Optimal. Leaf size=275 \[ \frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{2 a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 0.426936, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4944, 4946, 4962, 264, 4958, 4954} \[ \frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{i a^3 c \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{2 a^3 c \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
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Rule 4944
Rule 4946
Rule 4962
Rule 264
Rule 4958
Rule 4954
Rubi steps
\begin{align*} \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{x^4} \, dx &=-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{1}{3} (2 a) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx\\ &=-\frac{2 a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{1}{3} (2 a c) \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (2 a^2 c\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{2 a^2 \sqrt{c+a^2 c x^2}}{3 x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{1}{3} \left (a^2 c\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{3} \left (a^3 c\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2}}{3 x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}+\frac{\left (a^3 c \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{3 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a^2 \sqrt{c+a^2 c x^2}}{3 x}-\frac{a \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{3 x^2}-\frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}{3 c x^3}-\frac{2 a^3 c \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 \sqrt{c+a^2 c x^2}}+\frac{i a^3 c \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.66286, size = 239, normalized size = 0.87 \[ -\frac{c \sqrt{a^2 x^2+1} \left (-4 i a^3 x^3 \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )+4 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+\sqrt{a^2 x^2+1} \left (4 a^2 x^2+4 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+\tan ^{-1}(a x) \left (a x \left (-3 \sqrt{a^2 x^2+1} \log \left (1-e^{i \tan ^{-1}(a x)}\right )+3 \sqrt{a^2 x^2+1} \log \left (1+e^{i \tan ^{-1}(a x)}\right )+4\right )+\left (a^2 x^2+1\right ) \left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \sin \left (3 \tan ^{-1}(a x)\right )\right )\right )\right )}{12 x^3 \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.549, size = 195, normalized size = 0.7 \begin{align*} -{\frac{ \left ( \arctan \left ( ax \right ) \right ) ^{2}{x}^{2}{a}^{2}+{a}^{2}{x}^{2}+\arctan \left ( ax \right ) xa+ \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3\,{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{{\frac{i}{3}}{a}^{3}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \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 ){\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}}{x^{4}}, 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{\sqrt{c \left (a^{2} x^{2} + 1\right )} \operatorname{atan}^{2}{\left (a x \right )}}{x^{4}}\, 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{\sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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