Integrand size = 24, antiderivative size = 408 \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \sqrt {c}}+\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}} \]
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Time = 0.52 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.00, number of steps used = 24, 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} \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}+\frac {5 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {a^2 c x^2+c}}-\frac {5 i \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {a^2 c x^2+c}}-\frac {5 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {a^2 c x^2+c}}+\frac {5 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {a^2 c x^2+c}}-\frac {2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^4 c}-\frac {5 i \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^4 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^4 \sqrt {c}}-\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^3 c} \]
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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^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = -\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}+\frac {\int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^3}+\frac {2 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}+\frac {\int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^2} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^3}+\frac {\sqrt {1+a^2 x^2} \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^3 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 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^3}+\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 a^4 \sqrt {c+a^2 c x^2}}+\frac {\left (2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{a^4 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \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^4 \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^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \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^4 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \sqrt {c}}+\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \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^4 \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^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 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^4 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \sqrt {c}}+\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \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^4 \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^4 \sqrt {c+a^2 c x^2}}-\frac {\left (4 \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^4 \sqrt {c+a^2 c x^2}}+\frac {\left (4 \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^4 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \sqrt {c}}+\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.78 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\frac {12 \left (-5 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-\text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+5 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-5 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-5 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+5 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\left (1+a^2 x^2\right ) \arctan (a x) \left (-6+2 \arctan (a x)^2+6 \left (-1+\arctan (a x)^2\right ) \cos (2 \arctan (a x))+3 \arctan (a x) \sin (2 \arctan (a x))\right )\right )}{12 a^4 c} \]
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Time = 3.97 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.94
method | result | size |
default | \(\frac {\left (2 x^{2} \arctan \left (a x \right )^{2} a^{2}-3 x \arctan \left (a x \right ) a -4 \arctan \left (a x \right )^{2}+6\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,a^{4}}+\frac {5 \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}-\frac {5 \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}+\frac {2 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c}\) | \(382\) |
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\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Exception generated. \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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