Optimal. Leaf size=397 \[ -\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 c \sqrt{a^2 c x^2+c}}+\frac{11 i a^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 c \sqrt{a^2 c x^2+c}}-\frac{a^2 \sqrt{a^2 c x^2+c}}{3 c^2 x}+\frac{5 a^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x}-\frac{a \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{3 c^2 x^2}-\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{3 c^2 x^3}-\frac{2 a^4 x}{c \sqrt{a^2 c x^2+c}}+\frac{a^4 x \tan ^{-1}(a x)^2}{c \sqrt{a^2 c x^2+c}}+\frac{2 a^3 \tan ^{-1}(a x)}{c \sqrt{a^2 c x^2+c}}+\frac{22 a^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{3 c \sqrt{a^2 c x^2+c}} \]
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Rubi [A] time = 1.20162, antiderivative size = 397, normalized size of antiderivative = 1., 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 4966
Rule 4962
Rule 264
Rule 4958
Rule 4954
Rule 4944
Rule 4898
Rule 191
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*}
Mathematica [A] time = 3.41547, size = 270, normalized size = 0.68 \[ \frac{a^3 \sqrt{a^2 x^2+1} \left (\frac{\left (a^2 x^2+1\right )^{3/2} \left (\frac{88 i a^3 x^3 \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )}{\left (a^2 x^2+1\right )^{3/2}}+\tan ^{-1}(a x) \left (\frac{66 a x \left (\log \left (1+e^{i \tan ^{-1}(a x)}\right )-\log \left (1-e^{i \tan ^{-1}(a x)}\right )\right )}{\sqrt{a^2 x^2+1}}+8 \sin \left (2 \tan ^{-1}(a x)\right )-6 \sin \left (4 \tan ^{-1}(a x)\right )+22 \left (\log \left (1-e^{i \tan ^{-1}(a x)}\right )-\log \left (1+e^{i \tan ^{-1}(a x)}\right )\right ) \sin \left (3 \tan ^{-1}(a x)\right )\right )+28 \cos \left (2 \tan ^{-1}(a x)\right )-6 \cos \left (4 \tan ^{-1}(a x)\right )+\tan ^{-1}(a x)^2 \left (-36 \cos \left (2 \tan ^{-1}(a x)\right )+3 \cos \left (4 \tan ^{-1}(a x)\right )+25\right )-22\right )}{a^3 x^3}-88 i \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )\right )}{24 c \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.683, size = 318, normalized size = 0.8 \begin{align*}{\frac{{a}^{3} \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2+2\,i\arctan \left ( ax \right ) \right ) \left ( ax-i \right ) }{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( ax+i \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ){a}^{3}}{ \left ( 2\,{a}^{2}{x}^{2}+2 \right ){c}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{5\, \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\,{c}^{2}{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{{\frac{11\,i}{3}}{a}^{3}}{{c}^{2}} \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^{4} c^{2} x^{8} + 2 \, a^{2} c^{2} x^{6} + c^{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{\operatorname{atan}^{2}{\left (a x \right )}}{x^{4} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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