Integrand size = 10, antiderivative size = 68 \[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{8 a^5}-\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{16 a^5}-\frac {5 \operatorname {CosIntegral}(5 \arccos (a x))}{16 a^5} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4728, 3383} \[ \int \frac {x^4}{\arccos (a x)^2} \, dx=-\frac {\operatorname {CosIntegral}(\arccos (a x))}{8 a^5}-\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{16 a^5}-\frac {5 \operatorname {CosIntegral}(5 \arccos (a x))}{16 a^5}+\frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)} \]
[In]
[Out]
Rule 3383
Rule 4728
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}+\frac {\text {Subst}\left (\int \left (-\frac {\cos (x)}{8 x}-\frac {9 \cos (3 x)}{16 x}-\frac {5 \cos (5 x)}{16 x}\right ) \, dx,x,\arccos (a x)\right )}{a^5} \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arccos (a x)\right )}{8 a^5}-\frac {5 \text {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\arccos (a x)\right )}{16 a^5}-\frac {9 \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arccos (a x)\right )}{16 a^5} \\ & = \frac {x^4 \sqrt {1-a^2 x^2}}{a \arccos (a x)}-\frac {\operatorname {CosIntegral}(\arccos (a x))}{8 a^5}-\frac {9 \operatorname {CosIntegral}(3 \arccos (a x))}{16 a^5}-\frac {5 \operatorname {CosIntegral}(5 \arccos (a x))}{16 a^5} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {x^4}{\arccos (a x)^2} \, dx=-\frac {-\frac {16 a^4 x^4 \sqrt {1-a^2 x^2}}{\arccos (a x)}+2 \operatorname {CosIntegral}(\arccos (a x))+9 \operatorname {CosIntegral}(3 \arccos (a x))+5 \operatorname {CosIntegral}(5 \arccos (a x))}{16 a^5} \]
[In]
[Out]
Time = 0.70 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.19
method | result | size |
derivativedivides | \(\frac {\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )}-\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{16}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{16}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )}-\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\) | \(81\) |
default | \(\frac {\frac {3 \sin \left (3 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )}-\frac {9 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right )}{16}+\frac {\sin \left (5 \arccos \left (a x \right )\right )}{16 \arccos \left (a x \right )}-\frac {5 \,\operatorname {Ci}\left (5 \arccos \left (a x \right )\right )}{16}+\frac {\sqrt {-a^{2} x^{2}+1}}{8 \arccos \left (a x \right )}-\frac {\operatorname {Ci}\left (\arccos \left (a x \right )\right )}{8}}{a^{5}}\) | \(81\) |
[In]
[Out]
\[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\int \frac {x^{4}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\int { \frac {x^{4}}{\arccos \left (a x\right )^{2}} \,d x } \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.88 \[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\frac {\sqrt {-a^{2} x^{2} + 1} x^{4}}{a \arccos \left (a x\right )} - \frac {5 \, \operatorname {Ci}\left (5 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac {9 \, \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{16 \, a^{5}} - \frac {\operatorname {Ci}\left (\arccos \left (a x\right )\right )}{8 \, a^{5}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^4}{\arccos (a x)^2} \, dx=\int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \]
[In]
[Out]