Integrand size = 16, antiderivative size = 725 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=-\frac {\cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [3]{-1} \cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{2/3} \cot ^{-1}(a+b x) \log \left (\frac {2}{1-i (a+b x)}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {\cot ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{3 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{2/3} \cot ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [3]{-1} \cot ^{-1}(a+b x) \log \left (\frac {2 b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(-1)^{2/3} (i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{3 c^{2/3} \sqrt [3]{d}}-\frac {i \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {\sqrt [6]{-1} \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {(-1)^{5/6} \operatorname {PolyLog}\left (2,1-\frac {2}{1-i (a+b x)}\right )}{6 c^{2/3} \sqrt [3]{d}}+\frac {i \operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {\sqrt [6]{-1} \operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}-\sqrt [3]{-1} (i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{6 c^{2/3} \sqrt [3]{d}}-\frac {(-1)^{5/6} \operatorname {PolyLog}\left (2,1-\frac {2 b \left (\sqrt [3]{c}+(-1)^{2/3} \sqrt [3]{d} x\right )}{\left (b \sqrt [3]{c}+(-1)^{2/3} (i-a) \sqrt [3]{d}\right ) (1-i (a+b x))}\right )}{6 c^{2/3} \sqrt [3]{d}} \] Output:
-1/3*arccot(b*x+a)*ln(2/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)+1/3*(-1)^(1/3)*arcc ot(b*x+a)*ln(2/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)-1/3*(-1)^(2/3)*arccot(b*x+a) *ln(2/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)+1/3*arccot(b*x+a)*ln(2*b*(c^(1/3)+d^( 1/3)*x)/(b*c^(1/3)+(I-a)*d^(1/3))/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)+1/3*(-1)^ (2/3)*arccot(b*x+a)*ln(2*b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x)/(b*c^(1/3)-(-1)^ (1/3)*(I-a)*d^(1/3))/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)-1/3*(-1)^(1/3)*arccot( b*x+a)*ln(2*b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^(2/3)*(I-a)*d ^(1/3))/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)-1/6*I*polylog(2,1-2/(1-I*(b*x+a)))/ c^(2/3)/d^(1/3)+1/6*(-1)^(1/6)*polylog(2,1-2/(1-I*(b*x+a)))/c^(2/3)/d^(1/3 )+1/6*(-1)^(5/6)*polylog(2,1-2/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)+1/6*I*polylo g(2,1-2*b*(c^(1/3)+d^(1/3)*x)/(b*c^(1/3)+(I-a)*d^(1/3))/(1-I*(b*x+a)))/c^( 2/3)/d^(1/3)-1/6*(-1)^(1/6)*polylog(2,1-2*b*(c^(1/3)-(-1)^(1/3)*d^(1/3)*x) /(b*c^(1/3)-(-1)^(1/3)*(I-a)*d^(1/3))/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)-1/6*( -1)^(5/6)*polylog(2,1-2*b*(c^(1/3)+(-1)^(2/3)*d^(1/3)*x)/(b*c^(1/3)+(-1)^( 2/3)*(I-a)*d^(1/3))/(1-I*(b*x+a)))/c^(2/3)/d^(1/3)
Time = 0.81 (sec) , antiderivative size = 998, normalized size of antiderivative = 1.38 \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx =\text {Too large to display} \] Input:
Integrate[ArcCot[a + b*x]/(c + d*x^3),x]
Output:
((-I)*Log[-((d^(1/3)*(-I + a + b*x))/(b*c^(1/3) - (-I + a)*d^(1/3)))]*Log[ -c^(1/3) - d^(1/3)*x] + I*Log[(-I + a + b*x)/(a + b*x)]*Log[-c^(1/3) - d^( 1/3)*x] + I*Log[-((d^(1/3)*(I + a + b*x))/(b*c^(1/3) - (I + a)*d^(1/3)))]* Log[-c^(1/3) - d^(1/3)*x] - I*Log[(I + a + b*x)/(a + b*x)]*Log[-c^(1/3) - d^(1/3)*x] + (-1)^(1/6)*Log[((-1)^(1/3)*d^(1/3)*(-I + a + b*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] - (- 1)^(1/6)*Log[(-I + a + b*x)/(a + b*x)]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x ] - (-1)^(1/6)*Log[((-1)^(1/3)*d^(1/3)*(I + a + b*x))/(b*c^(1/3) + (-1)^(1 /3)*(I + a)*d^(1/3))]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] + (-1)^(1/6)*Lo g[(I + a + b*x)/(a + b*x)]*Log[-c^(1/3) + (-1)^(1/3)*d^(1/3)*x] + (-1)^(5/ 6)*Log[((-1)^(2/3)*d^(1/3)*(-I + a + b*x))/(-(b*c^(1/3)) + (-1)^(1/6)*(1 + I*a)*d^(1/3))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] - (-1)^(5/6)*Log[(-I + a + b*x)/(a + b*x)]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] - (-1)^(5/6)*Lo g[((-1)^(2/3)*d^(1/3)*(I + a + b*x))/(-(b*c^(1/3)) + (-1)^(2/3)*(I + a)*d^ (1/3))]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] + (-1)^(5/6)*Log[(I + a + b*x )/(a + b*x)]*Log[-c^(1/3) - (-1)^(2/3)*d^(1/3)*x] - I*PolyLog[2, (b*(c^(1/ 3) + d^(1/3)*x))/(b*c^(1/3) - (-I + a)*d^(1/3))] + I*PolyLog[2, (b*(c^(1/3 ) + d^(1/3)*x))/(b*c^(1/3) - (I + a)*d^(1/3))] + (-1)^(1/6)*PolyLog[2, (b* (c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) + (-1)^(1/3)*(-I + a)*d^(1/3) )] - (-1)^(1/6)*PolyLog[2, (b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1862\) vs. \(2(725)=1450\).
