Optimal. Leaf size=422 \[ -\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac {3 i a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 i a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {3 a^2 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {2 a^2}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 1.13, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 14, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4966, 4962, 4944, 266, 63, 208, 4958, 4956, 4183, 2531, 2282, 6589, 4930, 4894} \[ -\frac {3 i a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 i a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {3 a^2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}}-\frac {a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{c^2 x}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{2 c^2 x^2}-\frac {a^2 \tanh ^{-1}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{c^{3/2}}+\frac {2 a^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \tanh ^{-1}\left (e^{i \tan ^{-1}(a x)}\right )}{c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 63
Rule 208
Rule 266
Rule 2282
Rule 2531
Rule 4183
Rule 4894
Rule 4930
Rule 4944
Rule 4956
Rule 4958
Rule 4962
Rule 4966
Rule 6589
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x \left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^3 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+a^4 \int \frac {x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a \int \frac {\tan ^{-1}(a x)}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{2 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a^2 \int \frac {1}{x \sqrt {c+a^2 c x^2}} \, dx}{c}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)^2}{x \sqrt {1+a^2 x^2}} \, dx}{2 c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \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 {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {c+a^2 c x}} \, dx,x,x^2\right )}{2 c}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int x^2 \csc (x) \, dx,x,\tan ^{-1}(a x)\right )}{2 c \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \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 {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac {3 a^2 \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 {\operatorname {Subst}\left (\int \frac {1}{-\frac {1}{a^2}+\frac {x^2}{a^2 c}} \, dx,x,\sqrt {c+a^2 c x^2}\right )}{c^2}+\frac {\left (a^2 \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 (a^2 \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 (2 a^2 \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 (2 a^2 \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 {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac {3 a^2 \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 {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \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 {3 i a^2 \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 (i a^2 \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 (i a^2 \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 (2 i a^2 \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 (2 i a^2 \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 {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac {3 a^2 \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 {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \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 {3 i a^2 \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 (a^2 \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 (a^2 \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 (2 a^2 \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 (2 a^2 \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 {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \tan ^{-1}(a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}{c^2 x}-\frac {a^2 \tan ^{-1}(a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{2 c^2 x^2}+\frac {3 a^2 \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 {a^2 \tanh ^{-1}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \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 {3 i a^2 \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 {3 a^2 \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 {3 a^2 \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 = 2.39, size = 371, normalized size = 0.88 \[ \frac {a^2 \left (-24 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )+24 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )+24 \sqrt {a^2 x^2+1} \text {Li}_3\left (-e^{i \tan ^{-1}(a x)}\right )-24 \sqrt {a^2 x^2+1} \text {Li}_3\left (e^{i \tan ^{-1}(a x)}\right )-4 \sqrt {a^2 x^2+1} \tan \left (\frac {1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)-12 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1-e^{i \tan ^{-1}(a x)}\right )+12 \sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \log \left (1+e^{i \tan ^{-1}(a x)}\right )+8 \sqrt {a^2 x^2+1} \log \left (\tan \left (\frac {1}{2} \tan ^{-1}(a x)\right )\right )-\sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+\sqrt {a^2 x^2+1} \tan ^{-1}(a x)^2 \sec ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )-8 \tan ^{-1}(a x)^2+16 a x \tan ^{-1}(a x)-2 a x \tan ^{-1}(a x) \csc ^2\left (\frac {1}{2} \tan ^{-1}(a x)\right )+16\right )}{8 c \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}}, x\right ) \]
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 = 1.44, size = 376, normalized size = 0.89 \[ -\frac {a^{2} \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a^{2}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\left (2 a x +\arctan \left (a x \right )\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c^{2} x^{2}}-\frac {a^{2} \left (3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctanh \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \polylog \left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \polylog \left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c^{2}} \]
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 \[ \int \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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