Optimal. Leaf size=264 \[ \frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \text {ArcCos}(a x)^{3/2}}+\frac {4 x^5}{3 \text {ArcCos}(a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\text {ArcCos}(a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\text {ArcCos}(a x)}}+\frac {\sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{3 a^5} \]
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Rubi [A]
time = 0.27, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {4730, 4808,
4728, 3385, 3433} \begin {gather*} \frac {\sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a^5}-\frac {8 \sqrt {6 \pi } \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{3 a^5}-\frac {16 x^3}{15 a^2 \text {ArcCos}(a x)^{3/2}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\text {ArcCos}(a x)}}+\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\text {ArcCos}(a x)}}+\frac {4 x^5}{3 \text {ArcCos}(a x)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3433
Rule 4728
Rule 4730
Rule 4808
Rubi steps
\begin {align*} \int \frac {x^4}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+(2 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \cos ^{-1}(a x)^{3/2}}-\frac {20}{3} \int \frac {x^4}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac {16 \int \frac {x^2}{\cos ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\cos ^{-1}(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,\cos ^{-1}(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,\cos ^{-1}(a x)\right )}{3 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\cos ^{-1}(a x)}}-\frac {8 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}+\frac {5 \text {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{6 a^5}-\frac {24 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\cos ^{-1}(a x)}}-\frac {16 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{5 a^5}+\frac {10 \text {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}+\frac {25 \text {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}-\frac {48 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{5 a^5}+\frac {15 \text {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^5}\\ &=\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \cos ^{-1}(a x)^{3/2}}+\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\cos ^{-1}(a x)}}+\frac {\sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a^5}+\frac {5 \sqrt {\frac {3 \pi }{2}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{a^5}-\frac {8 \sqrt {6 \pi } C\left (\sqrt {\frac {6}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} C\left (\sqrt {\frac {10}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^5}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 7.24, size = 418, normalized size = 1.58 \begin {gather*} -\frac {2 \left (-6 \sqrt {1-a^2 x^2}-2 i e^{i \text {ArcCos}(a x)} \text {ArcCos}(a x) (-i+2 \text {ArcCos}(a x))-4 (-i \text {ArcCos}(a x))^{3/2} \text {ArcCos}(a x) \text {Gamma}\left (\frac {1}{2},-i \text {ArcCos}(a x)\right )+e^{-i \text {ArcCos}(a x)} \text {ArcCos}(a x) \left (-2+4 i \text {ArcCos}(a x)-4 e^{i \text {ArcCos}(a x)} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},i \text {ArcCos}(a x)\right )\right )\right )-5 \text {ArcCos}(a x) \left (2 e^{5 i \text {ArcCos}(a x)} (1+10 i \text {ArcCos}(a x))+20 \sqrt {5} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-5 i \text {ArcCos}(a x)\right )+e^{-5 i \text {ArcCos}(a x)} \left (2-20 i \text {ArcCos}(a x)+20 \sqrt {5} e^{5 i \text {ArcCos}(a x)} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},5 i \text {ArcCos}(a x)\right )\right )\right )+9 \left (-2 \text {ArcCos}(a x) \left (e^{3 i \text {ArcCos}(a x)} (1+6 i \text {ArcCos}(a x))+6 \sqrt {3} (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-3 i \text {ArcCos}(a x)\right )+e^{-3 i \text {ArcCos}(a x)} \left (1-6 i \text {ArcCos}(a x)+6 \sqrt {3} e^{3 i \text {ArcCos}(a x)} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},3 i \text {ArcCos}(a x)\right )\right )\right )-2 \sin (3 \text {ArcCos}(a x))\right )-6 \sin (5 \text {ArcCos}(a x))}{240 a^5 \text {ArcCos}(a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 225, normalized size = 0.85
method | result | size |
default | \(-\frac {-108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+8 \arccos \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+108 \arccos \left (a x \right )^{2} \sin \left (3 \arccos \left (a x \right )\right )+100 \arccos \left (a x \right )^{2} \sin \left (5 \arccos \left (a x \right )\right )-4 a x \arccos \left (a x \right )-18 \arccos \left (a x \right ) \cos \left (3 \arccos \left (a x \right )\right )-10 \arccos \left (a x \right ) \cos \left (5 \arccos \left (a x \right )\right )-6 \sqrt {-a^{2} x^{2}+1}-9 \sin \left (3 \arccos \left (a x \right )\right )-3 \sin \left (5 \arccos \left (a x \right )\right )}{120 a^{5} \arccos \left (a x \right )^{\frac {5}{2}}}\) | \(225\) |
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^{4}}{\operatorname {acos}^{\frac {7}{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 = -0.00 \begin {gather*} \int \frac {x^4}{{\mathrm {acos}\left (a\,x\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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