Integrand size = 22, antiderivative size = 160 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {11 \sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4} \]
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Time = 0.19 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5066, 5072, 327, 223, 212, 5050} \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{15 a^2}+\frac {1}{5} x^4 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {x^3 \sqrt {a^2 c x^2+c}}{20 a}-\frac {2 \arctan (a x) \sqrt {a^2 c x^2+c}}{15 a^4}+\frac {11 \sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{120 a^4}+\frac {x \sqrt {a^2 c x^2+c}}{24 a^3} \]
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Rule 212
Rule 223
Rule 327
Rule 5050
Rule 5066
Rule 5072
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{5} c \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{5} (a c) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {(2 c) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}-\frac {c \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a}+\frac {(3 c) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx}{20 a} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{30 a^3}-\frac {(3 c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^3}+\frac {(2 c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{30 a^3}-\frac {(3 c) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^3}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{15 a^3} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{24 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2}}{20 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {11 \sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\frac {a x \left (5-6 a^2 x^2\right ) \sqrt {c+a^2 c x^2}+8 \sqrt {c+a^2 c x^2} \left (-2+a^2 x^2+3 a^4 x^4\right ) \arctan (a x)+11 \sqrt {c} \log \left (a c x+\sqrt {c} \sqrt {c+a^2 c x^2}\right )}{120 a^4} \]
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Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 \arctan \left (a x \right ) a^{4} x^{4}-6 a^{3} x^{3}+8 a^{2} \arctan \left (a x \right ) x^{2}+5 a x -16 \arctan \left (a x \right )\right )}{120 a^{4}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-i\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}+\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+i\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(176\) |
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Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.59 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=-\frac {2 \, {\left (6 \, a^{3} x^{3} - 5 \, a x - 8 \, {\left (3 \, a^{4} x^{4} + a^{2} x^{2} - 2\right )} \arctan \left (a x\right )\right )} \sqrt {a^{2} c x^{2} + c} - 11 \, \sqrt {c} \log \left (-2 \, a^{2} c x^{2} - 2 \, \sqrt {a^{2} c x^{2} + c} a \sqrt {c} x - c\right )}{240 \, a^{4}} \]
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\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}{\left (a x \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.79 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=-\frac {1}{120} \, {\left (a {\left (\frac {3 \, {\left (\frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x}{a^{2}} - \frac {\sqrt {a^{2} x^{2} + 1} x}{a^{2}} - \frac {\operatorname {arsinh}\left (a x\right )}{a^{3}}\right )}}{a^{2}} - \frac {8 \, {\left (\sqrt {a^{2} x^{2} + 1} x + \frac {\operatorname {arsinh}\left (a x\right )}{a}\right )}}{a^{4}}\right )} - 8 \, {\left (\frac {3 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x^{2}}{a^{2}} - \frac {2 \, {\left (a^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{a^{4}}\right )} \arctan \left (a x\right )\right )} \sqrt {c} \]
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Exception generated. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x) \, dx=\int x^3\,\mathrm {atan}\left (a\,x\right )\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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