Integrand size = 16, antiderivative size = 649 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=-\frac {\coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \coth ^{-1}(a+b x) \log \left (\frac {2}{1+a+b x}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\coth ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \coth ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \coth ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,1-\frac {2}{1+a+b x}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(-1)^{2/3} (1-a) \sqrt [3]{d}\right ) (1+a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}} \] Output:
-1/3*arccoth(b*x+a)*ln(2/(b*x+a+1))/c^(2/3)/d^(1/3)+1/3*(-1)^(1/3)*arccoth (b*x+a)*ln(2/(b*x+a+1))/c^(2/3)/d^(1/3)-1/3*(-1)^(2/3)*arccoth(b*x+a)*ln(2 /(b*x+a+1))/c^(2/3)/d^(1/3)+1/3*arccoth(b*x+a)*ln(2*b*(c^(1/3)+d^(1/3)*x)/ (b*c^(1/3)+(1-a)*d^(1/3))/(b*x+a+1))/c^(2/3)/d^(1/3)+1/3*(-1)^(2/3)*arccot h(b*x+a)*ln(2*b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(1-a) *d^(1/3))/(b*x+a+1))/c^(2/3)/d^(1/3)-1/3*(-1)^(1/3)*arccoth(b*x+a)*ln(2*b* (c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(1-a)*d^(1/3))/(b*x+a +1))/c^(2/3)/d^(1/3)+1/6*polylog(2,1-2/(b*x+a+1))/c^(2/3)/d^(1/3)-1/6*(-1) ^(1/3)*polylog(2,1-2/(b*x+a+1))/c^(2/3)/d^(1/3)+1/6*(-1)^(2/3)*polylog(2,1 -2/(b*x+a+1))/c^(2/3)/d^(1/3)-1/6*polylog(2,1-2*b*(c^(1/3)+d^(1/3)*x)/(b*c ^(1/3)+(1-a)*d^(1/3))/(b*x+a+1))/c^(2/3)/d^(1/3)-1/6*(-1)^(2/3)*polylog(2, 1-2*b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^(1/3)*(1-a)*d^(1/3))/ (b*x+a+1))/c^(2/3)/d^(1/3)+1/6*(-1)^(1/3)*polylog(2,1-2*b*(c^(1/3)+(-1)^(2 /3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(1-a)*d^(1/3))/(b*x+a+1))/c^(2/3)/d^( 1/3)
Time = 0.75 (sec) , antiderivative size = 931, normalized size of antiderivative = 1.43 \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx =\text {Too large to display} \] Input:
Integrate[ArcCoth[a + b*x]/(c + d*x^3),x]
Output:
(Log[-((d^(1/3)*(-1 + a + b*x))/(b*c^(1/3) - (-1 + a)*d^(1/3)))]*Log[-c^(1 /3) - d^(1/3)*x] - Log[(-1 + a + b*x)/(a + b*x)]*Log[-c^(1/3) - d^(1/3)*x] - Log[-((d^(1/3)*(1 + a + b*x))/(b*c^(1/3) - (1 + a)*d^(1/3)))]*Log[-c^(1 /3) - d^(1/3)*x] + Log[(1 + a + b*x)/(a + b*x)]*Log[-c^(1/3) - d^(1/3)*x] + (-1)^(2/3)*Log[((-1)^(1/3)*d^(1/3)*(-1 + a + b*x))/(b*c^(1/3) + (-1)^(1/ 3)*(-1 + a)*d^(1/3))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] - (-1)^(2/3)*Lo g[(-1 + a + b*x)/(a + b*x)]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] - (-1)^(2 /3)*Log[((-1)^(1/3)*d^(1/3)*(1 + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(1 + a) *d^(1/3))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] + (-1)^(2/3)*Log[(1 + a + b*x)/(a + b*x)]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] - (-1)^(1/3)*Log[((-1 )^(2/3)*d^(1/3)*(-1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(-1 + a)*d^(1/3 ))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] + (-1)^(1/3)*Log[(-1 + a + b*x)/( a + b*x)]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] + (-1)^(1/3)*Log[((-1)^(2/3 )*d^(1/3)*(1 + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(1 + a)*d^(1/3))]*Log[ -c^(1/3) - (-1)^(2/3)*d^(1/3)*x] - (-1)^(1/3)*Log[(1 + a + b*x)/(a + b*x)] *Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] + PolyLog[2, (b*(c^(1/3) + d^(1/3)*x ))/(b*c^(1/3) - (-1 + a)*d^(1/3))] - PolyLog[2, (b*(c^(1/3) + d^(1/3)*x))/ (b*c^(1/3) - (1 + a)*d^(1/3))] + (-1)^(2/3)*PolyLog[2, (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)* PolyLog[2, (b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)...
