Integrand size = 22, antiderivative size = 373 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^2}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5050, 5000, 5010, 5008, 4266, 2611, 2320, 6724, 223, 212} \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^2 \sqrt {a^2 c x^2+c}}+\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}+\frac {i c \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {a^2 c x^2+c}}-\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a}+\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2} \]
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Rule 212
Rule 223
Rule 2320
Rule 2611
Rule 4266
Rule 5000
Rule 5008
Rule 5010
Rule 5050
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {c \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a}-\frac {c \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\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}-\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 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} 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^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} 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 \log \left (1-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{a^2 \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^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^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^2 \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^2 \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^2 \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^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^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^2 \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^2 \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^2 \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^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^2}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a}+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^2 \sqrt {c+a^2 c x^2}}+\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 a^2 c}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^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^2 \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^2 \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^2 \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^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.55 \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\frac {12 \left (i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-\text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+\left (1+a^2 x^2\right ) \arctan (a x) \left (6+4 \arctan (a x)^2+6 \cos (2 \arctan (a x))-3 \arctan (a x) \sin (2 \arctan (a x))\right )\right )}{12 a^2} \]
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Time = 3.95 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.99
method | result | size |
default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 x^{2} \arctan \left (a x \right )^{2} a^{2}-3 x \arctan \left (a x \right ) a +2 \arctan \left (a x \right )^{2}+6\right )}{6 a^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 a^{2} \sqrt {a^{2} x^{2}+1}}+\frac {2 i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2} \sqrt {a^{2} x^{2}+1}}\) | \(370\) |
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\[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3} \,d x } \]
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\[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
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\[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x \arctan \left (a x\right )^{3} \,d x } \]
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Exception generated. \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]
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