Optimal. Leaf size=264 \[ \frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^5}+\frac {8 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}} \]
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Rubi [A] time = 0.40, 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 = {4633, 4719, 4631, 3305, 3351} \[ \frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^5}+\frac {8 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3305
Rule 3351
Rule 4631
Rule 4633
Rule 4719
Rubi steps
\begin {align*} \int \frac {x^4}{\sin ^{-1}(a x)^{7/2}} \, dx &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}+\frac {8 \int \frac {x^3}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx}{5 a}-(2 a) \int \frac {x^5}{\sqrt {1-a^2 x^2} \sin ^{-1}(a x)^{5/2}} \, dx\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac {20}{3} \int \frac {x^4}{\sin ^{-1}(a x)^{3/2}} \, dx+\frac {16 \int \frac {x^2}{\sin ^{-1}(a x)^{3/2}} \, dx}{5 a^2}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}+\frac {32 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{4 \sqrt {x}}+\frac {3 \sin (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}-\frac {40 \operatorname {Subst}\left (\int \left (-\frac {\sin (x)}{8 \sqrt {x}}+\frac {9 \sin (3 x)}{16 \sqrt {x}}-\frac {5 \sin (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}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}-\frac {8 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}+\frac {5 \operatorname {Subst}\left (\int \frac {\sin (x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{3 a^5}+\frac {25 \operatorname {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{6 a^5}+\frac {24 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{5 a^5}-\frac {15 \operatorname {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {x}} \, dx,x,\sin ^{-1}(a x)\right )}{2 a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}-\frac {16 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}+\frac {10 \operatorname {Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {25 \operatorname {Subst}\left (\int \sin \left (5 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}+\frac {48 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}-\frac {15 \operatorname {Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt {\sin ^{-1}(a x)}\right )}{a^5}\\ &=-\frac {2 x^4 \sqrt {1-a^2 x^2}}{5 a \sin ^{-1}(a x)^{5/2}}-\frac {16 x^3}{15 a^2 \sin ^{-1}(a x)^{3/2}}+\frac {4 x^5}{3 \sin ^{-1}(a x)^{3/2}}-\frac {32 x^2 \sqrt {1-a^2 x^2}}{5 a^3 \sqrt {\sin ^{-1}(a x)}}+\frac {40 x^4 \sqrt {1-a^2 x^2}}{3 a \sqrt {\sin ^{-1}(a x)}}+\frac {\sqrt {2 \pi } S\left (\sqrt {\frac {2}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{15 a^5}-\frac {5 \sqrt {\frac {3 \pi }{2}} S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{a^5}+\frac {8 \sqrt {6 \pi } S\left (\sqrt {\frac {6}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{5 a^5}+\frac {5 \sqrt {\frac {5 \pi }{2}} S\left (\sqrt {\frac {10}{\pi }} \sqrt {\sin ^{-1}(a x)}\right )}{3 a^5}\\ \end {align*}
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Mathematica [C] time = 0.74, size = 417, normalized size = 1.58 \[ \frac {9 e^{3 i \sin ^{-1}(a x)} \left (-12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+1\right )+2 e^{i \sin ^{-1}(a x)} \left (4 \sin ^{-1}(a x)^2-2 i \sin ^{-1}(a x)-3\right )+e^{5 i \sin ^{-1}(a x)} \left (100 \sin ^{-1}(a x)^2-10 i \sin ^{-1}(a x)-3\right )-8 \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-i \sin ^{-1}(a x)\right )+108 \sqrt {3} \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-3 i \sin ^{-1}(a x)\right )-100 \sqrt {5} \sqrt {-i \sin ^{-1}(a x)} \sin ^{-1}(a x)^2 \Gamma \left (\frac {1}{2},-5 i \sin ^{-1}(a x)\right )+e^{-i \sin ^{-1}(a x)} \left (8 \sin ^{-1}(a x)^2+4 i \sin ^{-1}(a x)+8 e^{i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},i \sin ^{-1}(a x)\right )-6\right )-9 e^{-3 i \sin ^{-1}(a x)} \left (12 \sin ^{-1}(a x)^2+2 i \sin ^{-1}(a x)+12 \sqrt {3} e^{3 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},3 i \sin ^{-1}(a x)\right )-1\right )+e^{-5 i \sin ^{-1}(a x)} \left (100 \sin ^{-1}(a x)^2+10 i \sin ^{-1}(a x)+100 \sqrt {5} e^{5 i \sin ^{-1}(a x)} \left (i \sin ^{-1}(a x)\right )^{5/2} \Gamma \left (\frac {1}{2},5 i \sin ^{-1}(a x)\right )-3\right )}{240 a^5 \sin ^{-1}(a x)^{5/2}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\arcsin \left (a x\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 225, normalized size = 0.85 \[ -\frac {108 \sqrt {2}\, \sqrt {\pi }\, \sqrt {3}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-100 \sqrt {2}\, \sqrt {\pi }\, \sqrt {5}\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {5}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \sqrt {2}\, \sqrt {\pi }\, \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {\arcsin \left (a x \right )}}{\sqrt {\pi }}\right ) \arcsin \left (a x \right )^{\frac {5}{2}}-8 \arcsin \left (a x \right )^{2} \sqrt {-a^{2} x^{2}+1}+108 \arcsin \left (a x \right )^{2} \cos \left (3 \arcsin \left (a x \right )\right )-100 \arcsin \left (a x \right )^{2} \cos \left (5 \arcsin \left (a x \right )\right )-4 a x \arcsin \left (a x \right )+18 \arcsin \left (a x \right ) \sin \left (3 \arcsin \left (a x \right )\right )-10 \arcsin \left (a x \right ) \sin \left (5 \arcsin \left (a x \right )\right )+6 \sqrt {-a^{2} x^{2}+1}-9 \cos \left (3 \arcsin \left (a x \right )\right )+3 \cos \left (5 \arcsin \left (a x \right )\right )}{120 a^{5} \arcsin \left (a x \right )^{\frac {5}{2}}} \]
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 = -0.00 \[ \int \frac {x^4}{{\mathrm {asin}\left (a\,x\right )}^{7/2}} \,d x \]
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 \[ \int \frac {x^{4}}{\operatorname {asin}^{\frac {7}{2}}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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