Integrand size = 24, antiderivative size = 788 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
[Out]
Time = 1.33 (sec) , antiderivative size = 788, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5070, 5064, 5082, 272, 65, 214, 5078, 5076, 4268, 2611, 2320, 6724, 5010, 5008, 4266, 6744} \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{x}-\frac {a^2 c \arctan (a x) \sqrt {a^2 c x^2+c}}{x}-\frac {a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 x^2}-\frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{3 x^3}-\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {3 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {3 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 a^3 c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {7 a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {6 i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {6 i a^3 c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i a^3 c^2 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right ) \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 2320
Rule 2611
Rule 4266
Rule 4268
Rule 5008
Rule 5010
Rule 5064
Rule 5070
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^4} \, dx+\left (a^2 c\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{x^2} \, dx \\ & = -\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+(a c) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^3} \, dx+\left (a^2 c^2\right ) \int \frac {\arctan (a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^2\right ) \int \frac {\arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+\left (a c^2\right ) \int \frac {\arctan (a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\left (3 a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^4 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}+\left (a^2 c^2\right ) \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^3 c^2\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^3 \sec (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}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}+\left (a^3 c^2\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \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 ) \text {Subst}\left (\int x^2 \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 ) \text {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, 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 ) \text {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {1}{2} \left (a^3 c^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \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^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \csc (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {8 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {8 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+(a c) \text {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )-\frac {\left (2 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \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^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \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^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,-i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (3,i e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^{i x}\right ) \, dx,x,\arctan (a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \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^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,-i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 i a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(3,i x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (6 a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {8 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (a^3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ & = -\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 9.61 (sec) , antiderivative size = 1508, normalized size of antiderivative = 1.91 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\frac {a^3 c \sqrt {c \left (1+a^2 x^2\right )} \csc \left (\frac {1}{2} \arctan (a x)\right ) \left (-\frac {7 i a \pi ^4 x}{\sqrt {1+a^2 x^2}}-\frac {8 i a \pi ^3 x \arctan (a x)}{\sqrt {1+a^2 x^2}}+\frac {24 i a \pi ^2 x \arctan (a x)^2}{\sqrt {1+a^2 x^2}}-64 \arctan (a x)^3-\frac {32 i a \pi x \arctan (a x)^3}{\sqrt {1+a^2 x^2}}+\frac {16 i a x \arctan (a x)^4}{\sqrt {1+a^2 x^2}}+\frac {48 a \pi ^2 x \arctan (a x) \log \left (1-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {96 a \pi x \arctan (a x)^2 \log \left (1-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {8 a \pi ^3 x \log \left (1+i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {64 a x \arctan (a x)^3 \log \left (1+i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 a x \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {8 a \pi ^3 x \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {48 a \pi ^2 x \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {96 a \pi x \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {64 a x \arctan (a x)^3 \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 a x \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {8 a \pi ^3 x \log \left (\tan \left (\frac {1}{4} (\pi +2 \arctan (a x))\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {48 i a \pi x (\pi -4 \arctan (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {48 i a \pi ^2 x \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 i a \pi x \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 i a x \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {192 a \pi x \operatorname {PolyLog}\left (3,i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 a x \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {192 a \pi x \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 a x \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\frac {384 a x \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \operatorname {PolyLog}\left (4,-i e^{-i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\frac {384 i a x \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )}{128 \sqrt {1+a^2 x^2}}+\frac {a^3 c^2 \sqrt {1+a^2 x^2} \left (-12 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right )-3 \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {a x \arctan (a x)^3 \csc ^4\left (\frac {1}{2} \arctan (a x)\right )}{2 \sqrt {1+a^2 x^2}}+12 \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )-12 \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+24 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-24 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-24 \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+24 \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+3 \arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {8 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^3 \sin ^4\left (\frac {1}{2} \arctan (a x)\right )}{a^3 x^3}-12 \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )-2 \arctan (a x)^3 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{24 \sqrt {c \left (1+a^2 x^2\right )}} \]
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Time = 4.73 (sec) , antiderivative size = 557, normalized size of antiderivative = 0.71
method | result | size |
default | \(-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (8 x^{2} \arctan \left (a x \right )^{2} a^{2}+6 a^{2} x^{2}+3 x \arctan \left (a x \right ) a +2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}-\frac {i c \,a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 i \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-7 i \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-2 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+6 \arctan \left (a x \right )^{2} \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 )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 i \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-14 i \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-14 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+14 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+7 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+14 i \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\) | \(557\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^4} \,d x \]
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