Integrand size = 16, antiderivative size = 124 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {a b x^3}{6 c^3}+\frac {b^2 x^6}{36 c^2}+\frac {b^2 x^3 \arctan \left (c x^3\right )}{6 c^3}-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b^2 \log \left (1+c^2 x^6\right )}{9 c^4} \]
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Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4948, 4946, 5036, 272, 45, 4930, 266, 5004} \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}+\frac {a b x^3}{6 c^3}+\frac {b^2 x^3 \arctan \left (c x^3\right )}{6 c^3}+\frac {b^2 x^6}{36 c^2}-\frac {b^2 \log \left (c^2 x^6+1\right )}{9 c^4} \]
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Rule 45
Rule 266
Rule 272
Rule 4930
Rule 4946
Rule 4948
Rule 5004
Rule 5036
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int x^3 (a+b \arctan (c x))^2 \, dx,x,x^3\right ) \\ & = \frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {1}{6} (b c) \text {Subst}\left (\int \frac {x^4 (a+b \arctan (c x))}{1+c^2 x^2} \, dx,x,x^3\right ) \\ & = \frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b \text {Subst}\left (\int x^2 (a+b \arctan (c x)) \, dx,x,x^3\right )}{6 c}+\frac {b \text {Subst}\left (\int \frac {x^2 (a+b \arctan (c x))}{1+c^2 x^2} \, dx,x,x^3\right )}{6 c} \\ & = -\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2+\frac {1}{18} b^2 \text {Subst}\left (\int \frac {x^3}{1+c^2 x^2} \, dx,x,x^3\right )+\frac {b \text {Subst}\left (\int (a+b \arctan (c x)) \, dx,x,x^3\right )}{6 c^3}-\frac {b \text {Subst}\left (\int \frac {a+b \arctan (c x)}{1+c^2 x^2} \, dx,x,x^3\right )}{6 c^3} \\ & = \frac {a b x^3}{6 c^3}-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2+\frac {1}{36} b^2 \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^6\right )+\frac {b^2 \text {Subst}\left (\int \arctan (c x) \, dx,x,x^3\right )}{6 c^3} \\ & = \frac {a b x^3}{6 c^3}+\frac {b^2 x^3 \arctan \left (c x^3\right )}{6 c^3}-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2+\frac {1}{36} b^2 \text {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^6\right )-\frac {b^2 \text {Subst}\left (\int \frac {x}{1+c^2 x^2} \, dx,x,x^3\right )}{6 c^2} \\ & = \frac {a b x^3}{6 c^3}+\frac {b^2 x^6}{36 c^2}+\frac {b^2 x^3 \arctan \left (c x^3\right )}{6 c^3}-\frac {b x^9 \left (a+b \arctan \left (c x^3\right )\right )}{18 c}-\frac {\left (a+b \arctan \left (c x^3\right )\right )^2}{12 c^4}+\frac {1}{12} x^{12} \left (a+b \arctan \left (c x^3\right )\right )^2-\frac {b^2 \log \left (1+c^2 x^6\right )}{9 c^4} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.98 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {c x^3 \left (6 a b+b^2 c x^3-2 a b c^2 x^6+3 a^2 c^3 x^9\right )-2 b \left (b c x^3 \left (-3+c^2 x^6\right )+a \left (3-3 c^4 x^{12}\right )\right ) \arctan \left (c x^3\right )+3 b^2 \left (-1+c^4 x^{12}\right ) \arctan \left (c x^3\right )^2-4 b^2 \log \left (1+c^2 x^6\right )}{36 c^4} \]
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Time = 1.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.22
