Optimal. Leaf size=516 \[ \frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a}+\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {a^2 c x^2+c}}+\frac {17}{180} c^2 x \sqrt {a^2 c x^2+c}+\frac {5}{16} c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {5 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac {1}{60} c x \left (a^2 c x^2+c\right )^{3/2}+\frac {5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{36 a}+\frac {1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2-\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{15 a} \]
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Rubi [A] time = 0.39, antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 21, 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, 2282, 6589, 217, 206, 195} \[ \frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}-\frac {5 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}+\frac {5 c^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {a^2 c x^2+c}}+\frac {17}{180} c^2 x \sqrt {a^2 c x^2+c}-\frac {5 i c^3 \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {a^2 c x^2+c}}+\frac {5}{16} c^2 x \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2-\frac {5 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{8 a}+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{360 a}+\frac {1}{60} c x \left (a^2 c x^2+c\right )^{3/2}+\frac {5}{24} c x \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2-\frac {5 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{36 a}+\frac {1}{6} x \left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)^2-\frac {\left (a^2 c x^2+c\right )^{5/2} \tan ^{-1}(a x)}{15 a} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 2282
Rule 2531
Rule 4181
Rule 4880
Rule 4888
Rule 4890
Rule 6589
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2 \, dx &=-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {1}{15} c \int \left (c+a^2 c x^2\right )^{3/2} \, dx+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {1}{20} c^2 \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{36} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \, dx+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2 \, dx\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {1}{40} c^3 \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{72} \left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{16} \left (5 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx+\frac {1}{8} \left (5 c^3\right ) \int \frac {1}{\sqrt {c+a^2 c x^2}} \, dx\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {1}{40} c^3 \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 {1}{72} \left (5 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 {1}{8} \left (5 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 (5 c^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {c+a^2 c x^2}}\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {\left (5 c^3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{16 a \sqrt {c+a^2 c x^2}}\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}-\frac {\left (5 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 )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 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 )}{8 a \sqrt {c+a^2 c x^2}}\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {\left (5 i 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 )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 i 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 )}{8 a \sqrt {c+a^2 c x^2}}\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {\left (5 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 )}{8 a \sqrt {c+a^2 c x^2}}+\frac {\left (5 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 )}{8 a \sqrt {c+a^2 c x^2}}\\ &=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \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^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 1.66, size = 771, normalized size = 1.49 \[ \frac {c^2 \sqrt {a^2 c x^2+c} \left (-108 a^6 x^6 \sin \left (3 \tan ^{-1}(a x)\right )-705 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-52 a^6 x^6 \sin \left (5 \tan ^{-1}(a x)\right )+45 a^6 x^6 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+110 a^6 x^6 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-90 a^6 x^6 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+156 a^4 x^4 \sin \left (3 \tan ^{-1}(a x)\right )-2835 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-156 a^4 x^4 \sin \left (5 \tan ^{-1}(a x)\right )+135 a^4 x^4 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+1770 a^4 x^4 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-270 a^4 x^4 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )+424 a x \sqrt {a^2 x^2+1}+504 a^2 x^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+11970 a x \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2-11028 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+8288 \tanh ^{-1}\left (\frac {a x}{\sqrt {a^2 x^2+1}}\right )+636 a^2 x^2 \sin \left (3 \tan ^{-1}(a x)\right )-3555 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-156 a^2 x^2 \sin \left (5 \tan ^{-1}(a x)\right )+135 a^2 x^2 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )+3210 a^2 x^2 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-270 a^2 x^2 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )-56 a^5 x^5 \sqrt {a^2 x^2+1}+1170 a^5 x^5 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2+12 a^4 x^4 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+368 a^3 x^3 \sqrt {a^2 x^2+1}+7380 a^3 x^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2+7200 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-7200 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )-7200 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+7200 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )-7200 i \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2-1425 \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )+372 \sin \left (3 \tan ^{-1}(a x)\right )+45 \tan ^{-1}(a x)^2 \sin \left (5 \tan ^{-1}(a x)\right )-52 \sin \left (5 \tan ^{-1}(a x)\right )+1550 \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-90 \tan ^{-1}(a x) \cos \left (5 \tan ^{-1}(a x)\right )\right )}{11520 a \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.96, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 0.82, size = 342, normalized size = 0.66 \[ \frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 \arctan \left (a x \right )^{2} x^{5} a^{5}-48 \arctan \left (a x \right ) x^{4} a^{4}+390 \arctan \left (a x \right )^{2} x^{3} a^{3}+12 a^{3} x^{3}-196 \arctan \left (a x \right ) a^{2} x^{2}+495 \arctan \left (a x \right )^{2} x a +80 a x -598 \arctan \left (a x \right )\right )}{720 a}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (225 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-225 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 i \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 i \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1036 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a \sqrt {a^{2} x^{2}+1}} \]
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 \[ \int {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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 \[ \int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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