Optimal. Leaf size=171 \[ -\frac {\sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {3 \sqrt {\frac {3 \pi }{2}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^5}-\frac {5 \sqrt {\frac {5 \pi }{2}} C\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{6 a^5}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}} \]
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Rubi [A] time = 0.43, antiderivative size = 235, normalized size of antiderivative = 1.37, number of steps used = 19, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4633, 4719, 4635, 4406, 3304, 3352} \[ \frac {4 \sqrt {2 \pi } \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}-\frac {25 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^5}-\frac {5 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{6 a^5}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}} \]
Antiderivative was successfully verified.
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Rule 3304
Rule 3352
Rule 4406
Rule 4633
Rule 4635
Rule 4719
Rubi steps
\begin {align*} \int \frac {x^4}{\sin ^{-1}(a x)^{5/2}} \, dx &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}+\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx}{3 a}-\frac {1}{3} (10 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{3/2}} \, dx\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {100}{3} \int \frac {x^4}{\sqrt {\sin ^{-1}(a x)}} \, dx+\frac {16 \int \frac {x^2}{\sqrt {\sin ^{-1}(a x)}} \, dx}{a^2}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac {100 \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^4(x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}+\frac {16 \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{4 \sqrt {x}}-\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac {100 \operatorname {Subst}\left (\int \left (\frac {\cos (x)}{8 \sqrt {x}}-\frac {3 \cos (3 x)}{16 \sqrt {x}}+\frac {\cos (5 x)}{16 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {25 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{12 a^5}+\frac {4 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac {4 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{a^5}-\frac {25 \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{4 a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {25 \operatorname {Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{6 a^5}+\frac {8 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^5}-\frac {8 \operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^5}-\frac {25 \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {25 \operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{2 a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{3 a \sin ^{-1}(a x)^{3/2}}-\frac {16 x^3}{3 a^2 \sqrt {\sin ^{-1}(a x)}}+\frac {20 x^5}{3 \sqrt {\sin ^{-1}(a x)}}-\frac {25 \sqrt {\frac {\pi }{2}} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {4 \sqrt {2 \pi } C\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}+\frac {25 \sqrt {\frac {\pi }{6}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{2 a^5}-\frac {4 \sqrt {\frac {2 \pi }{3}} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}-\frac {5 \sqrt {\frac {5 \pi }{2}} C\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{6 a^5}\\ \end {align*}
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Mathematica [C] time = 0.30, size = 418, normalized size = 2.44 \[ \frac {\frac {i e^{i \sin ^{-1}(a x)} \left (-2 \sin ^{-1}(a x)+i\right )-2 \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac {e^{-i \sin ^{-1}(a x)} \left (-2 i \sin ^{-1}(a x)+2 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )+1\right )}{24 \sin ^{-1}(a x)^{3/2}}-\frac {i e^{3 i \sin ^{-1}(a x)} \left (-6 \sin ^{-1}(a x)+i\right )-6 \sqrt {3} \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac {e^{-3 i \sin ^{-1}(a x)} \left (-6 i \sin ^{-1}(a x)+6 \sqrt {3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )+1\right )}{16 \sin ^{-1}(a x)^{3/2}}+\frac {i e^{5 i \sin ^{-1}(a x)} \left (-10 \sin ^{-1}(a x)+i\right )-10 \sqrt {5} \left (-i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},-5 i \sin ^{-1}(a x)\right )}{48 \sin ^{-1}(a x)^{3/2}}-\frac {e^{-5 i \sin ^{-1}(a x)} \left (-10 i \sin ^{-1}(a x)+10 \sqrt {5} e^{5 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{3/2} \Gamma \left (\frac {1}{2},5 i \sin ^{-1}(a x)\right )+1\right )}{48 \sin ^{-1}(a x)^{3/2}}}{a^5} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 173, normalized size = 1.01 \[ -\frac {10 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}-18 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}+4 \sqrt {2}\, \sqrt {\pi }\, \FresnelC \left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {3}{2}}-4 a x \arcsin \left (a x \right )+18 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-10 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )+2 \sqrt {-a^{2} x^{2}+1}-3 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )}{24 a^{5} \arcsin \left (a x \right )^{\frac {3}{2}}} \]
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 \[ \int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{5/2}} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {asin}^{\frac {5}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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