Integrand size = 16, antiderivative size = 780 \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=-\frac {\log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (1-a-b x)}{b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,-\frac {(-1)^{2/3} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
-1/6*ln(-b*x-a+1)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)+(1-a)*d^(1/3)))/c^(2 /3)/d^(1/3)+1/6*ln(b*x+a+1)*ln(b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)-(1+a)*d^(1 /3)))/c^(2/3)/d^(1/3)-1/6*(-1)^(2/3)*ln(-b*x-a+1)*ln(b*(c^(1/3)-(-1)^(1/3) *d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(1-a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1) ^(2/3)*ln(b*x+a+1)*ln(b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(1/ 3)*(1+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(1/3)*ln(-b*x-a+1)*ln(b*(c^(1/ 3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(1-a)*d^(1/3)))/c^(2/3)/d^( 1/3)-1/6*(-1)^(1/3)*ln(b*x+a+1)*ln(b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^( 1/3)-(-1)^(2/3)*(1+a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*polylog(2,d^(1/3)*(-b* x-a+1)/(b*c^(1/3)+(1-a)*d^(1/3)))/c^(2/3)/d^(1/3)-1/6*(-1)^(2/3)*polylog(2 ,-(-1)^(1/3)*d^(1/3)*(-b*x-a+1)/(b*c^(1/3)-(-1)^(1/3)*(1-a)*d^(1/3)))/c^(2 /3)/d^(1/3)+1/6*(-1)^(1/3)*polylog(2,(-1)^(2/3)*d^(1/3)*(-b*x-a+1)/(b*c^(1 /3)+(-1)^(2/3)*(1-a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*polylog(2,-d^(1/3)*(b*x +a+1)/(b*c^(1/3)-(1+a)*d^(1/3)))/c^(2/3)/d^(1/3)+1/6*(-1)^(2/3)*polylog(2, (-1)^(1/3)*d^(1/3)*(b*x+a+1)/(b*c^(1/3)+(-1)^(1/3)*(1+a)*d^(1/3)))/c^(2/3) /d^(1/3)-1/6*(-1)^(1/3)*polylog(2,-(-1)^(2/3)*d^(1/3)*(b*x+a+1)/(b*c^(1/3) -(-1)^(2/3)*(1+a)*d^(1/3)))/c^(2/3)/d^(1/3)
Time = 0.53 (sec) , antiderivative size = 623, normalized size of antiderivative = 0.80 \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\frac {-\log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1+a) \sqrt [3]{d}}\right )+\log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )-(-1)^{2/3} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (-1+a) \sqrt [3]{d}}\right )+(-1)^{2/3} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )+\sqrt [3]{-1} \log (1-a-b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (-1+a) \sqrt [3]{d}}\right )-\sqrt [3]{-1} \log (1+a+b x) \log \left (\frac {b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )-\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (-1+a+b x)}{b \sqrt [3]{c}-(-1+a) \sqrt [3]{d}}\right )-(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (-1+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (-1+a) \sqrt [3]{d}}\right )+\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (-1+a+b x)}{-b \sqrt [3]{c}+(-1)^{2/3} (-1+a) \sqrt [3]{d}}\right )+\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}-(1+a) \sqrt [3]{d}}\right )+(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (1+a+b x)}{b \sqrt [3]{c}+\sqrt [3]{-1} (1+a) \sqrt [3]{d}}\right )-\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (1+a+b x)}{-b \sqrt [3]{c}+(-1)^{2/3} (1+a) \sqrt [3]{d}}\right )}{6 c^{2/3} \sqrt [3]{d}} \]
(-(Log[1 - a - b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (-1 + a)*d^ (1/3))]) + Log[1 + a + b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (1 + a)*d^(1/3))] - (-1)^(2/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)* d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(-1 + a)*d^(1/3))] + (-1)^(2/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3 )*(1 + a)*d^(1/3))] + (-1)^(1/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) + (-1)^( 2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(-1 + a)*d^(1/3))] - (-1)^(1/3)*L og[1 + a + b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1) ^(2/3)*(1 + a)*d^(1/3))] - PolyLog[2, -((d^(1/3)*(-1 + a + b*x))/(b*c^(1/3 ) - (-1 + a)*d^(1/3)))] - (-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(-1 + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(-1 + a)*d^(1/3))] + (-1)^(1/3)*PolyLog[ 2, ((-1)^(2/3)*d^(1/3)*(-1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(-1 + a) *d^(1/3))] + PolyLog[2, -((d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^( 1/3)))] + (-1)^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1 /3) + (-1)^(1/3)*(1 + a)*d^(1/3))] - (-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^( 1/3)*(1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(1 + a)*d^(1/3))])/(6*c^(2/ 3)*d^(1/3))
Time = 1.48 (sec) , antiderivative size = 800, normalized size of antiderivative = 1.03, 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.
