Optimal. Leaf size=171 \[ \frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7} \]
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Rubi [A]
time = 0.11, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4728, 3385,
3433} \begin {gather*} -\frac {5 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {14}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{16 a^7}+\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4728
Rubi steps
\begin {align*} \int \frac {x^6}{\cos ^{-1}(a x)^{3/2}} \, dx &=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}+\frac {2 \text {Subst}\left (\int \left (-\frac {5 \cos (x)}{64 \sqrt {x}}-\frac {27 \cos (3 x)}{64 \sqrt {x}}-\frac {25 \cos (5 x)}{64 \sqrt {x}}-\frac {7 \cos (7 x)}{64 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{a^7}\\ &=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}-\frac {27 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{32 a^7}\\ &=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {5 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {7 \text {Subst}\left (\int \cos \left (7 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {25 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {27 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}\\ &=\frac {2 x^6 \sqrt {1-a^2 x^2}}{a \sqrt {\cos ^{-1}(a x)}}-\frac {5 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {9 \sqrt {\frac {3 \pi }{2}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {5 \sqrt {\frac {5 \pi }{2}} C\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}-\frac {\sqrt {\frac {7 \pi }{2}} C\left (\sqrt {\frac {14}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{16 a^7}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 306, normalized size = 1.79 \begin {gather*} \frac {i \left (-10 i \sqrt {1-a^2 x^2}+5 \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )-5 \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )+9 \sqrt {3} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcCos}(a x)\right )-9 \sqrt {3} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcCos}(a x)\right )+5 \sqrt {5} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-5 i \text {ArcCos}(a x)\right )-5 \sqrt {5} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},5 i \text {ArcCos}(a x)\right )+\sqrt {7} \sqrt {-i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},-7 i \text {ArcCos}(a x)\right )-\sqrt {7} \sqrt {i \text {ArcCos}(a x)} \text {Gamma}\left (\frac {1}{2},7 i \text {ArcCos}(a x)\right )-18 i \sin (3 \text {ArcCos}(a x))-10 i \sin (5 \text {ArcCos}(a x))-2 i \sin (7 \text {ArcCos}(a x))\right )}{64 a^7 \sqrt {\text {ArcCos}(a x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 182, normalized size = 1.06
method | result | size |
default | \(\frac {-\sqrt {2}\, \sqrt {\pi }\, \sqrt {7}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {7}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {\arccos \left (a x \right )}-5 \sqrt {5}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-9 \sqrt {3}\, \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right )-5 \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \sqrt {2}\, \sqrt {\arccos \left (a x \right )}\, \sqrt {\pi }+5 \sqrt {-a^{2} x^{2}+1}+\sin \left (7 \arccos \left (a x \right )\right )+9 \sin \left (3 \arccos \left (a x \right )\right )+5 \sin \left (5 \arccos \left (a x \right )\right )}{32 a^{7} \sqrt {\arccos \left (a x \right )}}\) | \(182\) |
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 \frac {x^{6}}{\operatorname {acos}^{\frac {3}{2}}{\left (a x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.01 \begin {gather*} \int \frac {x^6}{{\mathrm {acos}\left (a\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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