Integrand size = 12, antiderivative size = 95 \[ \int x^3 \sqrt {\arccos (a x)} \, dx=-\frac {3 \sqrt {\arccos (a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{16 a^4} \]
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Time = 0.12 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4726, 4810, 3393, 3385, 3433} \[ \int x^3 \sqrt {\arccos (a x)} \, dx=-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{16 a^4}-\frac {3 \sqrt {\arccos (a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\arccos (a x)} \]
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Rule 3385
Rule 3393
Rule 3433
Rule 4726
Rule 4810
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \sqrt {\arccos (a x)}+\frac {1}{8} a \int \frac {x^4}{\sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}} \, dx \\ & = \frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos ^4(x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{8 a^4} \\ & = \frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\text {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,\arccos (a x)\right )}{8 a^4} \\ & = -\frac {3 \sqrt {\arccos (a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{64 a^4}-\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{16 a^4} \\ & = -\frac {3 \sqrt {\arccos (a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{32 a^4}-\frac {\text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{8 a^4} \\ & = -\frac {3 \sqrt {\arccos (a x)}}{32 a^4}+\frac {1}{4} x^4 \sqrt {\arccos (a x)}-\frac {\sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{64 a^4}-\frac {\sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{16 a^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.06 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.38 \[ \int x^3 \sqrt {\arccos (a x)} \, dx=\frac {i \left (4 \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-2 i \arccos (a x)\right )-4 \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},2 i \arccos (a x)\right )+\sqrt {-i \arccos (a x)} \Gamma \left (\frac {3}{2},-4 i \arccos (a x)\right )-\sqrt {i \arccos (a x)} \Gamma \left (\frac {3}{2},4 i \arccos (a x)\right )\right )}{128 a^4 \sqrt {\arccos (a x)}} \]
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Time = 0.86 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.96
method | result | size |
default | \(\frac {-\sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+16 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )+4 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-8 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )}{128 a^{4} \sqrt {\arccos \left (a x \right )}}\) | \(91\) |
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Exception generated. \[ \int x^3 \sqrt {\arccos (a x)} \, dx=\text {Exception raised: TypeError} \]
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\[ \int x^3 \sqrt {\arccos (a x)} \, dx=\int x^{3} \sqrt {\operatorname {acos}{\left (a x \right )}}\, dx \]
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Exception generated. \[ \int x^3 \sqrt {\arccos (a x)} \, dx=\text {Exception raised: RuntimeError} \]
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.61 \[ \int x^3 \sqrt {\arccos (a x)} \, dx=\frac {\left (i + 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{512 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{512 \, a^{4}} + \frac {\left (i + 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} - \frac {\left (i - 1\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{64 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\sqrt {\arccos \left (a x\right )} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} \]
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Timed out. \[ \int x^3 \sqrt {\arccos (a x)} \, dx=\int x^3\,\sqrt {\mathrm {acos}\left (a\,x\right )} \,d x \]
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