Integrand size = 24, antiderivative size = 531 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}+\frac {31 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}-\frac {19 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {41 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}} \]
[Out]
Time = 2.25 (sec) , antiderivative size = 531, normalized size of antiderivative = 1.00, number of steps used = 92, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072, 5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=-\frac {19 c x^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{180 a}+\frac {c x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{16 a^2}+\frac {1}{6} a^2 c x^5 \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {1}{15} a c x^4 \arctan (a x) \sqrt {a^2 c x^2+c}+\frac {7}{24} c x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}+\frac {c x \sqrt {a^2 c x^2+c}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {a^2 c x^2+c}-\frac {i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}-\frac {c^2 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {i c^2 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {a^2 c x^2+c}}+\frac {31 c \arctan (a x) \sqrt {a^2 c x^2+c}}{360 a^3}-\frac {41 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a^3} \]
[In]
[Out]
Rule 212
Rule 223
Rule 327
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rule 6724
Rubi steps \begin{align*} \text {integral}& = c \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = c^2 \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )+\left (a^4 c^2\right ) \int \frac {x^6 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx \\ & = \frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2}+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c^2 \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{2 a^2}-\frac {c^2 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a}+2 \left (\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{4} \left (3 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{2} \left (a c^2\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx\right )-\frac {1}{6} \left (5 a^2 c^2\right ) \int \frac {x^4 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a^3 c^2\right ) \int \frac {x^5 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx \\ & = -\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)}{a^3}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{8} \left (5 c^2\right ) \int \frac {x^2 \arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{a^2}+2 \left (-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (3 c^2\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+\frac {c^2 \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{3 a}+\frac {\left (3 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{4 a}\right )+\frac {1}{15} \left (4 a c^2\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{12} \left (5 a c^2\right ) \int \frac {x^3 \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{15} \left (a^2 c^2\right ) \int \frac {x^4}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{2 a^2 \sqrt {c+a^2 c x^2}} \\ & = \frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)}{a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {1}{20} c^2 \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{45} \left (4 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{36} \left (5 c^2\right ) \int \frac {x^2}{\sqrt {c+a^2 c x^2}} \, dx-\frac {\left (5 c^2\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{16 a^2}+\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{a^2}-\frac {\left (8 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{45 a}-\frac {\left (5 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{18 a}-\frac {\left (5 c^2\right ) \int \frac {x \arctan (a x)}{\sqrt {c+a^2 c x^2}} \, dx}{8 a}-\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{12 a^2}-\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{3 a^2}-\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{4 a^2}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{8 a^2 \sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}+\frac {c^2 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{40 a^2}+\frac {\left (2 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{72 a^2}+\frac {\left (8 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{45 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{18 a^2}+\frac {\left (5 c^2\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx}{8 a^2}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{12 a^2}-\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{3 a^2}-\frac {\left (3 c^2\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 )}{4 a^2}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{8 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 a^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \text {Subst}\left (\int \frac {1}{1-a^2 c x^2} \, dx,x,\frac {x}{\sqrt {c+a^2 c x^2}}\right )}{40 a^2}+\frac {\left (2 c^2\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 )}{45 a^2}+\frac {\left (5 c^2\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 )}{72 a^2}+\frac {\left (8 c^2\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 )}{45 a^2}+\frac {\left (5 c^2\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 )}{18 a^2}+\frac {\left (5 c^2\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 )}{8 a^2}+\frac {\left (i 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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (i 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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int x^2 \sec (x) \, dx,x,\arctan (a x)\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {\left (3 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 )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 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 )}{4 a^3 \sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 i 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 )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 i 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 )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (5 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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (c^2 \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 )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (c^2 \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 )}{a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (5 i 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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 i 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 )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (3 c^2 \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 a^3 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \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 a^3 \sqrt {c+a^2 c x^2}}\right ) \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (5 c^2 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {\left (5 c^2 \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 )}{8 a^3 \sqrt {c+a^2 c x^2}} \\ & = -\frac {5 c x \sqrt {c+a^2 c x^2}}{36 a^2}+\frac {1}{60} c x^3 \sqrt {c+a^2 c x^2}-\frac {749 c \sqrt {c+a^2 c x^2} \arctan (a x)}{360 a^3}+\frac {41 c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{180 a}-\frac {1}{15} a c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {13 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{16 a^2}-\frac {5}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {799 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a^3}-\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {13 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {13 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+2 \left (\frac {c x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {13 c \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}-\frac {3 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {7 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}+\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}
Time = 2.99 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.99 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (960 \left (3 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+32 \left (-45 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+19 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+45 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-45 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-45 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+45 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+120 \left (1+a^2 x^2\right )^{3/2} \left (\arctan (a x) \left (2+6 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))\right )-3 \arctan (a x)^2 \left (-7 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )+2 \left (a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )\right )+\left (1+a^2 x^2\right )^3 \left (-\frac {56 a x}{\sqrt {1+a^2 x^2}}+\arctan (a x) \left (\frac {12}{\sqrt {1+a^2 x^2}}+110 \cos (3 \arctan (a x))-90 \cos (5 \arctan (a x))\right )-108 \sin (3 \arctan (a x))-52 \sin (5 \arctan (a x))+15 \arctan (a x)^2 \left (\frac {78 a x}{\sqrt {1+a^2 x^2}}-47 \sin (3 \arctan (a x))+3 \sin (5 \arctan (a x))\right )\right )\right )}{11520 a^3 \sqrt {1+a^2 x^2}} \]
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Time = 1.43 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.64
method | result | size |
default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 a^{5} \arctan \left (a x \right )^{2} x^{5}-48 \arctan \left (a x \right ) a^{4} x^{4}+210 a^{3} \arctan \left (a x \right )^{2} x^{3}+12 a^{3} x^{3}-76 a^{2} \arctan \left (a x \right ) x^{2}+45 a \arctan \left (a x \right )^{2} x +20 a x +62 \arctan \left (a x \right )\right )}{720 a^{3}}-\frac {i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (45 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-45 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+90 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-90 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-164 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(338\) |
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\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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