Optimal. Leaf size=282 \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{25 a^5}+\frac{11 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{400 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{50 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2} \]
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Rubi [A] time = 0.514535, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4630, 4708, 4678, 4624, 3305, 3351, 4636, 4406} \[ \frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{25 a^5}+\frac{11 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{400 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{50 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2} \]
Antiderivative was successfully verified.
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Rule 4630
Rule 4708
Rule 4678
Rule 4624
Rule 3305
Rule 3351
Rule 4636
Rule 4406
Rubi steps
\begin{align*} \int x^4 \cos ^{-1}(a x)^{3/2} \, dx &=\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{1}{10} (3 a) \int \frac{x^5 \sqrt{\cos ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}-\frac{3}{100} \int \frac{x^4}{\sqrt{\cos ^{-1}(a x)}} \, dx+\frac{6 \int \frac{x^3 \sqrt{\cos ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{100 a^5}+\frac{4 \int \frac{x \sqrt{\cos ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{25 a^3}-\frac{\int \frac{x^2}{\sqrt{\cos ^{-1}(a x)}} \, dx}{25 a^2}\\ &=-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \left (\frac{\sin (x)}{8 \sqrt{x}}+\frac{3 \sin (3 x)}{16 \sqrt{x}}+\frac{\sin (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{25 a^5}-\frac{2 \int \frac{1}{\sqrt{\cos ^{-1}(a x)}} \, dx}{25 a^4}\\ &=-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{1600 a^5}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{800 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{1600 a^5}+\frac{\operatorname{Subst}\left (\int \left (\frac{\sin (x)}{4 \sqrt{x}}+\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{25 a^5}+\frac{2 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{25 a^5}\\ &=-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{3 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{3 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{400 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{100 a^5}+\frac{9 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{4 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{25 a^5}\\ &=-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{3 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{400 a^5}+\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{25 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{\operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{50 a^5}+\frac{\operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{50 a^5}\\ &=-\frac{4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^5}-\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{25 a^3}-\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \cos ^{-1}(a x)^{3/2}+\frac{11 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{400 a^5}+\frac{2 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{25 a^5}+\frac{\sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{50 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}+\frac{3 \sqrt{\frac{\pi }{10}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{800 a^5}\\ \end{align*}
Mathematica [C] time = 0.123592, size = 185, normalized size = 0.66 \[ -\frac{2250 \left (\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},i \cos ^{-1}(a x)\right )\right )+125 \sqrt{3} \left (\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},3 i \cos ^{-1}(a x)\right )\right )+9 \sqrt{5} \left (\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-5 i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},5 i \cos ^{-1}(a x)\right )\right )}{36000 a^5 \sqrt{\cos ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.104, size = 193, normalized size = 0.7 \begin{align*}{\frac{1}{24000\,{a}^{5}} \left ( 3000\,ax \left ( \arccos \left ( ax \right ) \right ) ^{2}+9\,\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +125\,\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +1500\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 3\,\arccos \left ( ax \right ) \right ) +300\, \left ( \arccos \left ( ax \right ) \right ) ^{2}\cos \left ( 5\,\arccos \left ( ax \right ) \right ) +2250\,\sqrt{2}\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -4500\,\arccos \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}-90\,\arccos \left ( ax \right ) \sin \left ( 5\,\arccos \left ( ax \right ) \right ) -750\,\arccos \left ( ax \right ) \sin \left ( 3\,\arccos \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arccos \left ( ax \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(-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 [B] time = 1.34783, size = 586, normalized size = 2.08 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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