Integrand size = 12, antiderivative size = 157 \[ \int x^3 \arccos (a x)^{3/2} \, dx=-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{64 a^4} \]
[Out]
Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4726, 4796, 4738, 4732, 4491, 12, 3386, 3432} \[ \int x^3 \arccos (a x)^{3/2} \, dx=\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{64 a^4}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}+\frac {1}{4} x^4 \arccos (a x)^{3/2} \]
[In]
[Out]
Rule 12
Rule 3386
Rule 3432
Rule 4491
Rule 4726
Rule 4732
Rule 4738
Rule 4796
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {1}{8} (3 a) \int \frac {x^4 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx \\ & = -\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}+\frac {1}{4} x^4 \arccos (a x)^{3/2}-\frac {3}{64} \int \frac {x^3}{\sqrt {\arccos (a x)}} \, dx+\frac {9 \int \frac {x^2 \sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{32 a} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{64 a^4}+\frac {9 \int \frac {\sqrt {\arccos (a x)}}{\sqrt {1-a^2 x^2}} \, dx}{64 a^3}-\frac {9 \int \frac {x}{\sqrt {\arccos (a x)}} \, dx}{128 a^2} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{64 a^4}+\frac {9 \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{128 a^4} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{512 a^4}+\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\arccos (a x)\right )}{128 a^4} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \text {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{256 a^4}+\frac {3 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{128 a^4}+\frac {9 \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{256 a^4} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{256 a^4}+\frac {9 \text {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{128 a^4} \\ & = -\frac {9 x \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{64 a^3}-\frac {3 x^3 \sqrt {1-a^2 x^2} \sqrt {\arccos (a x)}}{32 a}-\frac {3 \arccos (a x)^{3/2}}{32 a^4}+\frac {1}{4} x^4 \arccos (a x)^{3/2}+\frac {3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{512 a^4}+\frac {3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos (a x)}}{\sqrt {\pi }}\right )}{64 a^4} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.82 \[ \int x^3 \arccos (a x)^{3/2} \, dx=-\frac {8 \sqrt {2} \sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-2 i \arccos (a x)\right )+8 \sqrt {2} \sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},2 i \arccos (a x)\right )+\sqrt {-i \arccos (a x)} \Gamma \left (\frac {5}{2},-4 i \arccos (a x)\right )+\sqrt {i \arccos (a x)} \Gamma \left (\frac {5}{2},4 i \arccos (a x)\right )}{512 a^4 \sqrt {\arccos (a x)}} \]
[In]
[Out]
Time = 0.92 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.75
method | result | size |
default | \(\frac {128 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (2 \arccos \left (a x \right )\right )+32 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \cos \left (4 \arccos \left (a x \right )\right )+3 \pi \sqrt {2}\, \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-96 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sin \left (2 \arccos \left (a x \right )\right )-12 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \sin \left (4 \arccos \left (a x \right )\right )+48 \pi \,\operatorname {FresnelS}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )}{1024 a^{4} \sqrt {\pi }}\) | \(118\) |
[In]
[Out]
Exception generated. \[ \int x^3 \arccos (a x)^{3/2} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int x^3 \arccos (a x)^{3/2} \, dx=\int x^{3} \operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}\, dx \]
[In]
[Out]
Exception generated. \[ \int x^3 \arccos (a x)^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.43 \[ \int x^3 \arccos (a x)^{3/2} \, dx=\frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{16 \, a^{4}} + \frac {\arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4096 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{4096 \, a^{4}} + \frac {\left (3 i - 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{256 \, a^{4}} - \frac {\left (3 i + 3\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{256 \, a^{4}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (4 i \, \arccos \left (a x\right )\right )}}{512 \, a^{4}} + \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (2 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{64 \, a^{4}} - \frac {3 i \, \sqrt {\arccos \left (a x\right )} e^{\left (-4 i \, \arccos \left (a x\right )\right )}}{512 \, a^{4}} \]
[In]
[Out]
Timed out. \[ \int x^3 \arccos (a x)^{3/2} \, dx=\int x^3\,{\mathrm {acos}\left (a\,x\right )}^{3/2} \,d x \]
[In]
[Out]