Leaf count is larger than twice the leaf count of optimal. \(1811\) vs. \(2(649)=1298\).
Time = 2.98 (sec) , antiderivative size = 1811, normalized size of antiderivative = 2.79, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6666, 2993, 750, 16, 1142, 25, 27, 1082, 217, 1103, 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.
| \(\displaystyle \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 6666 |
| \(\displaystyle \frac {1}{2} \int \frac {\log \left (\frac {a+b x+1}{a+b x}\right )}{d x^3+c}dx-\frac {1}{2} \int \frac {\log \left (-\frac {-a-b x+1}{a+b x}\right )}{d x^3+c}dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \int \frac {1}{d x^3+c}dx-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \int \frac {1}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} \left (\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx\right )+\frac {1}{2} \left (\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (-\int \frac {\log (-a-b x+1)}{d x^3+c}dx+\int \frac {\log (a+b x)}{d x^3+c}dx+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\frac {1}{2} \left (-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+1)}{d x^3+c}dx+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} \left (-\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+\int \left (-\frac {\log (a+b x)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\frac {1}{2} \left (-\int \left (-\frac {\log (a+b x)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx+\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+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )\) |
| \(\Big \downarrow \) 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 {\log (a+b x) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-a \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 )}{b \sqrt [3]{c}-\sqrt [3]{-1} (1-a) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \log (a+b x) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} a+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 )}{(-1)^{2/3} \sqrt [3]{d} (1-a)+b \sqrt [3]{c}}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \log (a+b x) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(-1)^{2/3} a \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\left (\log (-a-b x+1)-\log \left (-\frac {-a-b x+1}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\log \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}\right )-\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}}+\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (a+b x)}{b \sqrt [3]{c}-a \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)}{\sqrt [3]{-1} \sqrt [3]{d} a+b \sqrt [3]{c}}\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)}{b \sqrt [3]{c}-(-1)^{2/3} a \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )+\frac {1}{2} \left (-\frac {\log (a+b x) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-a \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\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) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} 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 )}{\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) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(-1)^{2/3} 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 )}{b \sqrt [3]{c}-(-1)^{2/3} (a+1) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\left (\log (a+b x)-\log (a+b x+1)+\log \left (\frac {a+b x+1}{a+b x}\right )\right ) \left (\frac {\log \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}\right )-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt [3]{d} (a+b x)}{b \sqrt [3]{c}-a \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)}{\sqrt [3]{-1} \sqrt [3]{d} a+b \sqrt [3]{c}}\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)}{b \sqrt [3]{c}-(-1)^{2/3} a \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 )\) |
Input:
Int[ArcCoth[a + b*x]/(c + d*x^3),x]
Output:
(-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)) + (Log[a + b*x]*Log[(b*(c^(1/3) + d^(1/3)*x) )/(b*c^(1/3) - 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)^(2/3)*Log[a + b*x]*Log[(b*(c^(1 /3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*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)) - ((-1)^(1/3)*Log[a + b*x]*Log[(b*(c^(1/3) + (-1)^(2/3)*d^(1/3)*x))/(b*c^(1 /3) - (-1)^(2/3)*a*d^(1/3))])/(3*c^(2/3)*d^(1/3)) + (Log[1 - a - b*x] - Lo g[-((1 - a - b*x)/(a + b*x))] - Log[a + b*x])*(Log[c^(1/3) + d^(1/3)*x]/(3 *c^(2/3)*d^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3] ])/d^(1/3)) - Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2]/(2*d^(1/3)))/ (3*c^(2/3))) - 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)) + PolyLog[2, -((d^(1 /3)*(a + b*x))/(b*c^(1/3) - a*d^(1/3)))]/(3*c^(2/3)*d^(1/3)) + ((-1)^(2/3) *PolyLog[2, ((-1)^(1/3)*d^(1/3)*(a + b*x))/(b*c^(1/3) + (-1)^(1/3)*a*d^...
