Optimal. Leaf size=397 \[ -\frac {a^2 \sqrt {a^2 c x^2+c}}{3 c^2 x}+\frac {5 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac {a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^2}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac {2 a^4 x}{c \sqrt {a^2 c x^2+c}}+\frac {a^4 x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}-\frac {11 i a^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}}+\frac {11 i a^3 \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}}+\frac {2 a^3 \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}}+\frac {22 a^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 1.20, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4966, 4962, 264, 4958, 4954, 4944, 4898, 191} \[ -\frac {11 i a^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}}+\frac {11 i a^3 \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}}-\frac {a^2 \sqrt {a^2 c x^2+c}}{3 c^2 x}+\frac {5 a^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac {a \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^2}-\frac {\sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac {2 a^4 x}{c \sqrt {a^2 c x^2+c}}+\frac {a^4 x \tan ^{-1}(a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 a^3 \tan ^{-1}(a x)}{c \sqrt {a^2 c x^2+c}}+\frac {22 a^3 \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 264
Rule 4898
Rule 4944
Rule 4954
Rule 4958
Rule 4962
Rule 4966
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx &=-\left (a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx\right )+\frac {\int \frac {\tan ^{-1}(a x)^2}{x^4 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {\sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c^2 x^3}+a^4 \int \frac {\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {(2 a) \int \frac {\tan ^{-1}(a x)}{x^3 \sqrt {c+a^2 c x^2}} \, dx}{3 c}-\frac {\left (2 a^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{3 c}-\frac {a^2 \int \frac {\tan ^{-1}(a x)^2}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=\frac {2 a^3 \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)}{3 c^2 x^2}+\frac {a^4 x \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}{3 c^2 x^3}+\frac {5 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c^2 x}-\left (2 a^4\right ) \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx+\frac {a^2 \int \frac {1}{x^2 \sqrt {c+a^2 c x^2}} \, dx}{3 c}-\frac {a^3 \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{3 c}-\frac {\left (4 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{3 c}-\frac {\left (2 a^3\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {c+a^2 c x^2}} \, dx}{c}\\ &=-\frac {2 a^4 x}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c^2 x}+\frac {2 a^3 \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)}{3 c^2 x^2}+\frac {a^4 x \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}{3 c^2 x^3}+\frac {5 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac {\left (a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{3 c \sqrt {c+a^2 c x^2}}-\frac {\left (4 a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{3 c \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {\tan ^{-1}(a x)}{x \sqrt {1+a^2 x^2}} \, dx}{c \sqrt {c+a^2 c x^2}}\\ &=-\frac {2 a^4 x}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c^2 x}+\frac {2 a^3 \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)}{3 c^2 x^2}+\frac {a^4 x \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}{3 c^2 x^3}+\frac {5 a^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}{3 c^2 x}+\frac {22 a^3 \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}}-\frac {11 i a^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}}+\frac {11 i a^3 \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 3.60, size = 270, normalized size = 0.68 \[ \frac {a^3 \sqrt {a^2 x^2+1} \left (\frac {\left (a^2 x^2+1\right )^{3/2} \left (\tan ^{-1}(a x) \left (\frac {66 a x \left (\log \left (1+e^{i \tan ^{-1}(a x)}\right )-\log \left (1-e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt {a^2 x^2+1}}+8 \sin \left (2 \tan ^{-1}(a x)\right )-6 \sin \left (4 \tan ^{-1}(a x)\right )+22 \left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \sin \left (3 \tan ^{-1}(a x)\right )\right )+\frac {88 i a^3 x^3 \text {Li}_2\left (e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+28 \cos \left (2 \tan ^{-1}(a x)\right )-6 \cos \left (4 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x)^2 \left (-36 \cos \left (2 \tan ^{-1}(a x)\right )+3 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )-22\right )}{a^3 x^3}-88 i \text {Li}_2\left (-e^{i \tan ^{-1}(a x)}\right )\right )}{24 c \sqrt {a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{2} x^{4}}, x\right ) \]
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 = 2.44, size = 318, normalized size = 0.80 \[ \frac {a^{3} \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a^{3}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\left (5 \arctan \left (a x \right )^{2} x^{2} a^{2}-a^{2} x^{2}-\arctan \left (a x \right ) x a -\arctan \left (a x \right )^{2}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 x^{3} c^{2}}+\frac {11 i a^{3} \left (-i \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+i \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )+\polylog \left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\polylog \left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, c^{2}} \]
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 \[ \int \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{4} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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