Optimal. Leaf size=675 \[ -\frac {a^2 c^2 \sqrt {a^2 c x^2+c}}{3 x}-\frac {2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac {a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac {c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+\frac {1}{2} a^4 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2+a^3 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )+\frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 c x^2+c}}-\frac {26 a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) \]
[Out]
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Rubi [A] time = 2.31, antiderivative size = 675, normalized size of antiderivative = 1.00, number of steps used = 48, number of rules used = 16, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4950, 4944, 4946, 4962, 264, 4958, 4954, 4890, 4888, 4181, 2531, 2282, 6589, 4880, 217, 206} \[ \frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {13 i a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}+\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {5 a^3 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {a^2 c^2 \sqrt {a^2 c x^2+c}}{3 x}+\frac {1}{2} a^4 c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {5 i a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 c x^2+c}}-a^3 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {2 a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{x}-\frac {a c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}+a^3 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )-\frac {26 a^3 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {a^2 c x^2+c}}-\frac {c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 217
Rule 264
Rule 2282
Rule 2531
Rule 4181
Rule 4880
Rule 4888
Rule 4890
Rule 4944
Rule 4946
Rule 4950
Rule 4954
Rule 4958
Rule 4962
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{x^4} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^4} \, dx+\left (a^2 c\right ) \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x^2} \, dx\\ &=c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^2} \, dx\right )+\left (a^4 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=-a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+\frac {1}{3} \left (2 a c^2\right ) \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {\tan ^{-1}(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 {\tan ^{-1}(a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\right )\\ &=-a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {2 a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac {1}{3} \left (2 a c^3\right ) \int \frac {\tan ^{-1}(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 ) \operatorname {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 {\tan ^{-1}(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} \tan ^{-1}(a x)^2}{x}+\left (2 a^3 c^3\right ) \int \frac {\tan ^{-1}(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 {\tan ^{-1}(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} \tan ^{-1}(a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}+a^3 c^{5/2} \tanh ^{-1}\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 {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx+\frac {\left (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(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} \tan ^{-1}(a x)^2}{x}+\frac {\left (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(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 {\tan ^{-1}(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} \tan ^{-1}(a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}+a^3 c^{5/2} \tanh ^{-1}\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 {\tan ^{-1}(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 ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \text {Li}_2\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} \text {Li}_2\left (\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 ) \operatorname {Subst}\left (\int x \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(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} \tan ^{-1}(a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\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} \text {Li}_2\left (\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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (i a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\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} \text {Li}_2\left (\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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(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 ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(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} \tan ^{-1}(a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\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} \text {Li}_2\left (\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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\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} \text {Li}_2\left (\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 ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 a^3 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \tan ^{-1}(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} \tan ^{-1}(a x)-\frac {a c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2-\frac {c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 x^3}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {2 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\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} \text {Li}_2\left (\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} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+2 \left (-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {4 a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\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} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_2\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} \text {Li}_2\left (\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} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^3 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\right )\\ \end {align*}
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Mathematica [A] time = 4.76, size = 644, normalized size = 0.95 \[ -\frac {c^3 \sqrt {a^2 x^2+1} \left (-52 i a^3 x^3 \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )-60 i a^3 x^3 \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+60 i a^3 x^3 \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+52 i a^3 x^3 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )+60 a^3 x^3 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-60 a^3 x^3 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+12 i a^3 x^3 \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-51 a^3 x^3 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )-24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+24 a^3 x^3 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+51 a^3 x^3 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+2 \left (a^2 x^2+1\right )^{3/2}+24 a^2 x^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2+4 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2+2 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \sin \left (2 \tan ^{-1}(a x)\right )-2 \left (a^2 x^2+1\right )^{3/2} \cos \left (2 \tan ^{-1}(a x)\right )+\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )-\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \sin \left (3 \tan ^{-1}(a x)\right )-6 a^4 x^4 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2+12 a^3 x^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)-12 a^3 x^3 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )-3 a x \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )+3 a x \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )\right )}{12 x^3 \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.81, size = 401, normalized size = 0.59 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{2} x^{4} a^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}-14 \arctan \left (a x \right )^{2} x^{2} a^{2}-2 a^{2} x^{2}-2 \arctan \left (a x \right ) x a -2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}-\frac {i a^{3} c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-26 i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+30 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+12 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-26 \dilog \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-26 \dilog \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 \sqrt {a^{2} x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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