Integrand size = 24, antiderivative size = 919 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=-\frac {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+\frac {6 i a^2 c^2 \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}}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {3 a^2 c^2 \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 {6 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \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 a^2 c^2 \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 {9 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 {9 a^2 c^2 \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 {9 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
[Out]
Time = 1.42 (sec) , antiderivative size = 919, normalized size of antiderivative = 1.00, number of steps used = 50, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5070, 5082, 5064, 5078, 5074, 5076, 4268, 2611, 6744, 2320, 6724, 5050, 5010, 5008, 4266} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=-\frac {3 a^2 c^2 \sqrt {a^2 x^2+1} \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}+a^2 c \sqrt {a^2 c x^2+c} \arctan (a x)^3-\frac {c \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 x^2}+\frac {6 i a^2 c^2 \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}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right ) \arctan (a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right ) \arctan (a x)^2}{2 \sqrt {a^2 c x^2+c}}-\frac {3 a c \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 x}-\frac {6 a^2 c^2 \sqrt {a^2 x^2+1} \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}+\frac {6 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}+\frac {9 a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right ) \arctan (a x)}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a^2 c^2 \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 a^2 c^2 \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 {9 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {9 i a^2 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}} \]
[In]
[Out]
Rule 2320
Rule 2611
Rule 4266
Rule 4268
Rule 5008
Rule 5010
Rule 5050
Rule 5064
Rule 5070
Rule 5074
Rule 5076
Rule 5078
Rule 5082
Rule 6724
Rule 6744
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x} \, dx \\ & = c^2 \int \frac {\arctan (a x)^3}{x^3 \sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\frac {1}{2} \left (3 a c^2\right ) \int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c^2\right ) \int \frac {\arctan (a x)^3}{x \sqrt {c+a^2 c x^2}} \, dx-\left (3 a^3 c^2\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \frac {\left (a^2 c^2 \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}} \\ & = -\frac {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\left (3 a^2 c^2\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+2 \frac {\left (a^2 c^2 \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^3 c^2 \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}} \\ & = -\frac {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {\left (a^2 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \csc (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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}}\right )-\frac {\left (3 a^2 c^2 \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 {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+\frac {6 i a^2 c^2 \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}}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\frac {a^2 c^2 \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 {6 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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}}\right )+\frac {\left (3 a^2 c^2 \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 )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 a^2 c^2 \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 )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 a^2 c^2 \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 a^2 c^2 \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 {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+\frac {6 i a^2 c^2 \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}}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}+\frac {a^2 c^2 \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 {6 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (3 i a^2 c^2 \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 (3 i a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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}}+2 \left (-\frac {2 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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 a^2 c^2 \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}}\right ) \\ & = \text {Too large to display} \\ \end{align*}
Time = 6.82 (sec) , antiderivative size = 691, normalized size of antiderivative = 0.75 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\frac {a^2 c \sqrt {c+a^2 c x^2} \left (-12 \arctan (a x)^2-3 i \pi ^4 \cot \left (\frac {1}{2} \arctan (a x)\right )+6 i \arctan (a x)^4 \cot \left (\frac {1}{2} \arctan (a x)\right )-12 \arctan (a x)^2 \cot ^2\left (\frac {1}{2} \arctan (a x)\right )+8 a x \arctan (a x)^3 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+24 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{-i \arctan (a x)}\right )+48 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-48 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-i e^{i \arctan (a x)}\right )+48 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+i e^{i \arctan (a x)}\right )-48 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )-24 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+72 i \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{-i \arctan (a x)}\right )+24 i \left (2+3 \arctan (a x)^2\right ) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-96 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+96 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-48 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+144 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,e^{-i \arctan (a x)}\right )-144 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+96 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-96 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )-144 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (4,e^{-i \arctan (a x)}\right )-144 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )+2 \arctan (a x)^3 \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{16 \sqrt {1+a^2 x^2}} \]
[In]
[Out]
Time = 5.25 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (2 a^{2} \arctan \left (a x \right ) x^{2}-3 a x -\arctan \left (a x \right )\right )}{2 x^{2}}-\frac {3 c \,a^{2} \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 )+2 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \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 )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 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 \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \ln \left (1-\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 )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \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 )-4 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 )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\) | \(592\) |
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^3} \,d x \]
[In]
[Out]