Integrand size = 24, antiderivative size = 600 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
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Time = 0.49 (sec) , antiderivative size = 600, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5078, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5010, 5008, 4266} \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=-\frac {2 c \sqrt {a^2 x^2+1} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i c \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i c \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 i c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 c \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 i c \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\arctan (a x)^3 \sqrt {a^2 c x^2+c}+\frac {6 i c \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {a^2 c x^2+c}} \]
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Rule 2320
Rule 2611
Rule 4266
Rule 4268
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5076
Rule 5078
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \sqrt {c+a^2 c x^2} \arctan (a x)^3-(3 a c) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {\left (c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^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 )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i c \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 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 )}{\sqrt {c+a^2 c x^2}} \\ & = \frac {6 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+\sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {2 c \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (-i \pi ^4+8 \sqrt {1+a^2 x^2} \arctan (a x)^3+2 i \arctan (a x)^4+8 \arctan (a x)^3 \log \left (1-e^{-i \arctan (a x)}\right )-24 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+24 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-8 \arctan (a x)^3 \log \left (1+e^{i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+24 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-48 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+48 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+48 \arctan (a x) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-48 \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+48 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-48 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-48 i \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )\right )}{8 \sqrt {1+a^2 x^2}} \]
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Time = 5.57 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.76
method | result | size |
default | \(\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{3}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{\sqrt {a^{2} x^{2}+1}}\) | \(454\) |
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x} \,d x } \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\int \frac {\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}}{x}\, dx \]
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\[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\int { \frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c}}{x} \,d x \]
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