method | result | size |
default | \(\frac {a^{2} x^{12}}{12}+\frac {b^{2} x^{12} \arctan \left (c \,x^{3}\right )^{2}}{12}-\frac {b^{2} \arctan \left (c \,x^{3}\right ) x^{9}}{18 c}+\frac {b^{2} x^{3} \arctan \left (c \,x^{3}\right )}{6 c^{3}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{12 c^{4}}+\frac {b^{2} x^{6}}{36 c^{2}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}+\frac {a b \,x^{12} \arctan \left (c \,x^{3}\right )}{6}-\frac {a b \,x^{9}}{18 c}+\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}\) | \(151\) |
parts | \(\frac {a^{2} x^{12}}{12}+\frac {b^{2} x^{12} \arctan \left (c \,x^{3}\right )^{2}}{12}-\frac {b^{2} \arctan \left (c \,x^{3}\right ) x^{9}}{18 c}+\frac {b^{2} x^{3} \arctan \left (c \,x^{3}\right )}{6 c^{3}}-\frac {b^{2} \arctan \left (c \,x^{3}\right )^{2}}{12 c^{4}}+\frac {b^{2} x^{6}}{36 c^{2}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}+\frac {a b \,x^{12} \arctan \left (c \,x^{3}\right )}{6}-\frac {a b \,x^{9}}{18 c}+\frac {a b \,x^{3}}{6 c^{3}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}\) | \(151\) |
parallelrisch | \(-\frac {-3 b^{2} \arctan \left (c \,x^{3}\right )^{2} x^{12} c^{4}-6 a b \arctan \left (c \,x^{3}\right ) x^{12} c^{4}-3 c^{4} a^{2} x^{12}+2 b^{2} \arctan \left (c \,x^{3}\right ) x^{9} c^{3}+2 a b \,c^{3} x^{9}-x^{6} b^{2} c^{2}-6 b^{2} \arctan \left (c \,x^{3}\right ) x^{3} c -6 a b c \,x^{3}+3 b^{2} \arctan \left (c \,x^{3}\right )^{2}+4 b^{2} \ln \left (c^{2} x^{6}+1\right )+6 a b \arctan \left (c \,x^{3}\right )+b^{2}}{36 c^{4}}\) | \(155\) |
risch | \(-\frac {b^{2} \left (c^{4} x^{12}-1\right ) \ln \left (i c \,x^{3}+1\right )^{2}}{48 c^{4}}-\frac {i b \left (6 a \,c^{4} x^{12}+3 i b \,c^{4} x^{12} \ln \left (-i c \,x^{3}+1\right )-2 b \,c^{3} x^{9}+6 b c \,x^{3}-3 i b \ln \left (-i c \,x^{3}+1\right )\right ) \ln \left (i c \,x^{3}+1\right )}{72 c^{4}}+\frac {i a b \,x^{12} \ln \left (-i c \,x^{3}+1\right )}{12}-\frac {b^{2} x^{12} \ln \left (-i c \,x^{3}+1\right )^{2}}{48}+\frac {a^{2} x^{12}}{12}-\frac {i b^{2} x^{9} \ln \left (-i c \,x^{3}+1\right )}{36 c}-\frac {a b \,x^{9}}{18 c}+\frac {b^{2} x^{6}}{36 c^{2}}+\frac {i b^{2} x^{3} \ln \left (-i c \,x^{3}+1\right )}{12 c^{3}}+\frac {a b \,x^{3}}{6 c^{3}}+\frac {b^{2} \ln \left (-i c \,x^{3}+1\right )^{2}}{48 c^{4}}-\frac {a b \arctan \left (c \,x^{3}\right )}{6 c^{4}}-\frac {b^{2} \ln \left (c^{2} x^{6}+1\right )}{9 c^{4}}\) | \(280\) |
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Time = 0.26 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.04 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {3 \, a^{2} c^{4} x^{12} - 2 \, a b c^{3} x^{9} + b^{2} c^{2} x^{6} + 6 \, a b c x^{3} + 3 \, {\left (b^{2} c^{4} x^{12} - b^{2}\right )} \arctan \left (c x^{3}\right )^{2} - 4 \, b^{2} \log \left (c^{2} x^{6} + 1\right ) + 2 \, {\left (3 \, a b c^{4} x^{12} - b^{2} c^{3} x^{9} + 3 \, b^{2} c x^{3} - 3 \, a b\right )} \arctan \left (c x^{3}\right )}{36 \, c^{4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (112) = 224\).
