Integrand size = 24, antiderivative size = 385 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {11 \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}} \]
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Time = 1.03 (sec) , antiderivative size = 385, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5070, 5072, 267, 5010, 5006, 5050, 272, 45} \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{15 a^2}+\frac {1}{5} x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{10 a}-\frac {2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{15 a^4}-\frac {11 i c \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {a^2 c x^2+c}}+\frac {11 i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {a^2 c x^2+c}}-\frac {11 i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {a^2 c x^2+c}}+\frac {\left (a^2 c x^2+c\right )^{3/2}}{30 a^4 c}-\frac {11 \sqrt {a^2 c x^2+c}}{60 a^4}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{12 a^3} \]
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Rule 45
Rule 267
Rule 272
Rule 5006
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rubi steps \begin{align*} \text {integral}& = c \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x^5 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{5} (4 c) \int \frac {x^3 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {(2 c) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {(2 c) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}-\frac {1}{5} (2 a c) \int \frac {x^4 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{3 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{10} c \int \frac {x^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {(4 c) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^3}+\frac {c \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}+\frac {(8 c) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {(3 c) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{10 a}+\frac {(8 c) \int \frac {x^2 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a} \\ & = \frac {\sqrt {c+a^2 c x^2}}{3 a^4}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{20} c \text {Subst}\left (\int \frac {x}{\sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {(3 c) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^3}-\frac {(4 c) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {(16 c) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^3}-\frac {(3 c) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{20 a^2}-\frac {(4 c) \int \frac {x}{\sqrt {c+a^2 c x^2}} \, dx}{15 a^2}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (4 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2}}{12 a^4}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {10 i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {5 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 a^4 \sqrt {c+a^2 c x^2}}+\frac {1}{20} c \text {Subst}\left (\int \left (-\frac {1}{a^2 \sqrt {c+a^2 c x}}+\frac {\sqrt {c+a^2 c x}}{a^2 c}\right ) \, dx,x,x^2\right )-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (4 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (16 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{15 a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {11 \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 1.05 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.94 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {\left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (50-32 \arctan (a x)^2+72 \cos (2 \arctan (a x))+160 \arctan (a x)^2 \cos (2 \arctan (a x))+22 \cos (4 \arctan (a x))-\frac {110 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {110 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {176 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \arctan (a x) \sin (2 \arctan (a x))-22 \arctan (a x) \sin (4 \arctan (a x))\right )}{960 a^4} \]
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Time = 1.39 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+4 x^{2} \arctan \left (a x \right )^{2} a^{2}+2 a^{2} x^{2}+5 x \arctan \left (a x \right ) a -8 \arctan \left (a x \right )^{2}-9\right )}{60 a^{4}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(235\) |
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\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{2} \,d x } \]
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Exception generated. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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