Integrand size = 14, antiderivative size = 255 \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \arctan (c x)}{60 c^6}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {23 b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {23 i b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{30 c^6} \]
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Time = 0.64 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4946, 5036, 308, 209, 327, 5040, 4964, 2449, 2352, 4930, 5004} \[ \int x^5 (a+b \arctan (c x))^3 \, dx=-\frac {23 b^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \arctan (c x))}{15 c^6}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}+\frac {(a+b \arctan (c x))^3}{6 c^6}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}-\frac {19 b^3 \arctan (c x)}{60 c^6}-\frac {23 i b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i c x+1}\right )}{30 c^6}+\frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3} \]
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Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {1}{2} (b c) \int \frac {x^6 (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx \\ & = \frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {b \int x^4 (a+b \arctan (c x))^2 \, dx}{2 c}+\frac {b \int \frac {x^4 (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 c} \\ & = -\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {1}{6} x^6 (a+b \arctan (c x))^3+\frac {1}{5} b^2 \int \frac {x^5 (a+b \arctan (c x))}{1+c^2 x^2} \, dx+\frac {b \int x^2 (a+b \arctan (c x))^2 \, dx}{2 c^3}-\frac {b \int \frac {x^2 (a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 c^3} \\ & = \frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {b \int (a+b \arctan (c x))^2 \, dx}{2 c^5}+\frac {b \int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx}{2 c^5}+\frac {b^2 \int x^3 (a+b \arctan (c x)) \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^2}-\frac {b^2 \int \frac {x^3 (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c^2} \\ & = \frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {b^2 \int x (a+b \arctan (c x)) \, dx}{5 c^4}+\frac {b^2 \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{5 c^4}-\frac {b^2 \int x (a+b \arctan (c x)) \, dx}{3 c^4}+\frac {b^2 \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{3 c^4}+\frac {b^2 \int \frac {x (a+b \arctan (c x))}{1+c^2 x^2} \, dx}{c^4}-\frac {b^3 \int \frac {x^4}{1+c^2 x^2} \, dx}{20 c} \\ & = -\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {b^2 \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{5 c^5}-\frac {b^2 \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{3 c^5}-\frac {b^2 \int \frac {a+b \arctan (c x)}{i-c x} \, dx}{c^5}+\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{10 c^3}+\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{6 c^3}-\frac {b^3 \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{20 c} \\ & = \frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {23 b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{20 c^5}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{10 c^5}-\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{6 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^5}+\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^5} \\ & = \frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \arctan (c x)}{60 c^6}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {23 b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^6}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{3 c^6}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^6} \\ & = \frac {19 b^3 x}{60 c^5}-\frac {b^3 x^3}{60 c^3}-\frac {19 b^3 \arctan (c x)}{60 c^6}-\frac {4 b^2 x^2 (a+b \arctan (c x))}{15 c^4}+\frac {b^2 x^4 (a+b \arctan (c x))}{20 c^2}-\frac {23 i b (a+b \arctan (c x))^2}{30 c^6}-\frac {b x (a+b \arctan (c x))^2}{2 c^5}+\frac {b x^3 (a+b \arctan (c x))^2}{6 c^3}-\frac {b x^5 (a+b \arctan (c x))^2}{10 c}+\frac {(a+b \arctan (c x))^3}{6 c^6}+\frac {1}{6} x^6 (a+b \arctan (c x))^3-\frac {23 b^2 (a+b \arctan (c x)) \log \left (\frac {2}{1+i c x}\right )}{15 c^6}-\frac {23 i b^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{30 c^6} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.14 \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\frac {-19 a b^2-30 a^2 b c x+19 b^3 c x-16 a b^2 c^2 x^2+10 a^2 b c^3 x^3-b^3 c^3 x^3+3 a b^2 c^4 x^4-6 a^2 b c^5 x^5+10 a^3 c^6 x^6+2 b^2 \left (b \left (23 i-15 c x+5 c^3 x^3-3 c^5 x^5\right )+15 a \left (1+c^6 x^6\right )\right ) \arctan (c x)^2+10 b^3 \left (1+c^6 x^6\right ) \arctan (c x)^3+b \arctan (c x) \left (b^2 \left (-19-16 c^2 x^2+3 c^4 x^4\right )-4 a b c x \left (15-5 c^2 x^2+3 c^4 x^4\right )+30 a^2 \left (1+c^6 x^6\right )-92 b^2 \log \left (1+e^{2 i \arctan (c x)}\right )\right )+46 a b^2 \log \left (1+c^2 x^2\right )+46 i b^3 \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c x)}\right )}{60 c^6} \]
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Time = 2.52 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.58
method | result | size |
derivativedivides | \(\frac {\frac {a^{3} c^{6} x^{6}}{6}+b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )+3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )+3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(402\) |
default | \(\frac {\frac {a^{3} c^{6} x^{6}}{6}+b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )+3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )+3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(402\) |
parts | \(\frac {a^{3} x^{6}}{6}+\frac {b^{3} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{3}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )^{2}}{10}+\frac {c^{3} x^{3} \arctan \left (c x \right )^{2}}{6}-\frac {\arctan \left (c x \right )^{2} c x}{2}+\frac {\arctan \left (c x \right )^{3}}{6}+\frac {c^{4} x^{4} \arctan \left (c x \right )}{20}-\frac {4 c^{2} x^{2} \arctan \left (c x \right )}{15}+\frac {23 \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right )}{30}-\frac {c^{3} x^{3}}{60}+\frac {19 c x}{60}-\frac {19 \arctan \left (c x \right )}{60}+\frac {23 i \left (\ln \left (c x -i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (c x +i\right )}{2}\right )-\ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )\right )}{60}-\frac {23 i \left (\ln \left (c x +i\right ) \ln \left (c^{2} x^{2}+1\right )-\frac {\ln \left (c x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (c x -i\right )}{2}\right )-\ln \left (c x +i\right ) \ln \left (\frac {i \left (c x -i\right )}{2}\right )\right )}{60}\right )}{c^{6}}+\frac {3 a \,b^{2} \left (\frac {c^{6} x^{6} \arctan \left (c x \right )^{2}}{6}-\frac {c^{5} x^{5} \arctan \left (c x \right )}{15}+\frac {c^{3} x^{3} \arctan \left (c x \right )}{9}-\frac {c x \arctan \left (c x \right )}{3}+\frac {\arctan \left (c x \right )^{2}}{6}+\frac {c^{4} x^{4}}{60}-\frac {4 c^{2} x^{2}}{45}+\frac {23 \ln \left (c^{2} x^{2}+1\right )}{90}\right )}{c^{6}}+\frac {3 a^{2} b \left (\frac {c^{6} x^{6} \arctan \left (c x \right )}{6}-\frac {c^{5} x^{5}}{30}+\frac {c^{3} x^{3}}{18}-\frac {c x}{6}+\frac {\arctan \left (c x \right )}{6}\right )}{c^{6}}\) | \(404\) |
risch | \(\text {Expression too large to display}\) | \(1277\) |
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\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
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\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int x^{5} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]
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\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
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\[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int { {\left (b \arctan \left (c x\right ) + a\right )}^{3} x^{5} \,d x } \]
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Timed out. \[ \int x^5 (a+b \arctan (c x))^3 \, dx=\int x^5\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]
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