Optimal. Leaf size=605 \[ -\frac {149 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {2 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {2 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {a^2 c x^2+c}}-\frac {2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {29}{60} c^2 \sqrt {a^2 c x^2+c}+c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {29}{60} a c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)+\frac {1}{30} c \left (a^2 c x^2+c\right )^{3/2}+\frac {1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {1}{10} a c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2 \]
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Rubi [A] time = 1.26, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {4950, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4890, 4886, 4878} \[ -\frac {149 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {a^2 c x^2+c}}+\frac {2 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}-\frac {2 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {2 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {29}{60} c^2 \sqrt {a^2 c x^2+c}+\frac {149 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {a^2 c x^2+c}}+c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {29}{60} a c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)-\frac {2 c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {a^2 c x^2+c}}+\frac {1}{30} c \left (a^2 c x^2+c\right )^{3/2}+\frac {1}{3} c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {1}{10} a c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{5} \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 4878
Rule 4886
Rule 4890
Rule 4930
Rule 4950
Rule 4956
Rule 4958
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2}{x} \, dx &=c \int \frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{x} \, dx+\left (a^2 c\right ) \int x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {1}{5} (2 a c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x) \, dx+c^2 \int \frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{x} \, dx+\left (a^2 c^2\right ) \int x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {1}{10} \left (3 a c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx-\frac {1}{3} \left (2 a c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx+c^3 \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx+\left (a^2 c^3\right ) \int \frac {x \tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {1}{20} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\frac {1}{3} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx-\left (2 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {\left (c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}-\frac {\left (3 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{20 \sqrt {c+a^2 c x^2}}-\frac {\left (a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{3 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 i c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{\sqrt {c+a^2 c x^2}}\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}-\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {\left (2 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ &=\frac {29}{60} c^2 \sqrt {c+a^2 c x^2}+\frac {1}{30} c \left (c+a^2 c x^2\right )^{3/2}-\frac {29}{60} a c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {1}{10} a c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {1}{3} c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{5} \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {149 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}+\frac {149 i c^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 \sqrt {c+a^2 c x^2}}-\frac {2 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{\sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 7.14, size = 889, normalized size = 1.47 \[ \sqrt {c \left (a^2 x^2+1\right )} \left (\frac {\left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}+\tan ^{-1}(a x)^2+\frac {2 i \left (\text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )-\text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-\frac {2 \left (\tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )+i \left (\text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )\right )\right )}{\sqrt {a^2 x^2+1}}+\frac {2 \left (\text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )-\text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt {a^2 x^2+1}}\right ) c^2+\frac {1}{6} \left (a^2 x^2+1\right ) \sqrt {c \left (a^2 x^2+1\right )} \left (4 \tan ^{-1}(a x)^2-\cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac {3 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+\cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-2 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+2 \cos \left (2 \tan ^{-1}(a x)\right )-\frac {4 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\frac {4 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+2\right ) c^2-\frac {1}{960} \left (a^2 x^2+1\right )^2 \sqrt {c \left (a^2 x^2+1\right )} \left (160 \cos \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-32 \tan ^{-1}(a x)^2-55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)-\frac {110 \log \left (1-i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+55 \cos \left (3 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+11 \cos \left (5 \tan ^{-1}(a x)\right ) \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)+\frac {110 \log \left (1+i e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+4 \sin \left (2 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-22 \sin \left (4 \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)+72 \cos \left (2 \tan ^{-1}(a x)\right )+22 \cos \left (4 \tan ^{-1}(a x)\right )-\frac {176 i \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+\frac {176 i \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{5/2}}+50\right ) c^2 \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x}, x\right ) \]
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 = 1.06, size = 404, normalized size = 0.67 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 \arctan \left (a x \right )^{2} x^{4} a^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+44 \arctan \left (a x \right )^{2} x^{2} a^{2}+2 a^{2} x^{2}-35 \arctan \left (a x \right ) x a +92 \arctan \left (a x \right )^{2}+31\right )}{60}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (60 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-60 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+149 i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-149 i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+120 i \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 i \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+120 \arctan \left (a x \right ) \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-120 \arctan \left (a x \right ) \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+149 \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-149 \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x} \,d x \]
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 \[ \int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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