)*Poly Log[2, (c^(1/3)*(-1 + a + b*x))/((-1 + a)*c^(1/3) - b*d^(1/3))] - (-1)^(1/ 3)*b*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(-1 + a + b*x))/((-1)^(2/3)*(- 1 + a)*c^(1/3) - b*d^(1/3))] + (-1)^(2/3)*b*d^(1/3)*PolyLog[2, ((-1)^(1/3) *c^(1/3)*(-1 + a + b*x))/((-1)^(1/3)*(-1 + a)*c^(1/3) + b*d^(1/3))] - b*d^ (1/3)*PolyLog[2, (c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) - b*d^(1/3))] + (-1)^(1/3)*b*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(1 + a + b*x))/((-1)^( 2/3)*(1 + a)*c^(1/3) - b*d^(1/3))] - (-1)^(2/3)*b*d^(1/3)*PolyLog[2, ((-1) ^(1/3)*c^(1/3)*(1 + a + b*x))/((-1)^(1/3)*(1 + a)*c^(1/3) + b*d^(1/3))]...
Time = 1.57 (sec) , antiderivative size = 847, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6665, 2856, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx\) |
| \(\Big \downarrow \) 6665 |
| \(\displaystyle \frac {1}{2} \int \frac {\log (a+b x+1)}{c+\frac {d}{x^3}}dx-\frac {1}{2} \int \frac {\log (-a-b x+1)}{c+\frac {d}{x^3}}dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log (a+b x+1)}{c}-\frac {d \log (a+b x+1)}{c \left (c x^3+d\right )}\right )dx-\frac {1}{2} \int \left (\frac {\log (-a-b x+1)}{c}-\frac {d \log (-a-b x+1)}{c \left (c x^3+d\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {x}{c}+\frac {(-a-b x+1) \log (-a-b x+1)}{b c}+\frac {\sqrt [3]{d} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \log (-a-b x+1) \log \left (-\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (-a-b x+1) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}+\frac {(-1)^{2/3} \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{-1} (1-a) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}+\frac {\sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (-a-b x+1)}{\sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}-\frac {\sqrt [3]{-1} \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{c} (1-a)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}\right )+\frac {1}{2} \left (-\frac {x}{c}+\frac {(a+b x+1) \log (a+b x+1)}{b c}-\frac {\sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left (\sqrt [3]{c} x+\sqrt [3]{d}\right )}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{d}-\sqrt [3]{-1} \sqrt [3]{c} x\right )}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \log (a+b x+1) \log \left (-\frac {b \left ((-1)^{2/3} \sqrt [3]{c} x+\sqrt [3]{d}\right )}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}-\frac {\sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{c} (a+b x+1)}{(a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}+\frac {\sqrt [3]{-1} \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{c} (a+b x+1)}{(-1)^{2/3} (a+1) \sqrt [3]{c}-b \sqrt [3]{d}}\right )}{3 c^{4/3}}-\frac {(-1)^{2/3} \sqrt [3]{d} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{c} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{c} (a+1)+b \sqrt [3]{d}}\right )}{3 c^{4/3}}\right )\) |
Input:
Int[ArcTanh[a + b*x]/(c + d/x^3),x]
Output:
(x/c + ((1 - a - b*x)*Log[1 - a - b*x])/(b*c) + (d^(1/3)*Log[1 - a - b*x]* Log[(b*(d^(1/3) + c^(1/3)*x))/((1 - a)*c^(1/3) + b*d^(1/3))])/(3*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*Log[1 - a - b*x]*Log[-((b*(d^(1/3) - (-1)^(1/3)*c^(1 /3)*x))/((-1)^(1/3)*(1 - a)*c^(1/3) - b*d^(1/3)))])/(3*c^(4/3)) - ((-1)^(1 /3)*d^(1/3)*Log[1 - a - b*x]*Log[(b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1 )^(2/3)*(1 - a)*c^(1/3) + b*d^(1/3))])/(3*c^(4/3)) + ((-1)^(2/3)*d^(1/3)*P olyLog[2, ((-1)^(1/3)*c^(1/3)*(1 - a - b*x))/((-1)^(1/3)*(1 - a)*c^(1/3) - b*d^(1/3))])/(3*c^(4/3)) + (d^(1/3)*PolyLog[2, (c^(1/3)*(1 - a - b*x))/(( 1 - a)*c^(1/3) + b*d^(1/3))])/(3*c^(4/3)) - ((-1)^(1/3)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)*(1 - a - b*x))/((-1)^(2/3)*(1 - a)*c^(1/3) + b*d^(1/3 ))])/(3*c^(4/3)))/2 + (-(x/c) + ((1 + a + b*x)*Log[1 + a + b*x])/(b*c) - ( d^(1/3)*Log[1 + a + b*x]*Log[-((b*(d^(1/3) + c^(1/3)*x))/((1 + a)*c^(1/3) - b*d^(1/3)))])/(3*c^(4/3)) - ((-1)^(2/3)*d^(1/3)*Log[1 + a + b*x]*Log[(b* (d^(1/3) - (-1)^(1/3)*c^(1/3)*x))/((-1)^(1/3)*(1 + a)*c^(1/3) + b*d^(1/3)) ])/(3*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*Log[1 + a + b*x]*Log[-((b*(d^(1/3) + (-1)^(2/3)*c^(1/3)*x))/((-1)^(2/3)*(1 + a)*c^(1/3) - b*d^(1/3)))])/(3*c^(4 /3)) - (d^(1/3)*PolyLog[2, (c^(1/3)*(1 + a + b*x))/((1 + a)*c^(1/3) - b*d^ (1/3))])/(3*c^(4/3)) + ((-1)^(1/3)*d^(1/3)*PolyLog[2, ((-1)^(2/3)*c^(1/3)* (1 + a + b*x))/((-1)^(2/3)*(1 + a)*c^(1/3) - b*d^(1/3))])/(3*c^(4/3)) - (( -1)^(2/3)*d^(1/3)*PolyLog[2, ((-1)^(1/3)*c^(1/3)*(1 + a + b*x))/((-1)^(...
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_. )*(x_)^(r_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x) ^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x] && I GtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))
Int[ArcTanh[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Simp [1/2 Int[Log[1 + c + d*x]/(e + f*x^n), x], x] - Simp[1/2 Int[Log[1 - c - d*x]/(e + f*x^n), x], x] /; FreeQ[{c, d, e, f}, x] && RationalQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\ln \left (b x +a +1\right ) x}{2 c}+\frac {\ln \left (b x +a +1\right ) a}{2 b c}+\frac {\ln \left (b x +a +1\right )}{2 b c}-\frac {1}{b c}-\frac {b^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+\left (-3 a c -3 c \right ) \textit {\_Z}^{2}+\left (3 a^{2} c +6 a c +3 c \right ) \textit {\_Z} -a^{3} c +b^{3} d -3 a^{2} c -3 a c -c \right )}{\sum }\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}-2 \textit {\_R1} +2 a +1}\right )}{6 c^{2}}-\frac {\ln \left (-b x -a +1\right ) x}{2 c}-\frac {\ln \left (-b x -a +1\right ) a}{2 b c}+\frac {\ln \left (-b x -a +1\right )}{2 b c}+\frac {b^{2} d \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}+\left (3 a c -3 c \right ) \textit {\_Z}^{2}+\left (3 a^{2} c -6 a c +3 c \right ) \textit {\_Z} +a^{3} c -b^{3} d -3 a^{2} c +3 a c -c \right )}{\sum }\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {b x +\textit {\_R1} +a -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {b x +\textit {\_R1} +a -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}+2 \textit {\_R1} a +a^{2}-2 \textit {\_R1} -2 a +1}\right )}{6 c^{2}}\) | \(366\) |
| derivativedivides | \(\frac {\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{c}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) d \,b^{3}}{3 c^{2}}+\frac {\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}}{c}+\frac {d \,b^{3} \left (-\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )+3 c \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c -b^{3} d -3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 b^{3} d -3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 b^{3} d +3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -b^{3} d +3 a^{2} c +3 a c +c \right )}{\sum }\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -b^{3} d +2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c -b^{3} d -3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 b^{3} d -3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 b^{3} d +3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -b^{3} d +3 a^{2} c +3 a c +c \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {arctanh}\left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -b^{3} d +2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3}\right )\right )}{3 c^{2}}}{b}\) | \(784\) |
