Integrand size = 24, antiderivative size = 436 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.83 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{6 a}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{8 a^2}+\frac {1}{4} x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}+\frac {x \sqrt {a^2 c x^2+c}}{12 a^2}-\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{12 a^3}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{6 a^3} \]
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Rule 212
Rule 223
Rule 327
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rule 6724
Rubi steps \begin{align*} \text {integral}& = c \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} (3 c) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {c \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}-\frac {1}{2} (a c) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} c \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {(3 c) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+\frac {c \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {(3 c) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {(3 c) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\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 )}{a^2}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}-\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 )}{12 a^2}-\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 )}{3 a^2}-\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 )}{4 a^2}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 1.09 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (8 \left (3 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+\left (1+a^2 x^2\right )^{3/2} \left (\arctan (a x) \left (2+6 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))\right )-3 \arctan (a x)^2 \left (-7 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )+2 \left (a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )\right )\right )}{96 a^3 \sqrt {1+a^2 x^2}} \]
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Time = 1.40 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{3} \arctan \left (a x \right )^{2} x^{3}-4 a^{2} \arctan \left (a x \right ) x^{2}+3 a \arctan \left (a x \right )^{2} x +2 a x +2 \arctan \left (a x \right )\right )}{24 a^{3}}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-8 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(302\) |
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\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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