Optimal. Leaf size=760 \[ \frac {5 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {9 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 i c^2 \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 c x^2+c}}{4 a}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac {9 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac {1}{4} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) \]
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Rubi [A] time = 0.52, antiderivative size = 760, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 10, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {4880, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4886, 4878} \[ \frac {5 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {9 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {9 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 c^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {9 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}+\frac {9 i c^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {a^2 c x^2+c}}-\frac {3 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^2 \sqrt {a^2 x^2+1} \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right ) \tan ^{-1}(a x)}{a \sqrt {a^2 c x^2+c}}-\frac {c \sqrt {a^2 c x^2+c}}{4 a}+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^3+\frac {3}{8} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3-\frac {\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}-\frac {9 c \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{8 a}+\frac {1}{4} c x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4181
Rule 4878
Rule 4880
Rule 4886
Rule 4888
Rule 4890
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx &=-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{2} c \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x) \, dx+\frac {1}{4} (3 c) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {1}{4} c^2 \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (3 c^2\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{4} \left (9 c^2\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3+\frac {\left (c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{8 \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{4 \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}+\frac {\left (3 c^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{8 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \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 )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \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 )}{8 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {\left (9 i c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}+\frac {\left (9 i c^2 \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 )}{4 a \sqrt {c+a^2 c x^2}}\\ &=-\frac {c \sqrt {c+a^2 c x^2}}{4 a}+\frac {1}{4} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)-\frac {9 c \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{8 a}-\frac {\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{4 a}+\frac {3}{8} c x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3+\frac {1}{4} x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3-\frac {3 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{4 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \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 )}{a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^2 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a \sqrt {c+a^2 c x^2}}-\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 c^2 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}-\frac {9 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}+\frac {9 i c^2 \sqrt {1+a^2 x^2} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{4 a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [B] time = 12.84, size = 2105, normalized size = 2.77 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}, x\right ) \]
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 = 0.93, size = 466, normalized size = 0.61 \[ \frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{3} x^{3}-2 \arctan \left (a x \right )^{2} x^{2} a^{2}+5 \arctan \left (a x \right )^{3} x a +2 \arctan \left (a x \right ) x a -11 \arctan \left (a x \right )^{2}-2\right )}{8 a}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-9 i \arctan \left (a x \right )^{2} \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+9 i \arctan \left (a x \right )^{2} \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 \arctan \left (a x \right ) \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+18 i \polylog \left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 \arctan \left (a x \right ) \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-18 i \polylog \left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+20 i \dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-20 i \dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}\,{d x} \]
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 \[ \int {\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]
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 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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