Optimal. Leaf size=349 \[ \frac {2 x}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}+\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.34, antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4964, 4890, 4888, 4181, 2531, 2282, 6589, 4898, 191} \[ \frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \text {PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {2 \sqrt {a^2 x^2+1} \text {PolyLog}\left (3,i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {a^2 c x^2+c}}+\frac {2 x}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}\left (e^{i \tan ^{-1}(a x)}\right ) \tan ^{-1}(a x)^2}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 2282
Rule 2531
Rule 4181
Rule 4888
Rule 4890
Rule 4898
Rule 4964
Rule 6589
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2}+\frac {\int \frac {\tan ^{-1}(a x)^2}{\sqrt {c+a^2 c x^2}} \, dx}{a^2 c}\\ &=-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {2 \int \frac {1}{\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)^2}{\sqrt {1+a^2 x^2}} \, dx}{a^2 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{a^2 c \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int x^2 \sec (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{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)^2}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 \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^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{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)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 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^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{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)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 \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^3 c \sqrt {c+a^2 c x^2}}+\frac {\left (2 \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^3 c \sqrt {c+a^2 c x^2}}\\ &=\frac {2 x}{a^2 c \sqrt {c+a^2 c x^2}}-\frac {2 \tan ^{-1}(a x)}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {x \tan ^{-1}(a x)^2}{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)^2}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )}{a^3 c \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \text {Li}_3\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 = 0.36, size = 228, normalized size = 0.65 \[ -\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 a x}{\sqrt {a^2 x^2+1}}+\frac {a x \tan ^{-1}(a x)^2}{\sqrt {a^2 x^2+1}}+\frac {2 \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}-2 i \tan ^{-1}(a x) \text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )+2 i \tan ^{-1}(a x) \text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )+2 \text {Li}_3\left (-i e^{i \tan ^{-1}(a x)}\right )-2 \text {Li}_3\left (i e^{i \tan ^{-1}(a x)}\right )+\tan ^{-1}(a x)^2 \left (-\log \left (1-i e^{i \tan ^{-1}(a x)}\right )\right )+\tan ^{-1}(a x)^2 \log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )}{a^3 c \sqrt {c \left (a^2 x^2+1\right )}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}}, x\right ) \]
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 = 2.72, size = 0, normalized size = 0.00 \[ \int \frac {x^{2} \arctan \left (a x \right )^{2}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {x^{2} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^2}{{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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