Integrand size = 22, antiderivative size = 365 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)+\frac {27 c \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{32 a \sqrt {1-a^2 x^2}} \]
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Time = 0.23 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {4743, 4741, 4737, 4723, 4795, 30, 4767, 14} \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=-\frac {9 a c x^2 \arcsin (a x)^2 \sqrt {c-a^2 c x^2}}{16 \sqrt {1-a^2 x^2}}+\frac {1}{4} x \arcsin (a x)^3 \left (c-a^2 c x^2\right )^{3/2}+\frac {3}{8} c x \arcsin (a x)^3 \sqrt {c-a^2 c x^2}-\frac {45}{64} c x \arcsin (a x) \sqrt {c-a^2 c x^2}-\frac {3}{32} c x \left (1-a^2 x^2\right ) \arcsin (a x) \sqrt {c-a^2 c x^2}+\frac {3 c \arcsin (a x)^4 \sqrt {c-a^2 c x^2}}{32 a \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \arcsin (a x)^2 \sqrt {c-a^2 c x^2}}{16 a}+\frac {27 c \arcsin (a x)^2 \sqrt {c-a^2 c x^2}}{128 a \sqrt {1-a^2 x^2}}+\frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}} \]
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Rule 14
Rule 30
Rule 4723
Rule 4737
Rule 4741
Rule 4743
Rule 4767
Rule 4795
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {1}{4} (3 c) \int \sqrt {c-a^2 c x^2} \arcsin (a x)^3 \, dx-\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \arcsin (a x)^2 \, dx}{4 \sqrt {1-a^2 x^2}} \\ & = \frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3-\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \left (1-a^2 x^2\right )^{3/2} \arcsin (a x) \, dx}{8 \sqrt {1-a^2 x^2}}+\frac {\left (3 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}}-\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \arcsin (a x)^2 \, dx}{8 \sqrt {1-a^2 x^2}} \\ & = -\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \sqrt {1-a^2 x^2} \arcsin (a x) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int x \left (1-a^2 x^2\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a^2 c \sqrt {c-a^2 c x^2}\right ) \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{8 \sqrt {1-a^2 x^2}} \\ & = -\frac {45}{64} c x \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{32 a \sqrt {1-a^2 x^2}}-\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 c \sqrt {c-a^2 c x^2}\right ) \int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}} \, dx}{16 \sqrt {1-a^2 x^2}}+\frac {\left (3 a c \sqrt {c-a^2 c x^2}\right ) \int \left (x-a^2 x^3\right ) \, dx}{32 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{64 \sqrt {1-a^2 x^2}}+\frac {\left (9 a c \sqrt {c-a^2 c x^2}\right ) \int x \, dx}{16 \sqrt {1-a^2 x^2}} \\ & = \frac {51 a c x^2 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {3 a^3 c x^4 \sqrt {c-a^2 c x^2}}{128 \sqrt {1-a^2 x^2}}-\frac {45}{64} c x \sqrt {c-a^2 c x^2} \arcsin (a x)-\frac {3}{32} c x \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2} \arcsin (a x)+\frac {27 c \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{128 a \sqrt {1-a^2 x^2}}-\frac {9 a c x^2 \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 \sqrt {1-a^2 x^2}}+\frac {3 c \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2} \arcsin (a x)^2}{16 a}+\frac {3}{8} c x \sqrt {c-a^2 c x^2} \arcsin (a x)^3+\frac {1}{4} x \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3+\frac {3 c \sqrt {c-a^2 c x^2} \arcsin (a x)^4}{32 a \sqrt {1-a^2 x^2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.38 \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\frac {c \sqrt {c-a^2 c x^2} \left (96 \arcsin (a x)^4+24 \arcsin (a x)^2 (16 \cos (2 \arcsin (a x))+\cos (4 \arcsin (a x)))-3 (64 \cos (2 \arcsin (a x))+\cos (4 \arcsin (a x)))+32 \arcsin (a x)^3 (8 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x)))-12 \arcsin (a x) (32 \sin (2 \arcsin (a x))+\sin (4 \arcsin (a x)))\right )}{1024 a \sqrt {1-a^2 x^2}} \]
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Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.30
method | result | size |
default | \(-\frac {3 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )^{4} c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (-8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+8 a^{5} x^{5}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-12 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}+4 a x \right ) \left (24 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-3 i-12 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+2 a^{3} x^{3}-i \sqrt {-a^{2} x^{2}+1}-2 a x \right ) \left (-6 i \arcsin \left (a x \right )^{2}+4 \arcsin \left (a x \right )^{3}+3 i-6 \arcsin \left (a x \right )\right ) c}{32 a \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i a^{2} x^{2}-a x \sqrt {-a^{2} x^{2}+1}-i\right ) \left (408 i \arcsin \left (a x \right )^{2}+224 \arcsin \left (a x \right )^{3}-195 i-372 \arcsin \left (a x \right )\right ) \cos \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}+\frac {9 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (i \sqrt {-a^{2} x^{2}+1}\, a x +a^{2} x^{2}-1\right ) \left (40 i \arcsin \left (a x \right )^{2}+32 \arcsin \left (a x \right )^{3}-21 i-44 \arcsin \left (a x \right )\right ) \sin \left (3 \arcsin \left (a x \right )\right ) c}{2048 a \left (a^{2} x^{2}-1\right )}\) | \(474\) |
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\[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{3} \,d x } \]
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\[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\int \left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {asin}^{3}{\left (a x \right )}\, dx \]
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\[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\int { {\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arcsin \left (a x\right )^{3} \,d x } \]
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Exception generated. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c-a^2 c x^2\right )^{3/2} \arcsin (a x)^3 \, dx=\int {\mathrm {asin}\left (a\,x\right )}^3\,{\left (c-a^2\,c\,x^2\right )}^{3/2} \,d x \]
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