Optimal. Leaf size=205 \[ \frac {15 \sqrt {\frac {\pi }{2}} C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{4096 a^4}+\frac {15 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{256 a^4}-\frac {3 \cos ^{-1}(a x)^{5/2}}{32 a^4}+\frac {225 \sqrt {\cos ^{-1}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\cos ^{-1}(a x)}}{256 a^2}-\frac {5 x^3 \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{32 a}-\frac {15 x \sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}}{64 a^3}+\frac {1}{4} x^4 \cos ^{-1}(a x)^{5/2}-\frac {15}{256} x^4 \sqrt {\cos ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.59, antiderivative size = 205, normalized size of antiderivative = 1.00, 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.
[In]
[Out]
Rule 3304
Rule 3312
Rule 3352
Rule 4630
Rule 4642
Rule 4708
Rule 4724
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 = 0.15, size = 140, normalized size = 0.68 \[ -\frac {-16 \sqrt {2} \left (-i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {7}{2},-2 i \cos ^{-1}(a x)\right )-16 \sqrt {2} \left (i \cos ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {7}{2},2 i \cos ^{-1}(a x)\right )+\sqrt {\cos ^{-1}(a x)^2} \left (\sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {7}{2},-4 i \cos ^{-1}(a x)\right )+\sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {7}{2},4 i \cos ^{-1}(a x)\right )\right )}{2048 a^4 \cos ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 2.58, size = 345, normalized size = 1.68 \[ \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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 154, normalized size = 0.75 \[ \frac {1024 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \cos \left (2 \arccos \left (a x \right )\right )+256 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \cos \left (4 \arccos \left (a x \right )\right )-1280 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (2 \arccos \left (a x \right )\right )-160 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (4 \arccos \left (a x \right )\right )+15 \pi \sqrt {2}\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )+480 \pi \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-960 \cos \left (2 \arccos \left (a x \right )\right ) \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}-60 \sqrt {\pi }\, \sqrt {\arccos \left (a x \right )}\, \cos \left (4 \arccos \left (a x \right )\right )}{8192 a^{4} \sqrt {\pi }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{3} \operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________