Integrand size = 24, antiderivative size = 675 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {26 a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
[Out]
Time = 1.59 (sec) , antiderivative size = 675, 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, 5066, 5082, 270, 5078, 5074, 5010, 5008, 4266, 2611, 2320, 6724, 5000, 223, 212} \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=-\frac {2 a^2 c^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{x}-\frac {a c^2 \arctan (a x) \sqrt {a^2 c x^2+c}}{3 x^2}-\frac {c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 x^3}-\frac {a^2 c^2 \sqrt {a^2 c x^2+c}}{3 x}+\frac {1}{2} a^4 c^2 x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {26 a^3 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}-a^3 c^2 \arctan (a x) \sqrt {a^2 c x^2+c}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}} \]
[In]
[Out]
Rule 212
Rule 223
Rule 270
Rule 2320
Rule 2611
Rule 4266
Rule 5000
Rule 5008
Rule 5010
Rule 5064
Rule 5066
Rule 5070
Rule 5074
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^4} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^2} \, dx \\ & = c^2 \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^4} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{x^2} \, dx\right )+\left (a^4 c^2\right ) \int \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx \\ & = -a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}+\frac {1}{3} \left (2 a c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{x^3} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac {\arctan (a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {\arctan (a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right ) \\ & = -a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {2 a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {1}{3} \left (2 a c^3\right ) \int \frac {\arctan (a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{3} \left (2 a^2 c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 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 (a^4 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}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\left (2 a^3 c^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^4 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}}\right ) \\ & = -\frac {2 a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )-\frac {1}{3} \left (a^2 c^3\right ) \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\frac {1}{3} \left (a^3 c^3\right ) \int \frac {\arctan (a x)}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^3 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}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}+\frac {\left (a^3 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^3 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}}\right ) \\ & = -\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\arctan (a x)}{x \sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (a^3 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^3 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}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 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 {2 i a^3 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^3 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 (2 a^3 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^3 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}}\right ) \\ & = -\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 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 )}{3 \sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (i a^3 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^3 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}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 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 {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 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^3 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 (2 i a^3 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^3 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}}\right ) \\ & = -\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 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 )}{3 \sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (a^3 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^3 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}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 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 {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 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^3 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 (2 a^3 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^3 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}}\right ) \\ & = -\frac {a^2 c^2 \sqrt {c+a^2 c x^2}}{3 x}-a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 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 )}{3 \sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 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 {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 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^3 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 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}\right ) \\ \end{align*}
Time = 2.94 (sec) , antiderivative size = 644, normalized size of antiderivative = 0.95 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=-\frac {c^3 \sqrt {1+a^2 x^2} \left (2 \left (1+a^2 x^2\right )^{3/2}+12 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)+24 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)^2-6 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)^2+4 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2+12 i a^3 x^3 \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-12 a^3 x^3 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-2 \left (1+a^2 x^2\right )^{3/2} \cos (2 \arctan (a x))-3 a x \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-51 a^3 x^3 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-24 a^3 x^3 \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+24 a^3 x^3 \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+3 a x \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+51 a^3 x^3 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-52 i a^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-60 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+60 i a^3 x^3 \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+52 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+60 a^3 x^3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-60 a^3 x^3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+2 \left (1+a^2 x^2\right )^{3/2} \arctan (a x) \sin (2 \arctan (a x))+\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right ) \sin (3 \arctan (a x))-\left (1+a^2 x^2\right )^{3/2} \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right ) \sin (3 \arctan (a x))\right )}{12 x^3 \sqrt {c+a^2 c x^2}} \]
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Time = 4.44 (sec) , antiderivative size = 401, normalized size of antiderivative = 0.59
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}-14 x^{2} \arctan \left (a x \right )^{2} a^{2}-2 a^{2} x^{2}-2 x \arctan \left (a x \right ) a -2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}+\frac {i c^{2} a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+30 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-12 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+26 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+26 \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )\right )}{6 \sqrt {a^{2} x^{2}+1}}\) | \(401\) |
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
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\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{4}} \,d x } \]
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Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \]
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