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2) Int[1/ (Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2) Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
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[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) ^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r Int[RFx*Log[a + b*x], x], x] + (Si mp[q*r Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]) Int[RFx, x], x]) /; FreeQ[ {a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a *d, 0] && !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ m, n]]
Int[ArcCoth[(c_) + (d_.)*(x_)]/((e_) + (f_.)*(x_)^(n_.)), x_Symbol] :> Simp [1/2 Int[Log[(1 + c + d*x)/(c + d*x)]/(e + f*x^n), x], x] - Simp[1/2 In t[Log[(-1 + c + d*x)/(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 = 1.05 (sec) , antiderivative size = 260, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(-\frac {b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d +3 d \right ) \textit {\_Z}^{2}+\left (3 d \,a^{2}-6 a d +3 d \right ) \textit {\_Z} -a^{3} d +b^{3} c +3 d \,a^{2}-3 a d +d \right )}{\sum }\frac {\ln \left (b x +a -1\right ) \ln \left (\frac {-b x +\textit {\_R1} -a +1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x +\textit {\_R1} -a +1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}+2 \textit {\_R1} -2 a +1}\right )}{6 d}+\frac {b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 a d -3 d \right ) \textit {\_Z}^{2}+\left (3 d \,a^{2}+6 a d +3 d \right ) \textit {\_Z} -a^{3} d +b^{3} c -3 d \,a^{2}-3 a d -d \right )}{\sum }\frac {\ln \left (b x +a +1\right ) \ln \left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x +\textit {\_R1} -a -1}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} a +a^{2}-2 \textit {\_R1} +2 a +1}\right )}{6 d}\) | \(260\) |
| derivativedivides | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 d \,a^{2} \textit {\_Z} -a^{3} d +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 {arccoth}\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 a d \,\textit {\_Z}^{2}+3 d \,a^{2} \textit {\_Z} -a^{3} d +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 (a^{3} d -b^{3} c -3 d \,a^{2}+3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 b^{3} c -3 d \,a^{2}-3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 b^{3} c +3 d \,a^{2}-3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -b^{3} c +3 d \,a^{2}+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 +a^{3} d -b^{3} c +2 \textit {\_R1}^{2} d +d \,a^{2}-a d -d}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} d -b^{3} c -3 d \,a^{2}+3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 b^{3} c -3 d \,a^{2}-3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 b^{3} c +3 d \,a^{2}-3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -b^{3} c +3 d \,a^{2}+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 +a^{3} d -b^{3} c +2 \textit {\_R1}^{2} d +d \,a^{2}-a d -d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(744\) |
| default | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 d \,a^{2} \textit {\_Z} -a^{3} d +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 {arccoth}\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 a d \,\textit {\_Z}^{2}+3 d \,a^{2} \textit {\_Z} -a^{3} d +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 (a^{3} d -b^{3} c -3 d \,a^{2}+3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 b^{3} c -3 d \,a^{2}-3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 b^{3} c +3 d \,a^{2}-3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -b^{3} c +3 d \,a^{2}+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 +a^{3} d -b^{3} c +2 \textit {\_R1}^{2} d +d \,a^{2}-a d -d}\right )}{3}+\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} d -b^{3} c -3 d \,a^{2}+3 a d -d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d -3 b^{3} c -3 d \,a^{2}-3 a d +3 d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 b^{3} c +3 d \,a^{2}-3 a d -3 d \right ) \textit {\_Z}^{2}+a^{3} d -b^{3} c +3 d \,a^{2}+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 +a^{3} d -b^{3} c +2 \textit {\_R1}^{2} d +d \,a^{2}-a d -d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(744\) |
Input:
int(arccoth(b*x+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-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-a^3*d+b^3*c+3*d*a^2-3*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 -a^3*d+b^3*c-3*d*a^2-3*a*d-d))
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{3} + c} \,d x } \] Input:
integrate(arccoth(b*x+a)/(d*x^3+c),x, algorithm="fricas")
Output:
integral(arccoth(b*x + a)/(d*x^3 + c), x)
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\text {Timed out} \] Input:
integrate(acoth(b*x+a)/(d*x**3+c),x)
Output:
Timed out
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{3} + c} \,d x } \] Input:
integrate(arccoth(b*x+a)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(arccoth(b*x + a)/(d*x^3 + c), x)
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (b x + a\right )}{d x^{3} + c} \,d x } \] Input:
integrate(arccoth(b*x+a)/(d*x^3+c),x, algorithm="giac")
Output:
sage0*x
Timed out. \[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\int \frac {\mathrm {acoth}\left (a+b\,x\right )}{d\,x^3+c} \,d x \] Input:
int(acoth(a + b*x)/(c + d*x^3),x)
Output:
int(acoth(a + b*x)/(c + d*x^3), x)
\[ \int \frac {\coth ^{-1}(a+b x)}{c+d x^3} \, dx=\int \frac {\mathit {acoth} \left (b x +a \right )}{d \,x^{3}+c}d x \] Input:
int(acoth(b*x+a)/(d*x^3+c),x)
Output:
int(acoth(a + b*x)/(c + d*x**3),x)