Optimal. Leaf size=126 \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^4}+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}} \]
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Rubi [A] time = 0.33, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4634, 4720, 4636, 4406, 3305, 3351, 12} \[ \frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^4}+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}+\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3305
Rule 3351
Rule 4406
Rule 4634
Rule 4636
Rule 4720
Rubi steps
\begin {align*} \int \frac {x^3}{\cos ^{-1}(a x)^{5/2}} \, dx &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {2 \int \frac {x^2}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx}{a}+\frac {1}{3} (8 a) \int \frac {x^4}{\sqrt {1-a^2 x^2} \cos ^{-1}(a x)^{3/2}} \, dx\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {64}{3} \int \frac {x^3}{\sqrt {\cos ^{-1}(a x)}} \, dx+\frac {8 \int \frac {x}{\sqrt {\cos ^{-1}(a x)}} \, dx}{a^2}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {64 \operatorname {Subst}\left (\int \frac {\cos ^3(x) \sin (x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {64 \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {x}}+\frac {\sin (4 x)}{8 \sqrt {x}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {8 \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}-\frac {4 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{a^4}+\frac {16 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {x}} \, dx,x,\cos ^{-1}(a x)\right )}{3 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \sin \left (4 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^4}-\frac {8 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{a^4}+\frac {32 \operatorname {Subst}\left (\int \sin \left (2 x^2\right ) \, dx,x,\sqrt {\cos ^{-1}(a x)}\right )}{3 a^4}\\ &=\frac {2 x^3 \sqrt {1-a^2 x^2}}{3 a \cos ^{-1}(a x)^{3/2}}-\frac {4 x^2}{a^2 \sqrt {\cos ^{-1}(a x)}}+\frac {16 x^4}{3 \sqrt {\cos ^{-1}(a x)}}+\frac {4 \sqrt {2 \pi } S\left (2 \sqrt {\frac {2}{\pi }} \sqrt {\cos ^{-1}(a x)}\right )}{3 a^4}+\frac {4 \sqrt {\pi } S\left (\frac {2 \sqrt {\cos ^{-1}(a x)}}{\sqrt {\pi }}\right )}{3 a^4}\\ \end {align*}
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Mathematica [C] time = 0.98, size = 203, normalized size = 1.61 \[ -\frac {-\sin \left (4 \cos ^{-1}(a x)\right )-4 \cos ^{-1}(a x) \left (e^{-4 i \cos ^{-1}(a x)}+e^{4 i \cos ^{-1}(a x)}-2 \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-4 i \cos ^{-1}(a x)\right )-2 \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},4 i \cos ^{-1}(a x)\right )\right )-2 \left (\sin \left (2 \cos ^{-1}(a x)\right )+2 \cos ^{-1}(a x) \left (e^{-2 i \cos ^{-1}(a x)}+e^{2 i \cos ^{-1}(a x)}-\sqrt {2} \sqrt {-i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},-2 i \cos ^{-1}(a x)\right )-\sqrt {2} \sqrt {i \cos ^{-1}(a x)} \Gamma \left (\frac {1}{2},2 i \cos ^{-1}(a x)\right )\right )\right )}{12 a^4 \cos ^{-1}(a x)^{3/2}} \]
Warning: Unable to verify antiderivative.
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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.
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giac [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.
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maple [A] time = 0.22, size = 107, normalized size = 0.85 \[ \frac {16 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+16 \sqrt {\pi }\, \mathrm {S}\left (\frac {2 \sqrt {\arccos \left (a x \right )}}{\sqrt {\pi }}\right ) \arccos \left (a x \right )^{\frac {3}{2}}+8 \arccos \left (a x \right ) \cos \left (2 \arccos \left (a x \right )\right )+8 \arccos \left (a x \right ) \cos \left (4 \arccos \left (a x \right )\right )+2 \sin \left (2 \arccos \left (a x \right )\right )+\sin \left (4 \arccos \left (a x \right )\right )}{12 a^{4} \arccos \left (a x \right )^{\frac {3}{2}}} \]
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 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {x^3}{{\mathrm {acos}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{3}}{\operatorname {acos}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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