Optimal. Leaf size=400 \[ -\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.81751, antiderivative size = 400, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4964, 4930, 4890, 4886, 4894, 4940, 266, 43} \[ -\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{32}{9 a^6 c^2 \sqrt{a^2 c x^2+c}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{a^2 c x^2+c} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{4 i \sqrt{a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac{\sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{a^2 c x^2+c}}+\frac{2}{27 a^6 c \left (a^2 c x^2+c\right )^{3/2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4964
Rule 4930
Rule 4890
Rule 4886
Rule 4894
Rule 4940
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^5 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac{\int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac{\int \frac{x^3 \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{2 \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{9 a^2}+\frac{\int \frac{x \tan ^{-1}(a x)^2}{\sqrt{c+a^2 c x^2}} \, dx}{a^4 c^2}-\frac{2 \int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^4 c}-\frac{\int \frac{x \tan ^{-1}(a x)^2}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^4 c}\\ &=-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{\operatorname{Subst}\left (\int \frac{x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{9 a^2}-\frac{2 \int \frac{\tan ^{-1}(a x)}{\sqrt{c+a^2 c x^2}} \, dx}{a^5 c^2}-\frac{4 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^5 c}-\frac{2 \int \frac{\tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^5 c}\\ &=-\frac{10}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac{1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{9 a^2}-\frac{\left (2 \sqrt{1+a^2 x^2}\right ) \int \frac{\tan ^{-1}(a x)}{\sqrt{1+a^2 x^2}} \, dx}{a^5 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{2}{27 a^6 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{32}{9 a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 x^3 \tan ^{-1}(a x)}{9 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{10 x \tan ^{-1}(a x)}{3 a^5 c^2 \sqrt{c+a^2 c x^2}}+\frac{x^2 \tan ^{-1}(a x)^2}{3 a^4 c \left (c+a^2 c x^2\right )^{3/2}}+\frac{5 \tan ^{-1}(a x)^2}{3 a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{c+a^2 c x^2} \tan ^{-1}(a x)^2}{a^6 c^3}+\frac{4 i \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 )}{a^6 c^2 \sqrt{c+a^2 c x^2}}-\frac{2 i \sqrt{1+a^2 x^2} \text{Li}_2\left (-\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{c+a^2 c x^2}}+\frac{2 i \sqrt{1+a^2 x^2} \text{Li}_2\left (\frac{i \sqrt{1+i a x}}{\sqrt{1-i a x}}\right )}{a^6 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 1.31527, size = 229, normalized size = 0.57 \[ \frac{-432 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,-i e^{i \tan ^{-1}(a x)}\right )+432 i \sqrt{a^2 x^2+1} \text{PolyLog}\left (2,i e^{i \tan ^{-1}(a x)}\right )-9 \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2 \left (-20 \cos \left (2 \tan ^{-1}(a x)\right )+\cos \left (4 \tan ^{-1}(a x)\right )-45\right )+6 \tan ^{-1}(a x) \left (-72 \sqrt{a^2 x^2+1} \log \left (1-i e^{i \tan ^{-1}(a x)}\right )+72 \sqrt{a^2 x^2+1} \log \left (1+i e^{i \tan ^{-1}(a x)}\right )+\left (a^2 x^2+1\right ) \sin \left (4 \tan ^{-1}(a x)\right )-124 a x\right )+8 \left (\cos \left (2 \tan ^{-1}(a x)\right )-95\right )}{216 a^6 c^2 \sqrt{a^2 c x^2+c}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 1.515, size = 454, normalized size = 1.1 \begin{align*}{\frac{ \left ( 6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) \left ( i{x}^{3}{a}^{3}+3\,{a}^{2}{x}^{2}-3\,iax-1 \right ) }{216\, \left ({a}^{2}{x}^{2}+1 \right ) ^{2}{c}^{3}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}+{\frac{ \left ( 7\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-14+14\,i\arctan \left ( ax \right ) \right ) \left ( 1+iax \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( -7+7\,iax \right ) \left ( \left ( \arctan \left ( ax \right ) \right ) ^{2}-2-2\,i\arctan \left ( ax \right ) \right ) }{8\,{c}^{3}{a}^{6} \left ({a}^{2}{x}^{2}+1 \right ) }\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{ \left ( i{x}^{3}{a}^{3}-3\,{a}^{2}{x}^{2}-3\,iax+1 \right ) \left ( -6\,i\arctan \left ( ax \right ) +9\, \left ( \arctan \left ( ax \right ) \right ) ^{2}-2 \right ) }{216\,{c}^{3}{a}^{6} \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}{a}^{6}}\sqrt{c \left ( ax-i \right ) \left ( ax+i \right ) }}-{\frac{2\,i}{{c}^{3}{a}^{6}} \left ( i\arctan \left ( ax \right ) \ln \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -i\arctan \left ( ax \right ) \ln \left ( 1-{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) +{\it dilog} \left ( 1+{i \left ( 1+iax \right ){\frac{1}{\sqrt{{a}^{2}{x}^{2}+1}}}} \right ) -{\it dilog} \left ( 1-{i \left ( 1+iax \right ){\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} x^{5} \arctan \left (a x\right )^{2}}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, 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{x^{5} \operatorname{atan}^{2}{\left (a x \right )}}{\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{x^{5} \arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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