| \(\displaystyle \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 6665 |
| \(\displaystyle \frac {1}{2} \int \frac {\log (a+b x+1)}{d x^3+c}dx-\frac {1}{2} \int \frac {\log (-a-b x+1)}{d x^3+c}dx\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} \int \left (-\frac {\log (a+b x+1)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x+1)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x+1)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx-\frac {1}{2} \int \left (-\frac {\log (-a-b x+1)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-a-b x+1)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-a-b x+1)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \log (-a-b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \log (-a-b x+1) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{(-1)^{2/3} \sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt [3]{d} (-a-b x+1)}{\sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{-1} \sqrt [3]{d} (-a-b x+1)}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,\frac {(-1)^{2/3} \sqrt [3]{d} (-a-b x+1)}{(-1)^{2/3} \sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )+\frac {1}{2} \left (\frac {\log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (a+b x+1) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} (a+1)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (a+b x+1) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (a+b x+1)}{b \sqrt [3]{c}-(a+1) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,\frac {\sqrt [3]{-1} \sqrt [3]{d} (a+b x+1)}{\sqrt [3]{-1} \sqrt [3]{d} (a+1)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} \left ((-1)^{2/3} a+(-1)^{2/3} b x+(-1)^{2/3}\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\) |
(-1/3*(Log[1 - a - b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (1 - a) *d^(1/3))])/(c^(2/3)*d^(1/3)) - ((-1)^(2/3)*Log[1 - a - b*x]*Log[(b*(c^(1/ 3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*(1 - a)*d^(1/3))])/(3* c^(2/3)*d^(1/3)) + ((-1)^(1/3)*Log[1 - a - b*x]*Log[(b*(c^(1/3) + (-1)^(2/ 3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(1 - a)*d^(1/3))])/(3*c^(2/3)*d^(1/ 3)) - PolyLog[2, (d^(1/3)*(1 - a - b*x))/(b*c^(1/3) + (1 - a)*d^(1/3))]/(3 *c^(2/3)*d^(1/3)) - ((-1)^(2/3)*PolyLog[2, -(((-1)^(1/3)*d^(1/3)*(1 - a - b*x))/(b*c^(1/3) - (-1)^(1/3)*(1 - a)*d^(1/3)))])/(3*c^(2/3)*d^(1/3)) + (( -1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(1 - a - b*x))/(b*c^(1/3) + (-1)^ (2/3)*(1 - a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)))/2 + ((Log[1 + a + b*x]*Log[( b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) - (1 + a)*d^(1/3))])/(3*c^(2/3)*d^(1/3 )) + ((-1)^(2/3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x)) /(b*c^(1/3) + (-1)^(1/3)*(1 + a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)) - ((-1)^(1 /3)*Log[1 + a + b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(2/3)*(1 + a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)) + PolyLog[2, -((d^(1/3) *(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^(1/3)))]/(3*c^(2/3)*d^(1/3)) + ((-1 )^(2/3)*PolyLog[2, ((-1)^(1/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1/3) + (-1)^(1 /3)*(1 + a)*d^(1/3))])/(3*c^(2/3)*d^(1/3)) - ((-1)^(1/3)*PolyLog[2, -((d^( 1/3)*((-1)^(2/3) + (-1)^(2/3)*a + (-1)^(2/3)*b*x))/(b*c^(1/3) - (-1)^(2/3) *(1 + a)*d^(1/3)))])/(3*c^(2/3)*d^(1/3)))/2
3.1.53.3.1 Defintions of rubi rules used
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 = 0.42 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(-\frac {b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (3 a d -3 d \right ) \textit {\_Z}^{2}+\left (3 a^{2} d -6 a d +3 d \right ) \textit {\_Z} +d \,a^{3}-b^{3} c -3 a^{2} d +3 a d -d \right )}{\sum }\frac {\ln \left (-b x -a +1\right ) \ln \left (\frac {b x +\textit {\_R1} +a -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {b x +\textit {\_R1} +a -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}+2 \textit {\_R1} a +a^{2}-2 \textit {\_R1} -2 a +1}\right )}{6 d}+\frac {b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -3 d \right ) \textit {\_Z}^{2}+\left (3 a^{2} d +6 a d +3 d \right ) \textit {\_Z} -d \,a^{3}+b^{3} c -3 a^{2} d -3 a d -d \right )}{\sum }\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}-2 \textit {\_R1} +2 a +1}\right )}{6 d}\) | \(259\) |