Time = 3.12 (sec) , antiderivative size = 1862, normalized size of antiderivative = 2.57, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5575, 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.
| \(\displaystyle \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx\) |
| \(\Big \downarrow \) 5575 |
| \(\displaystyle \frac {1}{2} i \int \frac {\log \left (-\frac {-a-b x+i}{a+b x}\right )}{d x^3+c}dx-\frac {1}{2} i \int \frac {\log \left (\frac {a+b x+i}{a+b x}\right )}{d x^3+c}dx\) |
| \(\Big \downarrow \) 2993 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \int \frac {1}{d x^3+c}dx\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \int \frac {1}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )+\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} i \left (\int \frac {\log (-a-b x+i)}{d x^3+c}dx-\int \frac {\log (a+b x)}{d x^3+c}dx-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{a+b x}\right )-\log (a+b x)\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )\right )-\frac {1}{2} i \left (-\int \frac {\log (a+b x)}{d x^3+c}dx+\int \frac {\log (a+b x+i)}{d x^3+c}dx+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{a+b x}\right )\right ) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\right )\) |
| \(\Big \downarrow \) 2856 |
| \(\displaystyle \frac {1}{2} i \left (-\left (\left (\log (-a-b x+i)-\log \left (-\frac {-a-b x+i}{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 )\right )+\int \left (-\frac {\log (-a-b x+i)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-a-b x+i)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (-a-b x+i)}{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\right )-\frac {1}{2} i \left (\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{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 )-\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+i)}{3 c^{2/3} \left (-\sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x+i)}{3 c^{2/3} \left (\sqrt [3]{-1} \sqrt [3]{d} x-\sqrt [3]{c}\right )}-\frac {\log (a+b x+i)}{3 c^{2/3} \left (-(-1)^{2/3} \sqrt [3]{d} x-\sqrt [3]{c}\right )}\right )dx\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} i \left (\frac {\log (-a-b x+i) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{\sqrt [3]{d} (i-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+i) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-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+i) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{(-1)^{2/3} \sqrt [3]{d} (i-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+i)-\log \left (-\frac {-a-b x+i}{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+i)}{\sqrt [3]{d} (i-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+i)}{b \sqrt [3]{c}-\sqrt [3]{-1} (i-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+i)}{(-1)^{2/3} \sqrt [3]{d} (i-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} i \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+i) \log \left (\frac {b \left (\sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(a+i) \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+i) \log \left (\frac {b \left (\sqrt [3]{c}-\sqrt [3]{-1} \sqrt [3]{d} x\right )}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+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+i) \log \left (\frac {b \left ((-1)^{2/3} \sqrt [3]{d} x+\sqrt [3]{c}\right )}{b \sqrt [3]{c}-(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}+\left (\log (a+b x)-\log (a+b x+i)+\log \left (\frac {a+b x+i}{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+i)}{b \sqrt [3]{c}-(a+i) \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+i)}{\sqrt [3]{-1} \sqrt [3]{d} (a+i)+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+i)}{b \sqrt [3]{c}-(-1)^{2/3} (a+i) \sqrt [3]{d}}\right )}{3 c^{2/3} \sqrt [3]{d}}\right )\) |
Input:
Int[ArcCot[a + b*x]/(c + d*x^3),x]
Output:
(I/2)*((Log[I - a - b*x]*Log[(b*(c^(1/3) + d^(1/3)*x))/(b*c^(1/3) + (I - a )*d^(1/3))])/(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[I - a - b*x]*Log[(b*(c^(1/3) - (-1)^(1/3)*d^(1/3)*x))/(b*c^(1/3) - (-1)^(1/3)*( I - 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[I - a - b*x]*Log[(b*(c^(1/3) + (-1)^(2/3) *d^(1/3)*x))/(b*c^(1/3) + (-1)^(2/3)*(I - 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[I - a - b*x] - Log[-((I - 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)*(I - a - b*x))/(b*c^(1/3) + (I - a)* d^(1/3))]/(3*c^(2/3)*d^(1/3)) + ((-1)^(2/3)*PolyLog[2, -(((-1)^(1/3)*d^(1/ 3)*(I - a - b*x))/(b*c^(1/3) - (-1)^(1/3)*(I - a)*d^(1/3)))])/(3*c^(2/3)*d ^(1/3)) - ((-1)^(1/3)*PolyLog[2, ((-1)^(2/3)*d^(1/3)*(I - a - b*x))/(b*c^( 1/3) + (-1)^(2/3)*(I - 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...
