Optimal. Leaf size=377 \[ -\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac {6 i a \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 i a \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {6 a \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {6 a}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {a^2 c x^2+c}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.58, antiderivative size = 377, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4966, 4944, 4958, 4956, 4183, 2531, 2282, 6589, 4898, 4894} \[ \frac {6 i a \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 i a \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {6 a \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^3}{c^2 x}+\frac {6 a}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {a^2 c x^2+c}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}}-\frac {6 a \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 4183
Rule 4894
Rule 4898
Rule 4944
Rule 4956
Rule 4958
Rule 4966
Rule 6589
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^3}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^3}{\left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^3}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\left (6 a^2\right ) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {(3 a) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}+\frac {\left (3 a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac {6 a \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x \log \left (1+e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac {6 a \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 i a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 i a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac {6 a \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {\left (6 a \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}\\ &=\frac {6 a}{c \sqrt {c+a^2 c x^2}}+\frac {6 a^2 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {3 a \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 x \tan ^{-1}(a x)^3}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^3}{c^2 x}-\frac {6 a \sqrt {1+a^2 x^2} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 i a \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {6 a \sqrt {1+a^2 x^2} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {6 a \sqrt {1+a^2 x^2} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 1.60, size = 301, normalized size = 0.80 \[ \frac {a \left (12 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )-12 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )-12 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )+12 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )+6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )-6 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )-\frac {2 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^3 \sin ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )}{a x}-2 a x \tan ^{-1}(a x)^3-6 \tan ^{-1}(a x)^2+12 a x \tan ^{-1}(a x)-\frac {1}{2} a x \tan ^{-1}(a x)^3 \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+12\right )}{2 c \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{3}}{a^{4} c^{2} x^{6} + 2 \, a^{2} c^{2} x^{4} + c^{2} x^{2}}, x\right ) \]
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 [A] time = 0.99, size = 356, normalized size = 0.94 \[ -\frac {a \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )+3 i \arctan \left (a x \right )^{2}-6 i\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{3}-6 \arctan \left (a x \right )-3 i \arctan \left (a x \right )^{2}+6 i\right ) a}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\arctan \left (a x \right )^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{x \,c^{2}}+\frac {3 a \left (\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )-\arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )+2 \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{2}} \]
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 \[ \int \frac {\arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^2\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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