Integrand size = 16, antiderivative size = 355 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=-\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {8 b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right )^2 x}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}-\frac {8 b c d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{15 \sqrt {1-c} \left (1-c^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}+\frac {2 b (1+3 c) d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right ),-\frac {1-c}{1+c}\right )}{15 \sqrt {1-c} \left (1-c^2\right ) \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \]
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Time = 0.25 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4926, 12, 1137, 1295, 1216, 538, 435, 430} \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}+\frac {2 b (3 c+1) d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right ),-\frac {1-c}{c+1}\right )}{15 \sqrt {1-c} \left (1-c^2\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {8 b c d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {\frac {d x^2}{c+1}+1} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{c+1}\right )}{15 \sqrt {1-c} \left (1-c^2\right ) \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}-\frac {8 b c d^2 \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{15 \left (1-c^2\right )^2 x}-\frac {2 b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{15 \left (1-c^2\right ) x^3} \]
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Rule 12
Rule 430
Rule 435
Rule 538
Rule 1137
Rule 1216
Rule 1295
Rule 4926
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}+\frac {1}{5} b \int \frac {2 d}{x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx \\ & = -\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}+\frac {1}{5} (2 b d) \int \frac {1}{x^4 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx \\ & = -\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}+\frac {(2 b d) \int \frac {4 c d+d^2 x^2}{x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx}{15 \left (1-c^2\right )} \\ & = -\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {8 b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right )^2 x}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}-\frac {(2 b d) \int \frac {-\left (\left (1-c^2\right ) d^2\right )+4 c d^3 x^2}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx}{15 \left (1-c^2\right )^2} \\ & = -\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {8 b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right )^2 x}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}-\frac {\left (2 b d \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {-\left (\left (1-c^2\right ) d^2\right )+4 c d^3 x^2}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{15 \left (1-c^2\right )^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \\ & = -\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {8 b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right )^2 x}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}-\frac {\left (8 b c (1+c) d^3 \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}}}{\sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{15 \left (1-c^2\right )^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}+\frac {\left (2 b (1+c) (1+3 c) d^3 \sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}\right ) \int \frac {1}{\sqrt {1-\frac {2 d^2 x^2}{-2 d-2 c d}} \sqrt {1-\frac {2 d^2 x^2}{2 d-2 c d}}} \, dx}{15 \left (1-c^2\right )^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \\ & = -\frac {2 b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right ) x^3}-\frac {8 b c d^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{15 \left (1-c^2\right )^2 x}-\frac {a+b \arcsin \left (c+d x^2\right )}{5 x^5}-\frac {8 b c d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right )|-\frac {1-c}{1+c}\right )}{15 (1-c)^{3/2} (1+c) \sqrt {1-c^2-2 c d x^2-d^2 x^4}}+\frac {2 b (1+3 c) d^{5/2} \sqrt {1-\frac {d x^2}{1-c}} \sqrt {1+\frac {d x^2}{1+c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {1-c}}\right ),-\frac {1-c}{1+c}\right )}{15 (1-c)^{3/2} (1+c) \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.52 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.04 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\frac {\sqrt {\frac {d}{1+c}} \left (-3 a \left (-1+c^2\right )^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}+2 b d x^2 \left (-1-c^4+2 c^3 d x^2+d^2 x^4+c^2 \left (2+7 d^2 x^4\right )+c \left (-2 d x^2+4 d^3 x^6\right )\right )-3 b \left (-1+c^2\right )^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4} \arcsin \left (c+d x^2\right )\right )+8 i b (-1+c) c d^3 x^5 \sqrt {\frac {-1+c+d x^2}{-1+c}} \sqrt {\frac {1+c+d x^2}{1+c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {d}{1+c}} x\right )|\frac {1+c}{-1+c}\right )-2 i b \left (1-4 c+3 c^2\right ) d^3 x^5 \sqrt {\frac {-1+c+d x^2}{-1+c}} \sqrt {\frac {1+c+d x^2}{1+c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {d}{1+c}} x\right ),\frac {1+c}{-1+c}\right )}{15 \left (-1+c^2\right )^2 \sqrt {\frac {d}{1+c}} x^5 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \]
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Time = 0.84 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {a}{5 x^{5}}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{5 x^{5}}+\frac {2 d \left (\frac {\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{3 \left (c^{2}-1\right ) x^{3}}-\frac {4 c d \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{3 \left (c^{2}-1\right )^{2} x}-\frac {d^{2} \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )}{3 \left (c^{2}-1\right ) \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}+\frac {8 c \,d^{3} \left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )\right )}{3 \left (c^{2}-1\right )^{2} \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}\, \left (-2 c d +2 d \right )}\right )}{5}\right )\) | \(346\) |
parts | \(-\frac {a}{5 x^{5}}+b \left (-\frac {\arcsin \left (d \,x^{2}+c \right )}{5 x^{5}}+\frac {2 d \left (\frac {\sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{3 \left (c^{2}-1\right ) x^{3}}-\frac {4 c d \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{3 \left (c^{2}-1\right )^{2} x}-\frac {d^{2} \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )}{3 \left (c^{2}-1\right ) \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}+\frac {8 c \,d^{3} \left (-c^{2}+1\right ) \sqrt {1+\frac {d \,x^{2}}{-1+c}}\, \sqrt {1+\frac {d \,x^{2}}{1+c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{-1+c}}, \sqrt {-1+\frac {2 c}{1+c}}\right )\right )}{3 \left (c^{2}-1\right )^{2} \sqrt {-\frac {d}{-1+c}}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}\, \left (-2 c d +2 d \right )}\right )}{5}\right )\) | \(346\) |
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\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\int { \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{6}} \,d x } \]
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\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{6}}\, dx \]
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Exception generated. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\int { \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^6} \, dx=\int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^6} \,d x \]
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