Optimal. Leaf size=403 \[ \frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 x}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.52, antiderivative size = 403, 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, 4930, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191} \[ -\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {6 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{a^4 c^2}+\frac {6 x}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {a^2 c x^2+c}}+\frac {6 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {a^2 c x^2+c}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4898
Rule 4930
Rule 4964
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^3 \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {x \tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {x \tan ^{-1}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {3 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {3 \int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^3 c}\\ &=-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}+\frac {6 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}+\frac {\left (6 \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 )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 \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 )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\left (6 i \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 )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 i \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 )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\left (6 \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 )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {\left (6 \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 )}{a^4 c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 x}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {6 \tan ^{-1}(a x)}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {3 x \tan ^{-1}(a x)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\tan ^{-1}(a x)^3}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{a^4 c^2}-\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^4 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 308, normalized size = 0.76 \[ \frac {\sqrt {a^2 x^2+1} \left (\frac {6 a x}{\sqrt {a^2 x^2+1}}+\frac {3}{2} \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3-\frac {3 a x \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}-3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)+\frac {1}{2} \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^3 \cos \left (2 \tan ^{-1}(a x)\right )-3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \cos \left (2 \tan ^{-1}(a x)\right )-6 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+6 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+6 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-6 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )-3 \tan ^{-1}(a x)^2 \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+3 \tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )}{a^4 c \sqrt {c \left (a^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{3}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-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 [F] time = 3.35, size = 0, normalized size = 0.00 \[ \int \frac {x^{3} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{3} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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