Optimal. Leaf size=381 \[ \frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x}{27 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^3 x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a^2 x}{27 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.685573, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4966, 4944, 4958, 4954, 4898, 191, 4900, 192} \[ \frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}-\frac{2 i a \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{94 a^2 x}{27 c^2 \sqrt{a^2 c x^2+c}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{a^2 c x^2+c}}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{c^3 x}-\frac{4 a \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{a^2 c x^2+c}}+\frac{2 a^2 x}{27 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4966
Rule 4944
Rule 4958
Rule 4954
Rule 4898
Rule 191
Rule 4900
Rule 192
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\left (a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx\right )+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=-\frac{2 a \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{1}{9} \left (2 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{5/2}} \, dx+\frac{\int \frac{\tan ^{-1}(a x)^2}{x^2 \sqrt{c+a^2 c x^2}} \, dx}{c^2}-\frac{\left (2 a^2\right ) \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}-\frac{a^2 \int \frac{\tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^3 x}+\frac{(2 a) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx}{c^2}+\frac{\left (4 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{27 c}+\frac{\left (4 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 c}+\frac{\left (2 a^2\right ) \int \frac{1}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{c}\\ &=\frac{2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^3 x}+\frac{\left (2 a \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2 a^2 x}{27 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{94 a^2 x}{27 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 a \tan ^{-1}(a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 a \tan ^{-1}(a x)}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{a^2 x \tan ^{-1}(a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{5 a^2 x \tan ^{-1}(a x)^2}{3 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{c^3 x}-\frac{4 a \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 )}{c^2 \sqrt{c+a^2 c x^2}}+\frac{2 i a \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i a \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.60847, size = 296, normalized size = 0.78 \[ -\frac{a \left (-216 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+216 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+54 \sqrt{a^2 x^2+1} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \tan ^{-1}(a x)^2-216 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right )+216 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right )+9 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)^2 \sin \left (3 \tan ^{-1}(a x)\right )-2 \sqrt{a^2 x^2+1} \sin \left (3 \tan ^{-1}(a x)\right )+6 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \cos \left (3 \tan ^{-1}(a x)\right )-378 a x+189 a x \tan ^{-1}(a x)^2+378 \tan ^{-1}(a x)+27 a x \tan ^{-1}(a x)^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )}{108 c^2 \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.345, size = 433, normalized size = 1.1 \begin{align*}{\frac{a \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ({a}^{3}{x}^{3}-3\,i{a}^{2}{x}^{2}-3\,ax+i \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{7\,a \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( ax-i \right ) }{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( 7\,ax+7\,i \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) a}{8\,{c}^{3} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ({a}^{3}{x}^{3}+3\,i{a}^{2}{x}^{2}-3\,ax-i \right ) \left ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) a}{216\,{c}^{3} \left ({a}^{4}{x}^{4}+2\,{a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{{c}^{3}x}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{2\,ia}{{c}^{3}} \left ( i\arctan \left ( ax \right ) \ln \left ( 1-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it polylog} \left ( 2,{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it polylog} \left ( 2,-{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) \right ) \sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }{\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atan}^{2}{\left (a x \right )}}{x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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