Optimal. Leaf size=190 \[ \frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \text {ArcCos}(a x)^{3/2}}+\frac {16 x^4}{15 \text {ArcCos}(a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\text {ArcCos}(a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\text {ArcCos}(a x)}}+\frac {32 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{15 a^4} \]
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Rubi [A]
time = 0.22, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 12, 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 {32 \sqrt {2 \pi } \text {FresnelC}\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\text {ArcCos}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\text {ArcCos}(a x)}}{\sqrt {\pi }}\right )}{15 a^4}-\frac {4 x^2}{5 a^2 \text {ArcCos}(a x)^{3/2}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\text {ArcCos}(a x)}}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \text {ArcCos}(a x)^{5/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\text {ArcCos}(a x)}}+\frac {16 x^4}{15 \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^3}{\cos ^{-1}(a x)^{7/2}} \, dx &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {6 \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx}{5 a}+\frac {1}{5} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{5/2}} \, dx\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \cos ^{-1}(a x)^{3/2}}-\frac {64}{15} \int \frac {x^3}{\cos ^{-1}(a x)^{3/2}} \, dx+\frac {8 \int \frac {x}{\cos ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}-\frac {16 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{5 a^4}-\frac {128 \text {Subst}\left (\int \left (-\frac {\cos (2 x)}{2 \sqrt {x}}-\frac {\cos (4 x)}{2 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {64 \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}+\frac {64 \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{15 a^4}-\frac {32 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{5 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}-\frac {16 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{5 a^4}+\frac {128 \text {Subst}\left (\int \cos \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{15 a^4}+\frac {128 \text {Subst}\left (\int \cos \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{15 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{5 a \cos ^{-1}(a x)^{5/2}}-\frac {4 x^2}{5 a^2 \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{15 \cos ^{-1}(a x)^{3/2}}+\frac {16 x \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\cos ^{-1}(a x)}}-\frac {128 x^3 \sqrt {1-a^2 x^2}}{15 a \sqrt {\cos ^{-1}(a x)}}+\frac {32 \sqrt {2 \pi } C\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{15 a^4}+\frac {16 \sqrt {\pi } C\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{15 a^4}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 2.73, size = 264, normalized size = 1.39 \begin {gather*} -\frac {-4 e^{-2 i \text {ArcCos}(a x)} \left (1+e^{4 i \text {ArcCos}(a x)} (1+4 i \text {ArcCos}(a x))-4 i \text {ArcCos}(a x)\right ) \text {ArcCos}(a x)+\frac {16 \sqrt {2} \text {ArcCos}(a x)^3 \text {Gamma}\left (\frac {1}{2},-2 i \text {ArcCos}(a x)\right )}{\sqrt {-i \text {ArcCos}(a x)}}+16 i \sqrt {2} (i \text {ArcCos}(a x))^{5/2} \text {Gamma}\left (\frac {1}{2},2 i \text {ArcCos}(a x)\right )-2 \text {ArcCos}(a x) \left (2 e^{4 i \text {ArcCos}(a x)} (1+8 i \text {ArcCos}(a x))+32 (-i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},-4 i \text {ArcCos}(a x)\right )+2 e^{-4 i \text {ArcCos}(a x)} \left (1-8 i \text {ArcCos}(a x)+16 e^{4 i \text {ArcCos}(a x)} (i \text {ArcCos}(a x))^{3/2} \text {Gamma}\left (\frac {1}{2},4 i \text {ArcCos}(a x)\right )\right )\right )-6 \sin (2 \text {ArcCos}(a x))-3 \sin (4 \text {ArcCos}(a x))}{60 a^4 \text {ArcCos}(a x)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.23, size = 139, normalized size = 0.73
method | result | size |
default | \(-\frac {-128 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}-64 \sqrt {\pi }\, \FresnelC \left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {5}{2}}+32 \sin \left (2 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}+64 \sin \left (4 \arccos \left (a x \right )\right ) \arccos \left (a x \right )^{2}-8 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )-8 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )-6 \sin \left (2 \arccos \left (a x \right )\right )-3 \sin \left (4 \arccos \left (a x \right )\right )}{60 a^{4} \arccos \left (a x \right )^{\frac {5}{2}}}\) | \(139\) |
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^{3}}{\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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^3}{{\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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