| derivativedivides | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} a d +3 \textit {\_Z} \,a^{2} d -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) \operatorname {arctanh}\left (b x +a \right )}{3 d}+\frac {b^{3} \left (\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} a d +3 \textit {\_Z} \,a^{2} d -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 d \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}-b^{3} c -3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}-3 b^{3} c -3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 b^{3} c +3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+d \,a^{3}-b^{3} c +3 a^{2} d +3 a d +d \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} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +d \,a^{3}-b^{3} c +2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}-b^{3} c -3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}-3 b^{3} c -3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 b^{3} c +3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+d \,a^{3}-b^{3} c +3 a^{2} d +3 a d +d \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} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +d \,a^{3}-b^{3} c +2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(743\) |
| default | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} a d +3 \textit {\_Z} \,a^{2} d -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right ) \operatorname {arctanh}\left (b x +a \right )}{3 d}+\frac {b^{3} \left (\operatorname {arctanh}\left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 \textit {\_Z}^{2} a d +3 \textit {\_Z} \,a^{2} d -d \,a^{3}+b^{3} c \right )}{\sum }\frac {\ln \left (b x -\textit {\_R} +a \right )}{-\textit {\_R}^{2}+2 \textit {\_R} a -a^{2}}\right )-3 d \left (\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}-b^{3} c -3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}-3 b^{3} c -3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 b^{3} c +3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+d \,a^{3}-b^{3} c +3 a^{2} d +3 a d +d \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} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +d \,a^{3}-b^{3} c +2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (d \,a^{3}-b^{3} c -3 a^{2} d +3 a d -d \right ) \textit {\_Z}^{6}+\left (3 d \,a^{3}-3 b^{3} c -3 a^{2} d -3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 d \,a^{3}-3 b^{3} c +3 a^{2} d -3 a d -3 d \right ) \textit {\_Z}^{2}+d \,a^{3}-b^{3} c +3 a^{2} d +3 a d +d \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} d -\textit {\_R1}^{4} b^{3} c -3 \textit {\_R1}^{4} a^{2} d +3 \textit {\_R1}^{4} a d +2 \textit {\_R1}^{2} a^{3} d -2 \textit {\_R1}^{2} b^{3} c -\textit {\_R1}^{4} d -2 \textit {\_R1}^{2} a^{2} d -2 \textit {\_R1}^{2} a d +d \,a^{3}-b^{3} c +2 \textit {\_R1}^{2} d +a^{2} d -a d -d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(743\) |
-1/6*b^2/d*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(d*_Z^3+(3*a*d-3*d)*_Z^2+(3 *a^2*d-6*a*d+3*d)*_Z+d*a^3-b^3*c-3*a^2*d+3*a*d-d))+1/6*b^2/d*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(d*_Z^3+(-3*a*d-3*d)*_Z^2+(3*a^2*d+6*a*d+3*d)*_Z-d *a^3+b^3*c-3*a^2*d-3*a*d-d))
\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{3} + c} \,d x } \]
\[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {artanh}\left (b x + a\right )}{d x^{3} + c} \,d x } \]
Timed out. \[ \int \frac {\text {arctanh}(a+b x)}{c+d x^3} \, dx=\int \frac {\mathrm {atanh}\left (a+b\,x\right )}{d\,x^3+c} \,d x \]