Optimal. Leaf size=798 \[ \frac {85 c^2 x \sqrt {c+a^2 c x^2}}{12096 a^3}-\frac {c^2 x^3 \sqrt {c+a^2 c x^2}}{240 a}-\frac {1}{504} a c^2 x^5 \sqrt {c+a^2 c x^2}-\frac {6157 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{60480 a^4}-\frac {47 c^2 x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{30240 a^2}+\frac {67 c^2 x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}{2520}+\frac {1}{84} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)+\frac {47 c^2 x \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{896 a^3}-\frac {205 c^2 x^3 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{4032 a}-\frac {103 a c^2 x^5 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}{1008}-\frac {1}{24} a^3 c^2 x^7 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2-\frac {115 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{63 a^4}+\frac {c^2 x^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3}{63 a^2}+\frac {5}{21} c^2 x^4 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {19}{63} a^2 c^2 x^6 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1}{9} a^4 c^2 x^8 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^3+\frac {1433 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 i c^3 \sqrt {1+a^2 x^2} \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}-\frac {115 c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}}+\frac {115 c^3 \sqrt {1+a^2 x^2} \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 13.57, antiderivative size = 798, normalized size of antiderivative = 1.00, number of steps
used = 547, number of rules used = 12, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5070, 5072,
5050, 223, 212, 5010, 5008, 4266, 2611, 2320, 6724, 327} \begin {gather*} \frac {1}{9} a^4 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^8-\frac {1}{24} a^3 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x^7+\frac {19}{63} a^2 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^6+\frac {1}{84} a^2 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) x^6-\frac {103 a c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x^5}{1008}-\frac {1}{504} a c^2 \sqrt {a^2 c x^2+c} x^5+\frac {5}{21} c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^4+\frac {67 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) x^4}{2520}-\frac {205 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x^3}{4032 a}-\frac {c^2 \sqrt {a^2 c x^2+c} x^3}{240 a}+\frac {c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3 x^2}{63 a^2}-\frac {47 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x) x^2}{30240 a^2}+\frac {47 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^2 x}{896 a^3}+\frac {85 c^2 \sqrt {a^2 c x^2+c} x}{12096 a^3}-\frac {2 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)^3}{63 a^4}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}\left (e^{i \text {ArcTan}(a x)}\right ) \text {ArcTan}(a x)^2}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {6157 c^2 \sqrt {a^2 c x^2+c} \text {ArcTan}(a x)}{60480 a^4}+\frac {1433 c^{5/2} \tanh ^{-1}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{15120 a^4}+\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (-i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 i c^3 \sqrt {a^2 x^2+1} \text {ArcTan}(a x) \text {Li}_2\left (i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}-\frac {115 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}}+\frac {115 c^3 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \text {ArcTan}(a x)}\right )}{1344 a^4 \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 223
Rule 327
Rule 2320
Rule 2611
Rule 4266
Rule 5008
Rule 5010
Rule 5050
Rule 5070
Rule 5072
Rule 6724
Rubi steps
\begin {align*} \int x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3 \, dx &=c \int x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^5 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3 \, dx\\ \end {align*}
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Mathematica [A]
time = 7.48, size = 1466, normalized size = 1.84 \begin {gather*} \frac {c^2 \sqrt {c+a^2 c x^2} \left (-2419200 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x)+2370816 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)-657578 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)+516096 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x)^3+2101248 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^3+273408 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^3-3483648 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x) \cos (2 \text {ArcTan}(a x))+3606912 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x) \cos (2 \text {ArcTan}(a x))-1083168 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x) \cos (2 \text {ArcTan}(a x))-2580480 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x)^3 \cos (2 \text {ArcTan}(a x))-1032192 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^3 \cos (2 \text {ArcTan}(a x))-1092096 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^3 \cos (2 \text {ArcTan}(a