Optimal. Leaf size=372 \[ \frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{2} a^3 c^2 \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{13}{6} a^3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 x^3} \]
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Rubi [A] time = 0.975109, antiderivative size = 372, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {4950, 4944, 266, 47, 63, 208, 4890, 4886, 4878} \[ \frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{a^2 c x^2+c}}-\frac{1}{2} a^3 c^2 \sqrt{a^2 c x^2+c}-\frac{a c^2 \sqrt{a^2 c x^2+c}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)-\frac{5 i a^3 c^3 \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{a^2 c x^2+c}}-\frac{2 a^2 c^2 \sqrt{a^2 c x^2+c} \tan ^{-1}(a x)}{x}-\frac{13}{6} a^3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a^2 c x^2+c}}{\sqrt{c}}\right )-\frac{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)}{3 x^3} \]
Antiderivative was successfully verified.
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Rule 4950
Rule 4944
Rule 266
Rule 47
Rule 63
Rule 208
Rule 4890
Rule 4886
Rule 4878
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)}{x^4} \, dx &=c \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^4} \, dx+\left (a^2 c\right ) \int \frac{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{x^2} \, dx\\ &=c^2 \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^4} \, dx+2 \left (\left (a^2 c^2\right ) \int \frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x^2} \, dx\right )+\left (a^4 c^2\right ) \int \sqrt{c+a^2 c x^2} \tan ^{-1}(a x) \, dx\\ &=-\frac{1}{2} a^3 c^2 \sqrt{c+a^2 c x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac{1}{3} \left (a c^2\right ) \int \frac{\sqrt{c+a^2 c x^2}}{x^3} \, dx+\frac{1}{2} \left (a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx+2 \left (\left (a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{x^2 \sqrt{c+a^2 c x^2}} \, dx+\left (a^4 c^3\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx\right )\\ &=-\frac{1}{2} a^3 c^2 \sqrt{c+a^2 c x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}+\frac{1}{6} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{\sqrt{c+a^2 c x}}{x^2} \, dx,x,x^2\right )+\frac{\left (a^4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}+\left (a^3 c^3\right ) \int \frac{1}{x \sqrt{c+a^2 c x^2}} \, dx+\frac{\left (a^4 c^3 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{\sqrt{c+a^2 c x^2}}\right )\\ &=-\frac{1}{2} a^3 c^2 \sqrt{c+a^2 c x^2}-\frac{a c^2 \sqrt{c+a^2 c x^2}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{1}{12} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{1}{2} \left (a^3 c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{c+a^2 c x}} \, dx,x,x^2\right )\right )\\ &=-\frac{1}{2} a^3 c^2 \sqrt{c+a^2 c x^2}-\frac{a c^2 \sqrt{c+a^2 c x^2}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+\frac{1}{6} \left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}+\left (a c^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{a^2}+\frac{x^2}{a^2 c}} \, dx,x,\sqrt{c+a^2 c x^2}\right )\right )\\ &=-\frac{1}{2} a^3 c^2 \sqrt{c+a^2 c x^2}-\frac{a c^2 \sqrt{c+a^2 c x^2}}{6 x^2}+\frac{1}{2} a^4 c^2 x \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)-\frac{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}{3 x^3}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-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^3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{2 \sqrt{c+a^2 c x^2}}+2 \left (-\frac{a^2 c^2 \sqrt{c+a^2 c x^2} \tan ^{-1}(a x)}{x}-\frac{2 i a^3 c^3 \sqrt{1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-a^3 c^{5/2} \tanh ^{-1}\left (\frac{\sqrt{c+a^2 c x^2}}{\sqrt{c}}\right )+\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}-\frac{i a^3 c^3 \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{\sqrt{c+a^2 c x^2}}\right )\\ \end{align*}
Mathematica [A] time = 0.980372, size = 313, normalized size = 0.84 \[ \frac{c^2 \sqrt{a^2 c x^2+c} \left (15 i a^3 x^3 \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )-15 i a^3 x^3 \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-3 a^3 x^3 \sqrt{a^2 x^2+1}-a x \sqrt{a^2 x^2+1}+3 a^4 x^4 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-14 a^2 x^2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-2 \sqrt{a^2 x^2+1} \tan ^{-1}(a x)-a^3 x^3 \tanh ^{-1}\left (\sqrt{a^2 x^2+1}\right )+15 a^3 x^3 \tan ^{-1}(a x) \log \left (1-i e^{i \tan ^{-1}(a x)}\right )-15 a^3 x^3 \tan ^{-1}(a x) \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+12 a^3 x^3 \log \left (\sin \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )-12 a^3 x^3 \log \left (\cos \left (\frac{1}{2} \tan ^{-1}(a x)\right )\right )\right )}{6 x^3 \sqrt{a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.487, size = 270, normalized size = 0.7 \begin{align*}{\frac{{c}^{2} \left ( 3\,\arctan \left ( ax \right ){x}^{4}{a}^{4}-3\,{a}^{3}{x}^{3}-14\,\arctan \left ( ax \right ){a}^{2}{x}^{2}-ax-2\,\arctan \left ( ax \right ) \right ) }{6\,{x}^{3}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{{\frac{i}{6}}{a}^{3}{c}^{2}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) } \left ( 15\,i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -13\,i\ln \left ({(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}}-1 \right ) +13\,i\ln \left ( 1+{(1+iax){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -15\,{\it dilog} \left ( 1-{\frac{i \left ( 1+iax \right ) }{\sqrt{{a}^{2}{x}^{2}+1}}} \right ) +15\,{\it dilog} \left ( 1+{\frac{i \left ( 1+iax \right ) }{\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^{4}}, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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