| default | \(\frac {\frac {\operatorname {arctanh}\left (b x +a \right ) \left (b x +a \right )}{c}+\frac {\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) d \,b^{3}}{3 c^{2}}+\frac {\frac {\ln \left (b x +a -1\right )}{2}+\frac {\ln \left (b x +a +1\right )}{2}}{c}+\frac {d \,b^{3} \left (-\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{3}-3 a c \,\textit {\_Z}^{2}+3 a^{2} c \textit {\_Z} -a^{3} c +b^{3} d \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )+3 c \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c -b^{3} d -3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 b^{3} d -3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 b^{3} d +3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -b^{3} d +3 a^{2} c +3 a c +c \right )}{\sum }\frac {\operatorname {arctanh}\left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -b^{3} d +2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} c -b^{3} d -3 a^{2} c +3 a c -c \right ) \textit {\_Z}^{6}+\left (3 a^{3} c -3 b^{3} d -3 a^{2} c -3 a c +3 c \right ) \textit {\_Z}^{4}+\left (3 a^{3} c -3 b^{3} d +3 a^{2} c -3 a c -3 c \right ) \textit {\_Z}^{2}+a^{3} c -b^{3} d +3 a^{2} c +3 a c +c \right )}{\sum }\frac {\textit {\_R1}^{2} \left (\operatorname {arctanh}\left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {b x +a +1}{\sqrt {1-\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{4} a^{3} c -\textit {\_R1}^{4} b^{3} d -3 \textit {\_R1}^{4} a^{2} c +3 \textit {\_R1}^{4} a c +2 \textit {\_R1}^{2} a^{3} c -2 \textit {\_R1}^{2} b^{3} d -\textit {\_R1}^{4} c -2 \textit {\_R1}^{2} a^{2} c -2 \textit {\_R1}^{2} a c +a^{3} c -b^{3} d +2 \textit {\_R1}^{2} c +a^{2} c -a c -c}\right )}{3}\right )\right )}{3 c^{2}}}{b}\) | \(784\) |
Input:
int(arctanh(b*x+a)/(c+d/x^3),x,method=_RETURNVERBOSE)
Output:
1/2/c*ln(b*x+a+1)*x+1/2/b/c*ln(b*x+a+1)*a+1/2/b/c*ln(b*x+a+1)-1/b/c-1/6*b^ 2*d/c^2*sum(1/(_R1^2-2*_R1*a+a^2-2*_R1+2*a+1)*(ln(b*x+a+1)*ln((-b*x+_R1-a- 1)/_R1)+dilog((-b*x+_R1-a-1)/_R1)),_R1=RootOf(c*_Z^3+(-3*a*c-3*c)*_Z^2+(3* a^2*c+6*a*c+3*c)*_Z-a^3*c+b^3*d-3*a^2*c-3*a*c-c))-1/2/c*ln(-b*x-a+1)*x-1/2 /b/c*ln(-b*x-a+1)*a+1/2/b/c*ln(-b*x-a+1)+1/6*b^2*d/c^2*sum(1/(_R1^2+2*_R1* a+a^2-2*_R1-2*a+1)*(ln(-b*x-a+1)*ln((b*x+_R1+a-1)/_R1)+dilog((b*x+_R1+a-1) /_R1)),_R1=RootOf(c*_Z^3+(3*a*c-3*c)*_Z^2+(3*a^2*c-6*a*c+3*c)*_Z+a^3*c-b^3 *d-3*a^2*c+3*a*c-c))
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^3),x, algorithm="fricas")
Output:
integral(x^3*arctanh(b*x + a)/(c*x^3 + d), x)
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\text {Timed out} \] Input:
integrate(atanh(b*x+a)/(c+d/x**3),x)
Output:
Timed out
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^3),x, algorithm="maxima")
Output:
integrate(arctanh(b*x + a)/(c + d/x^3), x)
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{c + \frac {d}{x^{3}}} \,d x } \] Input:
integrate(arctanh(b*x+a)/(c+d/x^3),x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\int \frac {\mathrm {atanh}\left (a+b\,x\right )}{c+\frac {d}{x^3}} \,d x \] Input:
int(atanh(a + b*x)/(c + d/x^3),x)
Output:
int(atanh(a + b*x)/(c + d/x^3), x)
\[ \int \frac {\text {arctanh}(a+b x)}{c+\frac {d}{x^3}} \, dx=\int \frac {\mathit {atanh} \left (b x +a \right ) x^{3}}{c \,x^{3}+d}d x \] Input:
int(atanh(b*x+a)/(c+d/x^3),x)
Output:
int((atanh(a + b*x)*x**3)/(c*x**3 + d),x)