Integrand size = 24, antiderivative size = 655 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\frac {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {c+a^2 c x^2}}-\frac {4 a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {2 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {15 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {15 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 1.03 (sec) , antiderivative size = 655, normalized size of antiderivative = 1.00, number of steps used = 43, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5070, 5064, 5078, 5074, 5010, 5008, 4266, 2611, 2320, 6724, 5000, 223, 212, 201} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=-\frac {4 a c^3 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {15 i a c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 \sqrt {a^2 c x^2+c}}-\frac {15 i a c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 \sqrt {a^2 c x^2+c}}-\frac {15 a c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 \sqrt {a^2 c x^2+c}}+\frac {15 a c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 \sqrt {a^2 c x^2+c}}-\frac {15 i a c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {a^2 c x^2+c}}+\frac {7}{8} a^2 c^2 x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{x}-\frac {7}{4} a c^2 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {1}{4} a^2 c x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {1}{6} a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}+\frac {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {2 i a c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i a c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{12} a^2 c^2 x \sqrt {a^2 c x^2+c} \]
[In]
[Out]
Rule 201
Rule 212
Rule 223
Rule 2320
Rule 2611
Rule 4266
Rule 5000
Rule 5008
Rule 5010
Rule 5064
Rule 5070
Rule 5074
Rule 5078
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^2} \, dx+\left (a^2 c\right ) \int \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx \\ & = -\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx+\frac {1}{6} \left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{4} \left (3 a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+\left (a^2 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = \frac {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+c^3 \int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (a^2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 a^2 c^3\right ) \int \frac {\arctan (a 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}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (3 a^2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\left (2 a c^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {1}{4} \left (3 a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\left (a^2 c^3\right ) \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\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 ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}} \\ & = \frac {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {\left (a c^3 \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 {\left (2 a c^3 \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}} \\ & = \frac {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {c+a^2 c x^2}}-\frac {4 a c^3 \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 {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {2 i a c^3 \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 {2 i a c^3 \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 a c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \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 (a c^3 \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 (2 a c^3 \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 (2 a c^3 \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 {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {c+a^2 c x^2}}-\frac {4 a c^3 \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 {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {2 i a c^3 \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 {2 i a c^3 \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 c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i a c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (i a c^3 \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 (i a c^3 \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 (2 i a c^3 \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 (2 i a c^3 \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 {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {c+a^2 c x^2}}-\frac {4 a c^3 \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 {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {2 i a c^3 \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 {2 i a c^3 \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 a c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a c^3 \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 )}{4 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \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 (a c^3 \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 (2 a c^3 \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 (2 a c^3 \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 {1}{12} a^2 c^2 x \sqrt {c+a^2 c x^2}-\frac {7}{4} a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{6} a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {7}{8} a^2 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{4} a^2 c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 \sqrt {c+a^2 c x^2}}-\frac {4 a c^3 \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 {11}{6} a c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}-\frac {15 i a c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {2 i a c^3 \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 {2 i a c^3 \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 {15 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}}+\frac {15 a c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 1.61 (sec) , antiderivative size = 626, normalized size of antiderivative = 0.96 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (2 a^2 x^2 \sqrt {1+a^2 x^2}+2 a^4 x^4 \sqrt {1+a^2 x^2}-190 a x \sqrt {1+a^2 x^2} \arctan (a x)+2 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)-96 \sqrt {1+a^2 x^2} \arctan (a x)^2+117 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)^2+21 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)^2-168 i a x \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+176 a x \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+6 a x \arctan (a x) \cos (3 \arctan (a x))+12 a^3 x^3 \arctan (a x) \cos (3 \arctan (a x))+6 a^5 x^5 \arctan (a x) \cos (3 \arctan (a x))+192 a x \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )+96 a x \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )-96 a x \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )-192 a x \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+192 i a x \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+360 i a x \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-360 i a x \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-192 i a x \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-360 a x \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+360 a x \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+2 a x \sin (3 \arctan (a x))+4 a^3 x^3 \sin (3 \arctan (a x))+2 a^5 x^5 \sin (3 \arctan (a x))-3 a x \arctan (a x)^2 \sin (3 \arctan (a x))-6 a^3 x^3 \arctan (a x)^2 \sin (3 \arctan (a x))-3 a^5 x^5 \arctan (a x)^2 \sin (3 \arctan (a x))\right )}{96 x \sqrt {1+a^2 x^2}} \]
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Time = 2.93 (sec) , antiderivative size = 399, normalized size of antiderivative = 0.61
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{4} \arctan \left (a x \right )^{2} x^{4}-4 \arctan \left (a x \right ) x^{3} a^{3}+27 x^{2} \arctan \left (a x \right )^{2} a^{2}+2 a^{2} x^{2}-46 x \arctan \left (a x \right ) a -24 \arctan \left (a x \right )^{2}\right )}{24 x}+\frac {i c^{2} a \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (45 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+48 i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+90 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-88 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+48 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+48 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 \sqrt {a^{2} x^{2}+1}}\) | \(399\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{2}} \,d x } \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^2} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^2} \,d x \]
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