Integrand size = 14, antiderivative size = 168 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{70 a^7} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {200, 5033, 1824, 266} \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {d^2 x^4 \left (21 a^2 c-5 d\right )}{140 a^3}+\frac {d x^2 \left (35 a^4 c^2-21 a^2 c d+5 d^2\right )}{70 a^5}+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (a^2 x^2+1\right )}{70 a^7}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac {d^3 x^6}{42 a} \]
[In]
[Out]
Rule 200
Rule 266
Rule 1824
Rule 5033
Rubi steps \begin{align*} \text {integral}& = c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \frac {c^3 x+c^2 d x^3+\frac {3}{5} c d^2 x^5+\frac {d^3 x^7}{7}}{1+a^2 x^2} \, dx \\ & = c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+a \int \left (\frac {d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x}{35 a^6}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^3}{35 a^4}+\frac {d^3 x^5}{7 a^2}+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) x}{35 a^6 \left (1+a^2 x^2\right )}\right ) \, dx \\ & = \frac {d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \int \frac {x}{1+a^2 x^2} \, dx}{35 a^5} \\ & = \frac {d \left (35 a^4 c^2-21 a^2 c d+5 d^2\right ) x^2}{70 a^5}+\frac {\left (21 a^2 c-5 d\right ) d^2 x^4}{140 a^3}+\frac {d^3 x^6}{42 a}+c^3 x \cot ^{-1}(a x)+c^2 d x^3 \cot ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cot ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cot ^{-1}(a x)+\frac {\left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{70 a^7} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.89 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {a^2 d x^2 \left (30 d^2-3 a^2 d \left (42 c+5 d x^2\right )+a^4 \left (210 c^2+63 c d x^2+10 d^2 x^4\right )\right )+12 a^7 x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \cot ^{-1}(a x)+6 \left (35 a^6 c^3-35 a^4 c^2 d+21 a^2 c d^2-5 d^3\right ) \log \left (1+a^2 x^2\right )}{420 a^7} \]
[In]
[Out]
Time = 0.45 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99
method | result | size |
parts | \(\frac {d^{3} x^{7} \operatorname {arccot}\left (a x \right )}{7}+\frac {3 c \,d^{2} x^{5} \operatorname {arccot}\left (a x \right )}{5}+c^{2} d \,x^{3} \operatorname {arccot}\left (a x \right )+c^{3} x \,\operatorname {arccot}\left (a x \right )+\frac {a \left (\frac {d \left (\frac {5}{3} a^{4} d^{2} x^{6}+\frac {21}{2} a^{4} c d \,x^{4}+35 a^{4} c^{2} x^{2}-\frac {5}{2} a^{2} d^{2} x^{4}-21 a^{2} c d \,x^{2}+5 d^{2} x^{2}\right )}{2 a^{6}}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{8}}\right )}{35}\) | \(167\) |
derivativedivides | \(\frac {\operatorname {arccot}\left (a x \right ) c^{3} a x +a \,\operatorname {arccot}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccot}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}-\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{35 a^{6}}}{a}\) | \(175\) |
default | \(\frac {\operatorname {arccot}\left (a x \right ) c^{3} a x +a \,\operatorname {arccot}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\operatorname {arccot}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\operatorname {arccot}\left (a x \right ) d^{3} x^{7}}{7}+\frac {\frac {35 c^{2} a^{6} d \,x^{2}}{2}+\frac {21 c \,a^{6} d^{2} x^{4}}{4}-\frac {21 c \,a^{4} d^{2} x^{2}}{2}+\frac {5 d^{3} a^{6} x^{6}}{6}-\frac {5 d^{3} a^{4} x^{4}}{4}+\frac {5 d^{3} a^{2} x^{2}}{2}+\frac {\left (35 a^{6} c^{3}-35 a^{4} c^{2} d +21 a^{2} c \,d^{2}-5 d^{3}\right ) \ln \left (a^{2} x^{2}+1\right )}{2}}{35 a^{6}}}{a}\) | \(175\) |
parallelrisch | \(\frac {60 x^{7} \operatorname {arccot}\left (a x \right ) a^{7} d^{3}+252 x^{5} \operatorname {arccot}\left (a x \right ) a^{7} c \,d^{2}+10 d^{3} a^{6} x^{6}+420 x^{3} \operatorname {arccot}\left (a x \right ) a^{7} c^{2} d +63 c \,a^{6} d^{2} x^{4}+420 x \,\operatorname {arccot}\left (a x \right ) a^{7} c^{3}-15 d^{3} a^{4} x^{4}+210 c^{2} a^{6} d \,x^{2}+210 \ln \left (a^{2} x^{2}+1\right ) a^{6} c^{3}-126 c \,a^{4} d^{2} x^{2}-210 \ln \left (a^{2} x^{2}+1\right ) a^{4} c^{2} d +30 d^{3} a^{2} x^{2}+126 \ln \left (a^{2} x^{2}+1\right ) a^{2} c \,d^{2}-30 \ln \left (a^{2} x^{2}+1\right ) d^{3}}{420 a^{7}}\) | \(207\) |
