Optimal. Leaf size=364 \[ \frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
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Rubi [A] time = 1.14402, antiderivative size = 364, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4950, 4946, 4962, 264, 4958, 4954, 217, 206, 4930, 195} \[ \frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^2 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{6} a^3 c^2 x \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{2 x}+2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{2 x^2}-\frac{13}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{a^2 c x^2+c}}\right )-\frac{5 a^2 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tanh ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}+\frac{1}{3} a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x) \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4946
Rule 4962
Rule 264
Rule 4958
Rule 4954
Rule 217
Rule 206
Rule 4930
Rule 195
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x^3} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x} \, dx\\ &=c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^3} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x} \, dx\right )+\left (a^4 c^2\right ) \int x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{3} \left (a^3 c^2\right ) \int \sqrt{c+a^2 c x^2} \, dx-c^3 \int \frac{\tan ^{-1}(a x)}{x^3 \sqrt{c+a^2 c x^2}} \, dx+\left (a c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)+\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\left (a^3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx\right )\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{1}{2} \left (a c^3\right ) \int \frac{1}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\frac{1}{2} \left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{c+a^2 c x^2}} \, dx-\frac{1}{6} \left (a^3 c^3\right ) \int \frac{1}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{2 x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 a^2 c^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 )}{\sqrt{c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\right )-\frac{1}{6} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1-a^2 c x^2} \, dx,x,\frac{x}{\sqrt{c+a^2 c x^2}}\right )+\frac{\left (a^2 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{x \sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{a c^2 \sqrt{c+a^2 c x^2}}{2 x}-\frac{1}{6} a^3 c^2 x \sqrt{c+a^2 c x^2}-\frac{c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{2 x^2}+\frac{1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)-\frac{a^2 c^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 )}{\sqrt{c+a^2 c x^2}}-\frac{1}{6} a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+2 \left (a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{2 a^2 c^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 )}{\sqrt{c+a^2 c x^2}}-a^2 c^{5/2} \tanh ^{-1}\left (\frac{a \sqrt{c} x}{\sqrt{c+a^2 c x^2}}\right )+\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^2 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 2.02946, size = 361, normalized size = 0.99 \[ \frac{a^2 c^2 \sqrt{a^2 c x^2+c} \tan \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \left (60 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,-e^{i \tan ^{-1}(a x)}\right )-60 i \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \text{PolyLog}\left (2,e^{i \tan ^{-1}(a x)}\right )+4 a^3 x^3 \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-2 a^2 x^2 \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-6 \cot ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+28 a x \tan ^{-1}(a x) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )+60 \tan ^{-1}(a x) \log \left (1-e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-60 \tan ^{-1}(a x) \log \left (1+e^{i \tan ^{-1}(a x)}\right ) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )-3 \tan ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \csc ^2\left (\frac{1}{2} \tan ^{-1}(a x)\right )-4 \sinh ^{-1}(a x) \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right )+3 \tan ^{-1}(a x) \csc \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \sec \left (\frac{1}{2} \tan ^{-1}(a x)\right )+48 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )-\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-48 \cot \left (\frac{1}{2} \tan ^{-1}(a x)\right ) \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )+\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-6\right )}{24 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.356, size = 204, normalized size = 0.6 \begin{align*}{\frac{{c}^{2} \left ( 2\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-{a}^{3}{x}^{3}+14\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-3\,ax-3\,\arctan \left ( ax \right ) \right ) }{6\,{x}^{2}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{{a}^{2}{c}^{2}}{6}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 15\,\arctan \left ( ax \right ) \ln \left ( 1+{\frac{1+iax}{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) -26\,i\arctan \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,i{\it dilog} \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,i{\it dilog} \left ( 1+{(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{{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt{a^{2} c x^{2} + c} \arctan \left (a x\right )}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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