Optimal. Leaf size=171 \[ -\frac{\sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.434386, antiderivative size = 235, normalized size of antiderivative = 1.37, 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 = {4633, 4719, 4635, 4406, 3304, 3352} \[ \frac{4 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{25 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 4633
Rule 4719
Rule 4635
Rule 4406
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int \frac{x^4}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac{8 \int \frac{x^3}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac{1}{3} (10 a) \int \frac{x^5}{\sqrt{1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{100}{3} \int \frac{x^4}{\sqrt{\sin ^{-1}(a x)}} \, dx+\frac{16 \int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{100 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}+\frac{16 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{100 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 \sqrt{x}}-\frac{3 \cos (3 x)}{16 \sqrt{x}}+\frac{\cos (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^5}+\frac{4 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{4 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}+\frac{25 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}+\frac{8 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{8 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{25 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{25 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac{2 x^4 \sqrt{1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac{16 x^3}{3 a^2 \sqrt{\sin ^{-1}(a x)}}+\frac{20 x^5}{3 \sqrt{\sin ^{-1}(a x)}}-\frac{25 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{3 a^5}+\frac{4 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}+\frac{25 \sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{2 a^5}-\frac{4 \sqrt{\frac{2 \pi }{3}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{a^5}-\frac{5 \sqrt{\frac{5 \pi }{2}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{6 a^5}\\ \end{align*}
Mathematica [C] time = 0.328869, size = 418, normalized size = 2.44 \[ \frac{\frac{i e^{i \sin ^{-1}(a x)} \left (-2 \sin ^{-1}(a x)+i\right )-2 \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-i \sin ^{-1}(a x)\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac{e^{-i \sin ^{-1}(a x)} \left (2 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},i \sin ^{-1}(a x)\right )-2 i \sin ^{-1}(a x)+1\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac{i e^{3 i \sin ^{-1}(a x)} \left (-6 \sin ^{-1}(a x)+i\right )-6 \sqrt{3} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-3 i \sin ^{-1}(a x)\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac{e^{-3 i \sin ^{-1}(a x)} \left (6 \sqrt{3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},3 i \sin ^{-1}(a x)\right )-6 i \sin ^{-1}(a x)+1\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac{i e^{5 i \sin ^{-1}(a x)} \left (-10 \sin ^{-1}(a x)+i\right )-10 \sqrt{5} \left (-i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},-5 i \sin ^{-1}(a x)\right )}{48 \sin ^{-1}(a x)^{3/2}}-\frac{e^{-5 i \sin ^{-1}(a x)} \left (10 \sqrt{5} e^{5 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{1}{2},5 i \sin ^{-1}(a x)\right )-10 i \sin ^{-1}(a x)+1\right )}{48 \sin ^{-1}(a x)^{3/2}}}{a^5} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.07, size = 175, normalized size = 1. \begin{align*}{\frac{1}{24\,{a}^{5}} \left ( 18\,\sqrt{2}\sqrt{\pi }\sqrt{3}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{3}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}-10\,\sqrt{2}\sqrt{\pi }\sqrt{5}{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{5}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}-4\,\sqrt{2}\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \left ( \arcsin \left ( ax \right ) \right ) ^{3/2}+4\,ax\arcsin \left ( ax \right ) +10\,\arcsin \left ( ax \right ) \sin \left ( 5\,\arcsin \left ( ax \right ) \right ) -18\,\arcsin \left ( ax \right ) \sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -2\,\sqrt{-{a}^{2}{x}^{2}+1}-\cos \left ( 5\,\arcsin \left ( ax \right ) \right ) +3\,\cos \left ( 3\,\arcsin \left ( ax \right ) \right ) \right ) \left ( \arcsin \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{asin}^{\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}}{\arcsin \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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