risch | \(-\frac {i c^{2} d \,x^{3} \ln \left (-i a x +1\right )}{2}-\frac {i c^{3} x \ln \left (-i a x +1\right )}{2}+\frac {\pi \,d^{3} x^{7}}{14}-\frac {3 i c \,d^{2} x^{5} \ln \left (-i a x +1\right )}{10}+\frac {3 \pi c \,d^{2} x^{5}}{10}+\frac {i \left (5 d^{3} x^{7}+21 c \,d^{2} x^{5}+35 c^{2} d \,x^{3}+35 c^{3} x \right ) \ln \left (i a x +1\right )}{70}+\frac {d^{3} x^{6}}{42 a}+\frac {\pi \,c^{2} d \,x^{3}}{2}-\frac {i d^{3} x^{7} \ln \left (-i a x +1\right )}{14}+\frac {3 c \,d^{2} x^{4}}{20 a}+\frac {\pi \,c^{3} x}{2}+\frac {c^{2} d \,x^{2}}{2 a}-\frac {d^{3} x^{4}}{28 a^{3}}+\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{3}}{2 a}-\frac {3 c \,d^{2} x^{2}}{10 a^{3}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) c^{2} d}{2 a^{3}}+\frac {d^{3} x^{2}}{14 a^{5}}+\frac {3 \ln \left (-a^{2} x^{2}-1\right ) c \,d^{2}}{10 a^{5}}-\frac {\ln \left (-a^{2} x^{2}-1\right ) d^{3}}{14 a^{7}}\) | \(297\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.99 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {10 \, a^{6} d^{3} x^{6} + 3 \, {\left (21 \, a^{6} c d^{2} - 5 \, a^{4} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{6} c^{2} d - 21 \, a^{4} c d^{2} + 5 \, a^{2} d^{3}\right )} x^{2} + 12 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \operatorname {arccot}\left (a x\right ) + 6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{420 \, a^{7}} \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.45 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\begin {cases} c^{3} x \operatorname {acot}{\left (a x \right )} + c^{2} d x^{3} \operatorname {acot}{\left (a x \right )} + \frac {3 c d^{2} x^{5} \operatorname {acot}{\left (a x \right )}}{5} + \frac {d^{3} x^{7} \operatorname {acot}{\left (a x \right )}}{7} + \frac {c^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a} + \frac {c^{2} d x^{2}}{2 a} + \frac {3 c d^{2} x^{4}}{20 a} + \frac {d^{3} x^{6}}{42 a} - \frac {c^{2} d \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{2 a^{3}} - \frac {3 c d^{2} x^{2}}{10 a^{3}} - \frac {d^{3} x^{4}}{28 a^{3}} + \frac {3 c d^{2} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{10 a^{5}} + \frac {d^{3} x^{2}}{14 a^{5}} - \frac {d^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{14 a^{7}} & \text {for}\: a \neq 0 \\\frac {\pi \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right )}{2} & \text {otherwise} \end {cases} \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {1}{420} \, a {\left (\frac {10 \, a^{4} d^{3} x^{6} + 3 \, {\left (21 \, a^{4} c d^{2} - 5 \, a^{2} d^{3}\right )} x^{4} + 6 \, {\left (35 \, a^{4} c^{2} d - 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} x^{2}}{a^{6}} + \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (a^{2} x^{2} + 1\right )}{a^{8}}\right )} + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arccot}\left (a x\right ) \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.50 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=\frac {1}{420} \, {\left (\frac {12 \, {\left (5 \, d^{3} + \frac {21 \, c d^{2}}{x^{2}} + \frac {35 \, c^{2} d}{x^{4}} + \frac {35 \, c^{3}}{x^{6}}\right )} x^{7} \arctan \left (\frac {1}{a x}\right )}{a} + \frac {{\left (10 \, d^{3} + \frac {63 \, c d^{2}}{x^{2}} + \frac {210 \, c^{2} d}{x^{4}} - \frac {15 \, d^{3}}{a^{2} x^{2}} + \frac {385 \, c^{3}}{x^{6}} - \frac {126 \, c d^{2}}{a^{2} x^{4}} - \frac {385 \, c^{2} d}{a^{2} x^{6}} + \frac {30 \, d^{3}}{a^{4} x^{4}} + \frac {231 \, c d^{2}}{a^{4} x^{6}} - \frac {55 \, d^{3}}{a^{6} x^{6}}\right )} x^{6}}{a^{2}} + \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (\frac {1}{a^{2} x^{2}} + 1\right )}{a^{8}} - \frac {6 \, {\left (35 \, a^{6} c^{3} - 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} - 5 \, d^{3}\right )} \log \left (\frac {1}{a^{2} x^{2}}\right )}{a^{8}}\right )} a \]
[In]
[Out]
Time = 1.15 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.13 \[ \int \left (c+d x^2\right )^3 \cot ^{-1}(a x) \, dx=c^3\,x\,\mathrm {acot}\left (a\,x\right )+\frac {d^3\,x^7\,\mathrm {acot}\left (a\,x\right )}{7}+\frac {c^3\,\ln \left (a^2\,x^2+1\right )}{2\,a}-\frac {d^3\,\ln \left (a^2\,x^2+1\right )}{14\,a^7}+\frac {d^3\,x^6}{42\,a}-\frac {d^3\,x^4}{28\,a^3}+\frac {d^3\,x^2}{14\,a^5}-\frac {c^2\,d\,\ln \left (a^2\,x^2+1\right )}{2\,a^3}+\frac {3\,c\,d^2\,\ln \left (a^2\,x^2+1\right )}{10\,a^5}+\frac {c^2\,d\,x^2}{2\,a}+\frac {3\,c\,d^2\,x^4}{20\,a}-\frac {3\,c\,d^2\,x^2}{10\,a^3}+c^2\,d\,x^3\,\mathrm {acot}\left (a\,x\right )+\frac {3\,c\,d^2\,x^5\,\mathrm {acot}\left (a\,x\right )}{5} \]
[In]
[Out]