Optimal. Leaf size=495 \[ -\frac {x \tan ^{-1}(a x)^3}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {Li}_4\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.41, antiderivative size = 495, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4964, 4890, 4888, 4181, 2531, 6609, 2282, 6589, 4898, 4894} \[ \frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {3 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \text {PolyLog}\left (4,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^3}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4894
Rule 4898
Rule 4964
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {\tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {6 \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^3 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \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^3 c \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^3 c \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 x \tan ^{-1}(a x)}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {3 \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^3}{a^2 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \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^3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 1.75, size = 639, normalized size = 1.29 \[ -\frac {\sqrt {a^2 x^2+1} \left (-\frac {384}{\sqrt {a^2 x^2+1}}+\frac {64 a x \tan ^{-1}(a x)^3}{\sqrt {a^2 x^2+1}}+\frac {192 \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}-\frac {384 a x \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-192 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{-i \tan ^{-1}(a x)}\right )-192 i \tan ^{-1}(a x)^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+192 i \pi \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-384 \tan ^{-1}(a x) \text {Li}_3\left (-i e^{-i \tan ^{-1}(a x)}\right )+384 \tan ^{-1}(a x) \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-48 i \pi \left (\pi -4 \tan ^{-1}(a x)\right ) \text {Li}_2\left (i e^{-i \tan ^{-1}(a x)}\right )-48 i \pi ^2 \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+192 \pi \text {Li}_3\left (i e^{-i \tan ^{-1}(a x)}\right )-192 \pi \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )+384 i \text {Li}_4\left (-i e^{-i \tan ^{-1}(a x)}\right )+384 i \text {Li}_4\left (-i e^{i \tan ^{-1}(a x)}\right )-16 i \tan ^{-1}(a x)^4+32 i \pi \tan ^{-1}(a x)^3-24 i \pi ^2 \tan ^{-1}(a x)^2+8 i \pi ^3 \tan ^{-1}(a x)-64 \tan ^{-1}(a x)^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )+64 \tan ^{-1}(a x)^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+96 \pi \tan ^{-1}(a x)^2 \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )-96 \pi \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-48 \pi ^2 \tan ^{-1}(a x) \log \left (1-i e^{-i \tan ^{-1}(a x)}\right )+48 \pi ^2 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+8 \pi ^3 \log \left (1+i e^{-i \tan ^{-1}(a x)}\right )-8 \pi ^3 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )-8 \pi ^3 \log \left (\tan \left (\frac {1}{4} \left (2 \tan ^{-1}(a x)+\pi \right )\right )\right )+7 i \pi ^4\right )}{64 a^3 c \sqrt {c \left (a^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 3.40, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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