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[ArcCot[(a_) + (b_.)*(x_)]/((c_) + (d_.)*(x_)^(n_.)), x_Symbol] :> Simp[ I/2 Int[Log[(-I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] - Simp[I/2 In t[Log[(I + a + b*x)/(a + b*x)]/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d}, x ] && RationalQ[n]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.60 (sec) , antiderivative size = 536, normalized size of antiderivative = 0.74
| method | result | size |
| risch | \(\frac {i b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 d \right ) \textit {\_Z}^{2}+\left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 a^{2} d +3 d \right ) \textit {\_Z} -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} d +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} c +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d +3 a^{2} d -d \right )}{\sum }\frac {\ln \left (-b x i-a i+1\right ) \ln \left (\frac {b x i+a i+\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {b x i+a i+\textit {\_R1} -1}{\textit {\_R1}}\right )}{2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}-2 a i-2 \textit {\_R1} +1}\right )}{6 d}-\frac {b^{2} \pi \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 d \right ) \textit {\_Z}^{2}+\left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 a^{2} d +3 d \right ) \textit {\_Z} -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} d +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} c +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d +3 a^{2} d -d \right )}{\sum }\frac {\ln \left (-b x i-a i-\textit {\_R} +1\right )}{2 i a \textit {\_R} +\textit {\_R}^{2}-a^{2}-2 a i-2 \textit {\_R} +1}\right )}{6 d}-\frac {i b^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}+\left (-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 d \right ) \textit {\_Z}^{2}+\left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d -3 a^{2} d +3 d \right ) \textit {\_Z} +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a^{3} d -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) b^{3} c -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1, \operatorname {index} =1\right ) a d +3 a^{2} d -d \right )}{\sum }\frac {\ln \left (b x i+a i+1\right ) \ln \left (\frac {-b x i-a i+\textit {\_R1} -1}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-b x i-a i+\textit {\_R1} -1}{\textit {\_R1}}\right )}{-2 \textit {\_R1} a i+\textit {\_R1}^{2}-a^{2}+2 a i-2 \textit {\_R1} +1}\right )}{6 d}\) | \(536\) |
| derivativedivides | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \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 {arccot}\left (b x +a \right )}{3 d}-\frac {b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \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 +3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d +3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +a^{3} d -b^{3} c +i d -3 a d \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+a^{3} d -i a^{2} d -b^{3} c +a d -i d}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} d +3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d +3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +a^{3} d -b^{3} c +i d -3 a d \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+a^{3} d -i a^{2} d -b^{3} c +a d -i d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(787\) |
