Integrand size = 24, antiderivative size = 344 \[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 \sqrt {c}}-\frac {i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}} \]
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Time = 0.25 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724} \[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {a^2 c x^2+c}}+\frac {i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^3 c}+\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^3 \sqrt {c}} \]
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Rule 212
Rule 223
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 5072
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {\int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}-\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {\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 {\sqrt {1+a^2 x^2} \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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {i \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 {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 \sqrt {c}}+\frac {\sqrt {1+a^2 x^2} \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 {\sqrt {1+a^2 x^2} \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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {i \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 {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 \sqrt {c}}-\frac {i \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 \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 \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 \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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {i \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 {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 \sqrt {c}}-\frac {i \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 \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 {\sqrt {1+a^2 x^2} \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 {\sqrt {1+a^2 x^2} \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 {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^3 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2 c}+\frac {i \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 {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3 \sqrt {c}}-\frac {i \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 \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 {\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 {\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}} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.51 \[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\arctan (a x) (-2+a x \arctan (a x))+\frac {2 \left (i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+\text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}\right )}{2 a^3 c} \]
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Time = 1.21 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {\left (x \arctan \left (a x \right ) a -2\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c \,a^{3}}-\frac {i \left (i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, a^{3} c}\) | \(274\) |
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\[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{2} \arctan \left (a x\right )^{2}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^2}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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