x))-1064448 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x) \cos (4 \text {ArcTan}(a x))+1592064 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x) \cos (4 \text {ArcTan}(a x))-576936 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x) \cos (4 \text {ArcTan}(a x))+1290240 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^3 \cos (4 \text {ArcTan}(a x))+193536 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^3 \cos (4 \text {ArcTan}(a x))+355968 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x) \cos (6 \text {ArcTan}(a x))-184160 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x) \cos (6 \text {ArcTan}(a x))-161280 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^3 \cos (6 \text {ArcTan}(a x))-32814 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x) \cos (8 \text {ArcTan}(a x))+662400 \text {ArcTan}(a x)^2 \log \left (1-i e^{i \text {ArcTan}(a x)}\right )+662400 \pi \text {ArcTan}(a x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (1-i e^{i \text {ArcTan}(a x)}\right )\right )-662400 \text {ArcTan}(a x)^2 \log \left (1+i e^{i \text {ArcTan}(a x)}\right )-662400 \text {ArcTan}(a x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+662400 \pi \text {ArcTan}(a x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left (-i+e^{i \text {ArcTan}(a x)}\right )\right )+662400 \text {ArcTan}(a x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \text {ArcTan}(a x)} \left ((1+i)+(1-i) e^{i \text {ArcTan}(a x)}\right )\right )-662400 \pi \text {ArcTan}(a x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )-1467392 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+662400 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )-\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )+1467392 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-662400 \text {ArcTan}(a x)^2 \log \left (\cos \left (\frac {1}{2} \text {ArcTan}(a x)\right )+\sin \left (\frac {1}{2} \text {ArcTan}(a x)\right )\right )-662400 \pi \text {ArcTan}(a x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \text {ArcTan}(a x))\right )\right )+1324800 i \text {ArcTan}(a x) \text {PolyLog}\left (2,-i e^{i \text {ArcTan}(a x)}\right )-1324800 i \text {ArcTan}(a x) \text {PolyLog}\left (2,i e^{i \text {ArcTan}(a x)}\right )-1324800 \text {PolyLog}\left (3,-i e^{i \text {ArcTan}(a x)}\right )+1324800 \text {PolyLog}\left (3,i e^{i \text {ArcTan}(a x)}\right )-193536 \left (1+a^2 x^2\right )^{5/2} \sin (2 \text {ArcTan}(a x))+232704 \left (1+a^2 x^2\right )^{7/2} \sin (2 \text {ArcTan}(a x))-74932 \left (1+a^2 x^2\right )^{9/2} \sin (2 \text {ArcTan}(a x))-96768 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))-364608 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))-39222 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^2 \sin (2 \text {ArcTan}(a x))-96768 \left (1+a^2 x^2\right )^{5/2} \sin (4 \text {ArcTan}(a x))+202752 \left (1+a^2 x^2\right )^{7/2} \sin (4 \text {ArcTan}(a x))-77908 \left (1+a^2 x^2\right )^{9/2} \sin (4 \text {ArcTan}(a x))+532224 \left (1+a^2 x^2\right )^{5/2} \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))+103680 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))+80226 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^2 \sin (4 \text {ArcTan}(a x))+57600 \left (1+a^2 x^2\right )^{7/2} \sin (6 \text {ArcTan}(a x))-36612 \left (1+a^2 x^2\right )^{9/2} \sin (6 \text {ArcTan}(a x))-177984 \left (1+a^2 x^2\right )^{7/2} \text {ArcTan}(a x)^2 \sin (6 \text {ArcTan}(a x))-19086 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^2 \sin (6 \text {ArcTan}(a x))-7238 \left (1+a^2 x^2\right )^{9/2} \sin (8 \text {ArcTan}(a x))+16407 \left (1+a^2 x^2\right )^{9/2} \text {ArcTan}(a x)^2 \sin (8 \text {ArcTan}(a x))\right )}{15482880 a^4 \sqrt {1+a^2 x^2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 7.73, size = 525, normalized size = 0.66
method | result | size |
default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (13440 \arctan \left (a x \right )^{3} a^{8} x^{8}-5040 \arctan \left (a x \right )^{2} a^{7} x^{7}+36480 \arctan \left (a x \right )^{3} a^{6} x^{6}+1440 \arctan \left (a x \right ) a^{6} x^{6}-12360 \arctan \left (a x \right )^{2} a^{5} x^{5}+28800 \arctan \left (a x \right )^{3} a^{4} x^{4}-240 a^{5} x^{5}+3216 \arctan \left (a x \right ) a^{4} x^{4}-6150 \arctan \left (a x \right )^{2} a^{3} x^{3}+1920 \arctan \left (a x \right )^{3} a^{2} x^{2}-504 a^{3} x^{3}-188 \arctan \left (a x \right ) a^{2} x^{2}+6345 \arctan \left (a x \right )^{2} a x -3840 \arctan \left (a x \right )^{3}+850 a x -12314 \arctan \left (a x \right )\right )}{120960 a^{4}}+\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {115 c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8064 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {1433 i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{7560 a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(525\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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