| default | \(\frac {-\frac {b^{3} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \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 {arccot}\left (b x +a \right )}{3 d}-\frac {b^{3} \left (\arctan \left (b x +a \right ) \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (d \,\textit {\_Z}^{3}-3 a d \,\textit {\_Z}^{2}+3 a^{2} d \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 +3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d +3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +a^{3} d -b^{3} c +i d -3 a d \right )}{\sum }\frac {i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+a^{3} d -i a^{2} d -b^{3} c +a d -i d}\right )}{3}-\frac {2 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (\left (a^{3} d +3 i a^{2} d -b^{3} c -3 a d -i d \right ) \textit {\_Z}^{6}+\left (3 a^{3} d +3 i a^{2} d -3 b^{3} c +3 a d +3 i d \right ) \textit {\_Z}^{4}+\left (3 a^{3} d -3 i a^{2} d -3 b^{3} c +3 a d -3 i d \right ) \textit {\_Z}^{2}-3 i a^{2} d +a^{3} d -b^{3} c +i d -3 a d \right )}{\sum }\frac {\textit {\_R1}^{2} \left (i \arctan \left (b x +a \right ) \ln \left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -\frac {1+i \left (b x +a \right )}{\sqrt {1+\left (b x +a \right )^{2}}}}{\textit {\_R1}}\right )\right )}{a^{3} d \,\textit {\_R1}^{4}+3 i a^{2} d \,\textit {\_R1}^{4}-b^{3} c \,\textit {\_R1}^{4}-3 a d \,\textit {\_R1}^{4}-i d \,\textit {\_R1}^{4}+2 a^{3} d \,\textit {\_R1}^{2}+2 i a^{2} d \,\textit {\_R1}^{2}-2 b^{3} c \,\textit {\_R1}^{2}+2 a d \,\textit {\_R1}^{2}+2 i d \,\textit {\_R1}^{2}+a^{3} d -i a^{2} d -b^{3} c +a d -i d}\right )}{3}\right )\right )}{3 d}}{b}\) | \(787\) |
Input:
int(arccot(b*x+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/6*I*b^2/d*sum(1/(1+2*I*a*_R1-2*I*a+_R1^2-a^2-2*_R1)*(ln(1-I*a-I*b*x)*ln( (_R1+I*b*x+I*a-1)/_R1)+dilog((_R1+I*b*x+I*a-1)/_R1)),_R1=RootOf(d*_Z^3+(3* RootOf(_Z^2+1,index=1)*a*d-3*d)*_Z^2+(-6*RootOf(_Z^2+1,index=1)*a*d-3*a^2* d+3*d)*_Z-RootOf(_Z^2+1,index=1)*a^3*d+RootOf(_Z^2+1,index=1)*b^3*c+3*Root Of(_Z^2+1,index=1)*a*d+3*a^2*d-d))-1/6*b^2*Pi/d*sum(1/(1+2*I*a*_R-2*I*a+_R ^2-a^2-2*_R)*ln(-I*b*x-I*a+1-_R),_R=RootOf(d*_Z^3+(3*RootOf(_Z^2+1,index=1 )*a*d-3*d)*_Z^2+(-6*RootOf(_Z^2+1,index=1)*a*d-3*a^2*d+3*d)*_Z-RootOf(_Z^2 +1,index=1)*a^3*d+RootOf(_Z^2+1,index=1)*b^3*c+3*RootOf(_Z^2+1,index=1)*a* d+3*a^2*d-d))-1/6*I*b^2/d*sum(1/(1-2*I*a*_R1+2*I*a+_R1^2-a^2-2*_R1)*(ln(1+ I*a+I*b*x)*ln((_R1-I*b*x-I*a-1)/_R1)+dilog((_R1-I*b*x-I*a-1)/_R1)),_R1=Roo tOf(d*_Z^3+(-3*RootOf(_Z^2+1,index=1)*a*d-3*d)*_Z^2+(6*RootOf(_Z^2+1,index =1)*a*d-3*a^2*d+3*d)*_Z+RootOf(_Z^2+1,index=1)*a^3*d-RootOf(_Z^2+1,index=1 )*b^3*c-3*RootOf(_Z^2+1,index=1)*a*d+3*a^2*d-d))
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d x^{3} + c} \,d x } \] Input:
integrate(arccot(b*x+a)/(d*x^3+c),x, algorithm="fricas")
Output:
integral(arccot(b*x + a)/(d*x^3 + c), x)
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\text {Timed out} \] Input:
integrate(acot(b*x+a)/(d*x**3+c),x)
Output:
Timed out
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{d x^{3} + c} \,d x } \] Input:
integrate(arccot(b*x+a)/(d*x^3+c),x, algorithm="maxima")
Output:
integrate(arccot(b*x + a)/(d*x^3 + c), x)
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\text {Timed out} \] Input:
integrate(arccot(b*x+a)/(d*x^3+c),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{d\,x^3+c} \,d x \] Input:
int(acot(a + b*x)/(c + d*x^3),x)
Output:
int(acot(a + b*x)/(c + d*x^3), x)
\[ \int \frac {\cot ^{-1}(a+b x)}{c+d x^3} \, dx=\int \frac {\mathit {acot} \left (b x +a \right )}{d \,x^{3}+c}d x \] Input:
int(acot(b*x+a)/(d*x^3+c),x)
Output:
int(acot(a + b*x)/(c + d*x**3),x)