Optimal. Leaf size=205 \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4096 a^4}+\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)} \]
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Rubi [A] time = 0.58931, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 7, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {4630, 4708, 4642, 4724, 3312, 3304, 3352} \[ \frac{15 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4096 a^4}+\frac{15 \sqrt{\pi } \text{FresnelC}\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 4630
Rule 4708
Rule 4642
Rule 4724
Rule 3312
Rule 3304
Rule 3352
Rubi steps
\begin{align*} \int x^3 \cos ^{-1}(a x)^{5/2} \, dx &=\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{1}{8} (5 a) \int \frac{x^4 \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx\\ &=-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}-\frac{15}{64} \int x^3 \sqrt{\cos ^{-1}(a x)} \, dx+\frac{15 \int \frac{x^2 \cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{32 a}\\ &=-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \int \frac{\cos ^{-1}(a x)^{3/2}}{\sqrt{1-a^2 x^2}} \, dx}{64 a^3}-\frac{45 \int x \sqrt{\cos ^{-1}(a x)} \, dx}{128 a^2}-\frac{1}{512} (15 a) \int \frac{x^4}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx\\ &=-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos ^4(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}-\frac{45 \int \frac{x^2}{\sqrt{1-a^2 x^2} \sqrt{\cos ^{-1}(a x)}} \, dx}{512 a}\\ &=-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \left (\frac{3}{8 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}+\frac{\cos (4 x)}{8 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{\cos ^2(x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}\\ &=\frac{45 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (4 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{4096 a^4}+\frac{15 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{1024 a^4}+\frac{45 \operatorname{Subst}\left (\int \left (\frac{1}{2 \sqrt{x}}+\frac{\cos (2 x)}{2 \sqrt{x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{512 a^4}\\ &=\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \operatorname{Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{2048 a^4}+\frac{15 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{512 a^4}+\frac{45 \operatorname{Subst}\left (\int \frac{\cos (2 x)}{\sqrt{x}} \, dx,x,\cos ^{-1}(a x)\right )}{1024 a^4}\\ &=\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4096 a^4}+\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{1024 a^4}+\frac{45 \operatorname{Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt{\cos ^{-1}(a x)}\right )}{512 a^4}\\ &=\frac{225 \sqrt{\cos ^{-1}(a x)}}{2048 a^4}-\frac{45 x^2 \sqrt{\cos ^{-1}(a x)}}{256 a^2}-\frac{15}{256} x^4 \sqrt{\cos ^{-1}(a x)}-\frac{15 x \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}-\frac{5 x^3 \sqrt{1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac{3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac{1}{4} x^4 \cos ^{-1}(a x)^{5/2}+\frac{15 \sqrt{\frac{\pi }{2}} C\left (2 \sqrt{\frac{2}{\pi }} \sqrt{\cos ^{-1}(a x)}\right )}{4096 a^4}+\frac{15 \sqrt{\pi } C\left (\frac{2 \sqrt{\cos ^{-1}(a x)}}{\sqrt{\pi }}\right )}{256 a^4}\\ \end{align*}
Mathematica [C] time = 0.126707, size = 140, normalized size = 0.68 \[ -\frac{-16 \sqrt{2} \left (-i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},-2 i \cos ^{-1}(a x)\right )-16 \sqrt{2} \left (i \cos ^{-1}(a x)\right )^{3/2} \text{Gamma}\left (\frac{7}{2},2 i \cos ^{-1}(a x)\right )+\sqrt{\cos ^{-1}(a x)^2} \left (\sqrt{i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},-4 i \cos ^{-1}(a x)\right )+\sqrt{-i \cos ^{-1}(a x)} \text{Gamma}\left (\frac{7}{2},4 i \cos ^{-1}(a x)\right )\right )}{2048 a^4 \cos ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.094, size = 154, normalized size = 0.8 \begin{align*}{\frac{1}{8192\,{a}^{4}\sqrt{\pi }} \left ( 1024\, \left ( \arccos \left ( ax \right ) \right ) ^{5/2}\sqrt{\pi }\cos \left ( 2\,\arccos \left ( ax \right ) \right ) +256\, \left ( \arccos \left ( ax \right ) \right ) ^{5/2}\sqrt{\pi }\cos \left ( 4\,\arccos \left ( ax \right ) \right ) -1280\, \left ( \arccos \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }\sin \left ( 2\,\arccos \left ( ax \right ) \right ) -160\, \left ( \arccos \left ( ax \right ) \right ) ^{3/2}\sqrt{\pi }\sin \left ( 4\,\arccos \left ( ax \right ) \right ) +15\,\pi \,\sqrt{2}{\it FresnelC} \left ( 2\,{\frac{\sqrt{2}\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -960\,\sqrt{\pi }\sqrt{\arccos \left ( ax \right ) }\cos \left ( 2\,\arccos \left ( ax \right ) \right ) -60\,\sqrt{\arccos \left ( ax \right ) }\sqrt{\pi }\cos \left ( 4\,\arccos \left ( ax \right ) \right ) +480\,\pi \,{\it FresnelC} \left ( 2\,{\frac{\sqrt{\arccos \left ( ax \right ) }}{\sqrt{\pi }}} \right ) \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 = 1.32306, size = 466, normalized size = 2.27 \begin{align*} \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (4 \, i \arccos \left (a x\right )\right )}}{512 \, a^{4}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (4 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{16 \, a^{4}} - \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{16 \, a^{4}} - \frac{5 \, i \arccos \left (a x\right )^{\frac{3}{2}} e^{\left (-4 \, i \arccos \left (a x\right )\right )}}{512 \, a^{4}} + \frac{\arccos \left (a x\right )^{\frac{5}{2}} e^{\left (-4 \, i \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac{15 \, \sqrt{2} \sqrt{\pi } i \operatorname{erf}\left (-\sqrt{2}{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{16384 \, a^{4}{\left (i - 1\right )}} - \frac{15 \, \sqrt{\pi } i \operatorname{erf}\left (-{\left (i + 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{512 \, a^{4}{\left (i - 1\right )}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (4 \, i \arccos \left (a x\right )\right )}}{4096 \, a^{4}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (2 \, i \arccos \left (a x\right )\right )}}{256 \, a^{4}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (-2 \, i \arccos \left (a x\right )\right )}}{256 \, a^{4}} - \frac{15 \, \sqrt{\arccos \left (a x\right )} e^{\left (-4 \, i \arccos \left (a x\right )\right )}}{4096 \, a^{4}} + \frac{15 \, \sqrt{2} \sqrt{\pi } \operatorname{erf}\left (\sqrt{2}{\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{16384 \, a^{4}{\left (i - 1\right )}} + \frac{15 \, \sqrt{\pi } \operatorname{erf}\left ({\left (i - 1\right )} \sqrt{\arccos \left (a x\right )}\right )}{512 \, a^{4}{\left (i - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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