Integrand size = 12, antiderivative size = 264 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \]
[Out]
Time = 0.29 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4730, 4808, 4728, 3385, 3433} \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}} \]
[In]
[Out]
Rule 3385
Rule 3433
Rule 4728
Rule 4730
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \arccos (a x)^{5/2}} \, dx \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}-\frac {20}{3} \int \frac {x^4}{\arccos (a x)^{3/2}} \, dx+\frac {16 \int \frac {x^2}{\arccos (a x)^{3/2}} \, dx}{5 a^2} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {32 \text {Subst}\left (\int \left (-\frac {\cos (x)}{4 \sqrt {x}}-\frac {3 \cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{5 a^5}-\frac {40 \text {Subst}\left (\int \left (-\frac {\cos (x)}{8 \sqrt {x}}-\frac {9 \cos (3 x)}{16 \sqrt {x}}-\frac {5 \cos (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\arccos (a x)\right )}{3 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}-\frac {8 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{6 a^5}-\frac {24 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\arccos (a x)\right )}{2 a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}-\frac {16 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {10 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{3 a^5}-\frac {48 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\arccos (a x)}\right )}{a^5} \\ & = \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \arccos (a x)^{5/2}}-\frac {16 x^3}{15 a^2 \arccos (a x)^{3/2}}+\frac {4 x^5}{3 \arccos (a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\arccos (a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\arccos (a x)}}+\frac {\sqrt {2 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arccos (a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arccos (a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\arccos (a x)}\right )}{3 a^5} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.37 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.58 \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=-\frac {2 \left (-6 \sqrt {1-a^2 x^2}-2 i e^{i \arccos (a x)} \arccos (a x) (-i+2 \arccos (a x))-4 (-i \arccos (a x))^{3/2} \arccos (a x) \Gamma \left (\frac {1}{2},-i \arccos (a x)\right )+e^{-i \arccos (a x)} \arccos (a x) \left (-2+4 i \arccos (a x)-4 e^{i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},i \arccos (a x)\right )\right )\right )-5 \arccos (a x) \left (2 e^{5 i \arccos (a x)} (1+10 i \arccos (a x))+20 \sqrt {5} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-5 i \arccos (a x)\right )+e^{-5 i \arccos (a x)} \left (2-20 i \arccos (a x)+20 \sqrt {5} e^{5 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},5 i \arccos (a x)\right )\right )\right )+9 \left (-2 \arccos (a x) \left (e^{3 i \arccos (a x)} (1+6 i \arccos (a x))+6 \sqrt {3} (-i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},-3 i \arccos (a x)\right )+e^{-3 i \arccos (a x)} \left (1-6 i \arccos (a x)+6 \sqrt {3} e^{3 i \arccos (a x)} (i \arccos (a x))^{3/2} \Gamma \left (\frac {1}{2},3 i \arccos (a x)\right )\right )\right )-2 \sin (3 \arccos (a x))\right )-6 \sin (5 \arccos (a x))}{240 a^5 \arccos (a x)^{5/2}} \]
[In]
[Out]
Time = 1.07 (sec) , antiderivative size = 225, normalized size of antiderivative = 0.85
method | result | size |
default | \(-\frac {-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+8 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+100 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )+108 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )-4 \arccos \left (a x \right ) a x -10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )-18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-6 \sqrt {-a^{2} x^{2}+1}-3 \sin \left (5 \arccos \left (a x \right )\right )-9 \sin \left (3 \arccos \left (a x \right )\right )}{120 a^{5} \arccos \left (a x \right )^{\frac {5}{2}}}\) | \(225\) |
[In]
[Out]
Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
\[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{\frac {7}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x^4}{\arccos (a x)^{7/2}} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \]
[In]
[Out]