Optimal. Leaf size=296 \[ \frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (a^2 x^2+1\right )}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3} \]
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Rubi [A] time = 0.368857, antiderivative size = 296, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.524, Rules used = {4900, 4892, 4930, 4970, 4406, 12, 3305, 3351, 4904, 3312, 3296} \[ \frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (a^2 x^2+1\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (a^2 x^2+1\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (a^2 x^2+1\right )^2}-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (a^2 x^2+1\right )}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a c^3}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3} \]
Antiderivative was successfully verified.
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Rule 4900
Rule 4892
Rule 4930
Rule 4970
Rule 4406
Rule 12
Rule 3305
Rule 3351
Rule 4904
Rule 3312
Rule 3296
Rubi steps
\begin{align*} \int \frac{\tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^3} \, dx &=\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}-\frac{15}{64} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^3} \, dx+\frac{3 \int \frac{\tan ^{-1}(a x)^{5/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{4 c}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cos ^4(x) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac{(15 a) \int \frac{x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^2} \, dx}{16 c}\\ &=\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{15 \operatorname{Subst}\left (\int \left (\frac{3 \sqrt{x}}{8}+\frac{1}{2} \sqrt{x} \cos (2 x)+\frac{1}{8} \sqrt{x} \cos (4 x)\right ) \, dx,x,\tan ^{-1}(a x)\right )}{64 a c^3}-\frac{45 \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^2} \, dx}{64 c}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cos (4 x) \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}-\frac{15 \operatorname{Subst}\left (\int \sqrt{x} \cos (2 x) \, dx,x,\tan ^{-1}(a x)\right )}{128 a c^3}+\frac{(45 a) \int \frac{x}{\left (c+a^2 c x^2\right )^2 \sqrt{\tan ^{-1}(a x)}} \, dx}{256 c}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin (4 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{4096 a c^3}+\frac{15 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}+\frac{45 \operatorname{Subst}\left (\int \frac{\cos (x) \sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{256 a c^3}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3}+\frac{15 \operatorname{Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{2048 a c^3}+\frac{15 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a c^3}+\frac{45 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{2 \sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{256 a c^3}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{512 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3}+\frac{45 \operatorname{Subst}\left (\int \frac{\sin (2 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{512 a c^3}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{512 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3}+\frac{45 \operatorname{Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{256 a c^3}\\ &=-\frac{45 x \sqrt{\tan ^{-1}(a x)}}{128 c^3 \left (1+a^2 x^2\right )}-\frac{75 \tan ^{-1}(a x)^{3/2}}{256 a c^3}+\frac{5 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )^2}+\frac{15 \tan ^{-1}(a x)^{3/2}}{32 a c^3 \left (1+a^2 x^2\right )}+\frac{x \tan ^{-1}(a x)^{5/2}}{4 c^3 \left (1+a^2 x^2\right )^2}+\frac{3 x \tan ^{-1}(a x)^{5/2}}{8 c^3 \left (1+a^2 x^2\right )}+\frac{3 \tan ^{-1}(a x)^{7/2}}{28 a c^3}+\frac{15 \sqrt{\frac{\pi }{2}} S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{4096 a c^3}+\frac{15 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{128 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (2 \tan ^{-1}(a x)\right )}{256 a c^3}-\frac{15 \sqrt{\tan ^{-1}(a x)} \sin \left (4 \tan ^{-1}(a x)\right )}{2048 a c^3}\\ \end{align*}
Mathematica [A] time = 0.361495, size = 162, normalized size = 0.55 \[ \frac{\frac{16 \sqrt{\tan ^{-1}(a x)} \left (-105 a x \left (15 a^2 x^2+17\right )+384 \left (a^2 x^2+1\right )^2 \tan ^{-1}(a x)^3+448 a x \left (3 a^2 x^2+5\right ) \tan ^{-1}(a x)^2-70 \left (15 a^4 x^4+6 a^2 x^2-17\right ) \tan ^{-1}(a x)\right )}{\left (a^2 x^2+1\right )^2}+105 \sqrt{2 \pi } S\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )+6720 \sqrt{\pi } S\left (\frac{2 \sqrt{\tan ^{-1}(a x)}}{\sqrt{\pi }}\right )}{57344 a c^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.117, size = 194, normalized size = 0.7 \begin{align*}{\frac{3}{28\,a{c}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{7}{2}}}}+{\frac{\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{4\,a{c}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{32\,a{c}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{5}{2}}}}+{\frac{5\,\cos \left ( 2\,\arctan \left ( ax \right ) \right ) }{16\,a{c}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}+{\frac{5\,\cos \left ( 4\,\arctan \left ( ax \right ) \right ) }{256\,a{c}^{3}} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{15\,\sin \left ( 2\,\arctan \left ( ax \right ) \right ) }{64\,a{c}^{3}}\sqrt{\arctan \left ( ax \right ) }}-{\frac{15\,\sin \left ( 4\,\arctan \left ( ax \right ) \right ) }{2048\,a{c}^{3}}\sqrt{\arctan \left ( ax \right ) }}+{\frac{15\,\sqrt{2}\sqrt{\pi }}{8192\,a{c}^{3}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) }+{\frac{15\,\sqrt{\pi }}{128\,a{c}^{3}}{\it FresnelS} \left ( 2\,{\frac{\sqrt{\arctan \left ( ax \right ) }}{\sqrt{\pi }}} \right ) } \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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*} \frac{\int \frac{\operatorname{atan}^{\frac{5}{2}}{\left (a x \right )}}{a^{6} x^{6} + 3 a^{4} x^{4} + 3 a^{2} x^{2} + 1}\, dx}{c^{3}} \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{\arctan \left (a x\right )^{\frac{5}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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