Optimal. Leaf size=205 \[ \frac {225 \sqrt {\text {ArcCos}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {ArcCos}(a x)}}{256 a^2}-\frac {15}{256} x^4 \sqrt {\text {ArcCos}(a x)}-\frac {15 x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{64 a^3}-\frac {5 x^3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{32 a}-\frac {3 \text {ArcCos}(a x)^{5/2}}{32 a^4}+\frac {1}{4} x^4 \text {ArcCos}(a x)^{5/2}+\frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4096 a^4}+\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{256 a^4} \]
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Rubi [A]
time = 0.40, 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 = {4726, 4796,
4738, 4810, 3393, 3385, 3433} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{4096 a^4}+\frac {15 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{256 a^4}-\frac {3 \text {ArcCos}(a x)^{5/2}}{32 a^4}+\frac {225 \sqrt {\text {ArcCos}(a x)}}{2048 a^4}-\frac {45 x^2 \sqrt {\text {ArcCos}(a x)}}{256 a^2}-\frac {5 x^3 \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{32 a}-\frac {15 x \sqrt {1-a^2 x^2} \text {ArcCos}(a x)^{3/2}}{64 a^3}+\frac {1}{4} x^4 \text {ArcCos}(a x)^{5/2}-\frac {15}{256} x^4 \sqrt {\text {ArcCos}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3393
Rule 3433
Rule 4726
Rule 4738
Rule 4796
Rule 4810
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 \text {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 \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,\cos ^{-1}(a x)\right )}{512 a^4}+\frac {45 \text {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 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{4096 a^4}+\frac {15 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{1024 a^4}+\frac {45 \text {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 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{2048 a^4}+\frac {15 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{512 a^4}+\frac {45 \text {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 \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.09, size = 140, normalized size = 0.68 \begin {gather*} -\frac {-16 \sqrt {2} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},-2 i \text {ArcCos}(a x)\right )-16 \sqrt {2} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {7}{2},2 i \text {ArcCos}(a x)\right )+\sqrt {\text {ArcCos}(a x)^2} \left (\sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {7}{2},-4 i \text {ArcCos}(a x)\right )+\sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {7}{2},4 i \text {ArcCos}(a x)\right )\right )}{2048 a^4 \text {ArcCos}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 154, normalized size = 0.75
method | result | size |
default | \(\frac {1024 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \cos \left (2 \arccos \left (a x \right )\right )+256 \arccos \left (a x \right )^{\frac {5}{2}} \sqrt {\pi }\, \cos \left (4 \arccos \left (a x \right )\right )-1280 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (2 \arccos \left (a x \right )\right )-160 \arccos \left (a x \right )^{\frac {3}{2}} \sqrt {\pi }\, \sin \left (4 \arccos \left (a x \right )\right )+15 \pi \sqrt {2}\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-960 \cos \left (2 \arccos \left (a x \right )\right ) \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }-60 \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \cos \left (4 \arccos \left (a x \right )\right )+480 \pi \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )}{8192 a^{4} \sqrt {\pi }}\) | \(154\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{3} \operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.49, size = 297, normalized size = 1.45 \begin {gather*} \frac {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (4 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 {\arccos \left (a x\right )^{\frac {5}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{16 \, 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 (4 i \, \arccos \left (a x\right )\right )}}{512 \, 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 {5 i \, \arccos \left (a x\right )^{\frac {3}{2}} e^{\left (-2 i \, \arccos \left (a x\right )\right )}}{64 \, 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 {\left (15 i + 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32768 \, a^{4}} + \frac {\left (15 i - 15\right ) \, \sqrt {2} \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {2} \sqrt {\arccos \left (a x\right )}\right )}{32768 \, a^{4}} - \frac {\left (15 i + 15\right ) \, \sqrt {\pi } \operatorname {erf}\left (\left (i - 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{1024 \, a^{4}} + \frac {\left (15 i - 15\right ) \, \sqrt {\pi } \operatorname {erf}\left (-\left (i + 1\right ) \, \sqrt {\arccos \left (a x\right )}\right )}{1024 \, 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 {\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^3\,{\mathrm {acos}\left (a\,x\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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