Time = 147.67 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.96 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\begin {cases} \frac {a^{2} x^{12}}{12} + \frac {a b x^{12} \operatorname {atan}{\left (c x^{3} \right )}}{6} - \frac {a b x^{9}}{18 c} + \frac {a b x^{3}}{6 c^{3}} - \frac {a b \operatorname {atan}{\left (c x^{3} \right )}}{6 c^{4}} + \frac {b^{2} x^{12} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{12} - \frac {b^{2} x^{9} \operatorname {atan}{\left (c x^{3} \right )}}{18 c} + \frac {b^{2} x^{6}}{36 c^{2}} + \frac {b^{2} x^{3} \operatorname {atan}{\left (c x^{3} \right )}}{6 c^{3}} + \frac {2 b^{2} \sqrt {- \frac {1}{c^{2}}} \operatorname {atan}{\left (c x^{3} \right )}}{9 c^{3}} - \frac {2 b^{2} \log {\left (x - \sqrt [6]{- \frac {1}{c^{2}}} \right )}}{9 c^{4}} - \frac {2 b^{2} \log {\left (4 x^{2} + 4 x \sqrt [6]{- \frac {1}{c^{2}}} + 4 \sqrt [3]{- \frac {1}{c^{2}}} \right )}}{9 c^{4}} - \frac {b^{2} \operatorname {atan}^{2}{\left (c x^{3} \right )}}{12 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} x^{12}}{12} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.36 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {1}{12} \, b^{2} x^{12} \arctan \left (c x^{3}\right )^{2} + \frac {1}{12} \, a^{2} x^{12} + \frac {1}{18} \, {\left (3 \, x^{12} \arctan \left (c x^{3}\right ) - c {\left (\frac {c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac {3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )}\right )} a b - \frac {1}{36} \, {\left (2 \, c {\left (\frac {c^{2} x^{9} - 3 \, x^{3}}{c^{4}} + \frac {3 \, \arctan \left (c x^{3}\right )}{c^{5}}\right )} \arctan \left (c x^{3}\right ) - \frac {c^{2} x^{6} + 3 \, \arctan \left (c x^{3}\right )^{2} - 3 \, \log \left (18 \, c^{7} x^{6} + 18 \, c^{5}\right ) - \log \left (c^{2} x^{6} + 1\right )}{c^{4}}\right )} b^{2} \]
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Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.17 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {3 \, a^{2} c x^{12} + 2 \, {\left (3 \, c x^{12} \arctan \left (c x^{3}\right ) - \frac {3 \, \arctan \left (c x^{3}\right )}{c^{3}} - \frac {c^{9} x^{9} - 3 \, c^{7} x^{3}}{c^{9}}\right )} a b + {\left (3 \, c x^{12} \arctan \left (c x^{3}\right )^{2} - \frac {2 \, c^{3} x^{9} \arctan \left (c x^{3}\right ) - c^{2} x^{6} - 6 \, c x^{3} \arctan \left (c x^{3}\right ) + 3 \, \arctan \left (c x^{3}\right )^{2} + 4 \, \log \left (c^{2} x^{6} + 1\right )}{c^{3}}\right )} b^{2}}{36 \, c} \]
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Time = 1.17 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21 \[ \int x^{11} \left (a+b \arctan \left (c x^3\right )\right )^2 \, dx=\frac {a^2\,x^{12}}{12}-\frac {b^2\,{\mathrm {atan}\left (c\,x^3\right )}^2}{12\,c^4}+\frac {b^2\,x^{12}\,{\mathrm {atan}\left (c\,x^3\right )}^2}{12}-\frac {b^2\,\ln \left (c^2\,x^6+1\right )}{9\,c^4}+\frac {b^2\,x^6}{36\,c^2}+\frac {b^2\,x^3\,\mathrm {atan}\left (c\,x^3\right )}{6\,c^3}-\frac {b^2\,x^9\,\mathrm {atan}\left (c\,x^3\right )}{18\,c}+\frac {a\,b\,x^3}{6\,c^3}-\frac {a\,b\,x^9}{18\,c}-\frac {a\,b\,\mathrm {atan}\left (c\,x^3\right )}{6\,c^4}+\frac {a\,b\,x^{12}\,\mathrm {atan}\left (c\,x^3\right )}{6} \]
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