Integrand size = 24, antiderivative size = 661 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)+2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {26 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {13 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
[Out]
Time = 1.84 (sec) , antiderivative size = 661, normalized size of antiderivative = 1.00, number of steps used = 57, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {5070, 5082, 5064, 272, 65, 214, 5078, 5076, 4268, 2611, 2320, 6724, 5050, 5010, 5006, 4998} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=-\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^2 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {5 a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {26 i a^2 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+2 a^2 c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {a c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{x}-\frac {c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 x^2}+\frac {1}{3} a^2 c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-a^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )-\frac {13 i a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {13 i a^2 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {1}{3} a^2 c^2 \sqrt {a^2 c x^2+c}-\frac {1}{3} a^3 c^2 x \arctan (a x) \sqrt {a^2 c x^2+c} \]
[In]
[Out]
Rule 65
Rule 214
Rule 272
Rule 2320
Rule 2611
Rule 4268
Rule 4998
Rule 5006
Rule 5010
Rule 5050
Rule 5064
Rule 5070
Rule 5076
Rule 5078
Rule 5082
Rule 6724
Rubi steps \begin{align*} \text {integral}& = c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x} \, dx \\ & = c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^3} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x} \, dx\right )+\left (a^4 c^2\right ) \int x \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = \frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {1}{3} \left (2 a^3 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x) \, dx+c^3 \int \frac {\arctan (a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {x \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right ) \\ & = \frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\left (a c^3\right ) \int \frac {\arctan (a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a^2 c^3\right ) \int \frac {\arctan (a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c^3 \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}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\left (2 a^3 c^3\right ) \int \frac {\arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^2 c^3 \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}}\right ) \\ & = \frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\left (a^2 c^3\right ) \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx-\frac {\left (a^2 c^3 \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^2 c^3 \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 (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {\left (a^2 c^3 \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 (2 a^3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\right ) \\ & = \frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {1}{2} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )-\frac {\left (a^2 c^3 \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^2 c^3 \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^2 c^3 \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}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {4 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 c^3 \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^2 c^3 \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}}\right ) \\ & = \frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \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 {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+c^2 \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^2 c^3 \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^2 c^3 \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}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {4 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (2 i a^2 c^3 \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^2 c^3 \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}}\right )+\frac {\left (a^2 c^3 \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^2 c^3 \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 {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^2 c^3 \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^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^2 c^3 \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 {i a^2 c^3 \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 {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {\left (i a^2 c^3 \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^2 c^3 \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 a^2 c^3 \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^2 c^3 \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}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {4 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 c^3 \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^2 c^3 \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}}\right ) \\ & = \frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^2 c^3 \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^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^2 c^3 \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 {i a^2 c^3 \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 {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {4 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 c^3 \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^2 c^3 \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 {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^2 c^3 \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^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {i a^2 c^3 \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 {i a^2 c^3 \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 {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {4 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \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 {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}
Time = 7.59 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=2 a^2 c^2 \sqrt {c \left (1+a^2 x^2\right )} \left (\arctan (a x)^2+\frac {\arctan (a x)^2 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\frac {2 \left (\arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}\right )+\frac {1}{12} a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2+4 \arctan (a x)^2+2 \cos (2 \arctan (a x))-\frac {3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )+\frac {a^2 c^2 \sqrt {c \left (1+a^2 x^2\right )} \left (-4 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+4 \arctan (a x)^2 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )+8 \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+8 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )+8 \left (-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )+\arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-4 \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 \sqrt {1+a^2 x^2}} \]
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Time = 3.09 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 a^{4} \arctan \left (a x \right )^{2} x^{4}-2 \arctan \left (a x \right ) x^{3} a^{3}+14 x^{2} \arctan \left (a x \right )^{2} a^{2}+2 a^{2} x^{2}-6 x \arctan \left (a x \right ) a -3 \arctan \left (a x \right )^{2}\right )}{6 x^{2}}-\frac {c^{2} a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-15 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-26 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-26 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+6 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+30 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-30 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 \sqrt {a^{2} x^{2}+1}}\) | \(454\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^3} \,d x \]
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