Optimal. Leaf size=235 \[ -\frac{4 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{6 a^5}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}} \]
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Rubi [A] time = 0.4096, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4634, 4720, 4636, 4406, 3305, 3351} \[ -\frac{4 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{6 a^5}+\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4634
Rule 4720
Rule 4636
Rule 4406
Rule 3305
Rule 3351
Rubi steps
\begin{align*} \int \frac{x^4}{\cos ^{-1}(a x)^{5/2}} \, dx &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx}{3 a}+\frac{1}{3} (10 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{100}{3} \int \frac{x^4}{\sqrt{\cos ^{-1}(a x)}} \, dx+\frac{16 \int \frac{x^2}{\sqrt{\cos ^{-1}(a x)}} \, dx}{a^2}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{16 \operatorname{Subst}\left (\int \frac{\cos ^2(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac{100 \operatorname{Subst}\left (\int \frac{\cos ^4(x) \sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}-\frac{16 \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 )}{a^5}+\frac{100 \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 )}{3 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int \frac{\sin (5 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{12 a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{25 \operatorname{Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{6 a^5}-\frac{8 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^5}-\frac{8 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2 a^5}\\ &=\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\cos ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\cos ^{-1}(a x)}}+\frac{25 \sqrt{\frac{\pi }{2}} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{3 a^5}-\frac{4 \sqrt{2 \pi } S\left (\sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{2 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{a^5}+\frac{5 \sqrt{\frac{5 \pi }{2}} S\left (\sqrt{\frac{10}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{6 a^5}\\ \end{align*}
Mathematica [C] time = 1.63439, size = 322, normalized size = 1.37 \[ -\frac{2 \left (\sqrt{-i \cos ^{-1}(a x)} \cos ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},-i \cos ^{-1}(a x)\right )+\sqrt{i \cos ^{-1}(a x)} \cos ^{-1}(a x) \text{Gamma}\left (\frac{1}{2},i \cos ^{-1}(a x)\right )-\sqrt{1-a^2 x^2}-e^{-i \cos ^{-1}(a x)} \cos ^{-1}(a x)-e^{i \cos ^{-1}(a x)} \cos ^{-1}(a x)\right )-5 \cos ^{-1}(a x) \left (-\sqrt{5} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-5 i \cos ^{-1}(a x)\right )-\sqrt{5} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},5 i \cos ^{-1}(a x)\right )+e^{-5 i \cos ^{-1}(a x)}+e^{5 i \cos ^{-1}(a x)}\right )-3 \left (\sin \left (3 \cos ^{-1}(a x)\right )+3 \cos ^{-1}(a x) \left (-\sqrt{3} \sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \cos ^{-1}(a x)\right )-\sqrt{3} \sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \cos ^{-1}(a x)\right )+e^{-3 i \cos ^{-1}(a x)}+e^{3 i \cos ^{-1}(a x)}\right )\right )-\sin \left (5 \cos ^{-1}(a x)\right )}{24 a^5 \cos ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.102, size = 173, normalized size = 0.7 \begin{align*}{\frac{1}{24\,{a}^{5}} \left ( 18\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+10\,\sqrt{2}\sqrt{\pi }\sqrt{5}{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{5}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+4\,\sqrt{2}\sqrt{\pi }{\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arccos \left ( ax \right ) \right ) ^{3/2}+4\,ax\arccos \left ( ax \right ) +18\,\arccos \left ( ax \right ) \cos \left ( 3\,\arccos \left ( ax \right ) \right ) +10\,\arccos \left ( ax \right ) \cos \left ( 5\,\arccos \left ( ax \right ) \right ) +2\,\sqrt{-{a}^{2}{x}^{2}+1}+3\,\sin \left ( 3\,\arccos \left ( ax \right ) \right ) +\sin \left ( 5\,\arccos \left ( ax \right ) \right ) \right ) \left ( \arccos \left ( ax \right ) \right ) ^{-{\frac{3}{2}}}} \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*} \int \frac{x^{4}}{\operatorname{acos}^{\frac{5}{2}}{\left (a x \right )}}\, 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^{4}}{\arccos \left (a x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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