Integrand size = 14, antiderivative size = 145 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b^2 c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )+\frac {3}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \]
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Time = 0.21 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4946, 5038, 5044, 4988, 2497, 5004} \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=-3 b^2 c^2 \log \left (2-\frac {2}{1-\frac {i c}{x}}\right ) \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )+\frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,\frac {2}{1-\frac {i c}{x}}-1\right ) \]
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Rule 2497
Rule 4946
Rule 4948
Rule 4988
Rule 5004
Rule 5038
Rule 5044
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {(a+b \arctan (c x))^3}{x^3} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\frac {1}{2} (3 b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2 \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\frac {1}{2} (3 b c) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{x^2} \, dx,x,\frac {1}{x}\right )+\frac {1}{2} \left (3 b c^3\right ) \text {Subst}\left (\int \frac {(a+b \arctan (c x))^2}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\left (3 b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x \left (1+c^2 x^2\right )} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-\left (3 i b^2 c^2\right ) \text {Subst}\left (\int \frac {a+b \arctan (c x)}{x (i+c x)} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b^2 c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )+\left (3 b^3 c^3\right ) \text {Subst}\left (\int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx,x,\frac {1}{x}\right ) \\ & = \frac {3}{2} i b c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {3}{2} b c x \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^2+\frac {1}{2} c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3+\frac {1}{2} x^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right )^3-3 b^2 c^2 \left (a+b \cot ^{-1}\left (\frac {x}{c}\right )\right ) \log \left (2-\frac {2}{1-\frac {i c}{x}}\right )+\frac {3}{2} i b^3 c^2 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-\frac {i c}{x}}\right ) \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.20 \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\frac {1}{2} \left (3 b^2 \left (b c (i c+x)+a \left (c^2+x^2\right )\right ) \arctan \left (\frac {c}{x}\right )^2+b^3 \left (c^2+x^2\right ) \arctan \left (\frac {c}{x}\right )^3+3 b \arctan \left (\frac {c}{x}\right ) \left (a \left (2 b c x+a \left (c^2+x^2\right )\right )-2 b^2 c^2 \log \left (1-e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right )+a \left (a x (3 b c+a x)-6 b^2 c^2 \log \left (\frac {c}{\sqrt {1+\frac {c^2}{x^2}} x}\right )\right )+3 i b^3 c^2 \operatorname {PolyLog}\left (2,e^{2 i \arctan \left (\frac {c}{x}\right )}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 404 vs. \(2 (131 ) = 262\).
Time = 8.16 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.79
method | result | size |
derivativedivides | \(-c^{2} \left (-\frac {a^{3} x^{2}}{2 c^{2}}+b^{3} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )+3 a \,b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+3 a^{2} b \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )\right )\) | \(405\) |
default | \(-c^{2} \left (-\frac {a^{3} x^{2}}{2 c^{2}}+b^{3} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )+3 a \,b^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+3 a^{2} b \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )}{2 c^{2}}-\frac {x}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )}{2}\right )\right )\) | \(405\) |
parts | \(\frac {a^{3} x^{2}}{2}-b^{3} c^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{3}}{2 c^{2}}-\frac {3 x \arctan \left (\frac {c}{x}\right )^{2}}{2 c}-\frac {\arctan \left (\frac {c}{x}\right )^{3}}{2}+3 \ln \left (\frac {c}{x}\right ) \arctan \left (\frac {c}{x}\right )-\frac {3 \arctan \left (\frac {c}{x}\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \ln \left (\frac {c}{x}\right ) \ln \left (1-\frac {i c}{x}\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+\frac {i c}{x}\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-\frac {i c}{x}\right )}{2}-\frac {3 i \left (\ln \left (\frac {c}{x}-i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}-i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )-\ln \left (\frac {c}{x}-i\right ) \ln \left (-\frac {i \left (\frac {c}{x}+i\right )}{2}\right )\right )}{4}+\frac {3 i \left (\ln \left (\frac {c}{x}+i\right ) \ln \left (1+\frac {c^{2}}{x^{2}}\right )-\frac {\ln \left (\frac {c}{x}+i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )-\ln \left (\frac {c}{x}+i\right ) \ln \left (\frac {i \left (\frac {c}{x}-i\right )}{2}\right )\right )}{4}\right )-3 a \,b^{2} c^{2} \left (-\frac {x^{2} \arctan \left (\frac {c}{x}\right )^{2}}{2 c^{2}}-\frac {\arctan \left (\frac {c}{x}\right )^{2}}{2}-\frac {x \arctan \left (\frac {c}{x}\right )}{c}-\frac {\ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2}+\ln \left (\frac {c}{x}\right )\right )+\frac {3 x^{2} a^{2} b \arctan \left (\frac {c}{x}\right )}{2}-\frac {3 a^{2} b \,c^{2} \arctan \left (\frac {x}{c}\right )}{2}+\frac {3 a^{2} b c x}{2}\) | \(407\) |
risch | \(\text {Expression too large to display}\) | \(233342\) |
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\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
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\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x \left (a + b \operatorname {atan}{\left (\frac {c}{x} \right )}\right )^{3}\, dx \]
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\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
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\[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int { {\left (b \arctan \left (\frac {c}{x}\right ) + a\right )}^{3} x \,d x } \]
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Timed out. \[ \int x \left (a+b \arctan \left (\frac {c}{x}\right )\right )^3 \, dx=\int x\,{\left (a+b\,\mathrm {atan}\left (\frac {c}{x}\right )\right )}^3 \,d x \]
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