Optimal. Leaf size=622 \[ -\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {68}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{27 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 1.00, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4964, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4898, 4894, 4944, 4940, 4930, 266, 43} \[ \frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {68}{9 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {a^2 c x^2+c}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{27 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4894
Rule 4898
Rule 4930
Rule 4940
Rule 4944
Rule 4964
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^4 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\int \frac {x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a}+\frac {\int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac {\int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {3 \tan ^{-1}(a x)^2}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a}+\frac {6 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}+\frac {2 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^3 c}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {6 x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\operatorname {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{9 a}+\frac {4 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {22}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac {1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{9 a}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \log \left (1+i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {68}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {68}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (-i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_3\left (i e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {68}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(i x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {68}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \tan ^{-1}(a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \tan ^{-1}(a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \tan ^{-1}(a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \tan ^{-1}(a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 2.92, size = 691, normalized size = 1.11 \[ -\frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {12960}{\sqrt {a^2 x^2+1}}+\frac {2160 a x \tan ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}+\frac {6480 \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}-\frac {12960 a x \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-5184 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{-i \tan ^{-1}(a x)}\right )-5184 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+5184 i \pi \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-10368 \tan ^{-1}(a x) \text {Li}_3\left (-i e^{-i \tan ^{-1}(a x)}\right )+10368 \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-1296 i \pi \left (\pi -4 \tan ^{-1}(a x)\right ) \text {Li}_2\left (i e^{-i \tan ^{-1}(a x)}\right )-1296 i \pi ^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+5184 \pi \text {Li}_3\left (i e^{-i \tan ^{-1}(a x)}\right )-5184 \pi \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+10368 i \text {Li}_4\left (-i e^{-i \tan ^{-1}(a x)}\right )+10368 i \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )-432 i \tan ^{-1}(a x)^4+864 i \pi \tan ^{-1}(a x)^3-648 i \pi ^2 \tan ^{-1}(a x)^2+216 i \pi ^3 \tan ^{-1}(a x)-1728 \tan ^{-1}(a x)^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )+1728 \tan ^{-1}(a x)^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+2592 \pi \tan ^{-1}(a x)^2 \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )-2592 \pi \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-1296 \pi ^2 \tan ^{-1}(a x) \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )+1296 \pi ^2 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+216 \pi ^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )-216 \pi ^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-216 \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (2 \tan ^{-1}(a x)+\pi \right )\right )\right )-144 \tan ^{-1}(a x)^3 \sin \left (3 \tan ^{-1}(a x)\right )+96 \tan ^{-1}(a x) \sin \left (3 \tan ^{-1}(a x)\right )-144 \tan ^{-1}(a x)^2 \cos \left (3 \tan ^{-1}(a x)\right )+32 \cos \left (3 \tan ^{-1}(a x)\right )+189 i \pi ^4\right )}{1728 a^5 c^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{4} \arctan \left (a x\right )^{3}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.31, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {x^{4} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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