Integrand size = 16, antiderivative size = 137 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=-\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}-\frac {b c d^2 \text {arctanh}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{4 \left (1-c^2\right )^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4926, 12, 1128, 744, 738, 212} \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=-\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}-\frac {b c d^2 \text {arctanh}\left (\frac {-c^2-c d x^2+1}{\sqrt {1-c^2} \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}\right )}{4 \left (1-c^2\right )^{3/2}}-\frac {b d \sqrt {-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (1-c^2\right ) x^2} \]
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Rule 12
Rule 212
Rule 738
Rule 744
Rule 1128
Rule 4926
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}+\frac {1}{4} b \int \frac {2 d}{x^3 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx \\ & = -\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}+\frac {1}{2} (b d) \int \frac {1}{x^3 \sqrt {1-c^2-2 c d x^2-d^2 x^4}} \, dx \\ & = -\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}+\frac {1}{4} (b d) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right ) \\ & = -\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}+\frac {\left (b c d^2\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{4 \left (1-c^2\right )} \\ & = -\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}-\frac {\left (b c d^2\right ) \text {Subst}\left (\int \frac {1}{4 \left (1-c^2\right )-x^2} \, dx,x,\frac {2 \left (1-c^2-c d x^2\right )}{\sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{2 \left (1-c^2\right )} \\ & = -\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac {a+b \arcsin \left (c+d x^2\right )}{4 x^4}-\frac {b c d^2 \text {arctanh}\left (\frac {1-c^2-c d x^2}{\sqrt {1-c^2} \sqrt {1-c^2-2 c d x^2-d^2 x^4}}\right )}{4 \left (1-c^2\right )^{3/2}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.08 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=-\frac {a}{4 x^4}+\frac {b d \sqrt {1-c^2-2 c d x^2-d^2 x^4}}{4 \left (-1+c^2\right ) x^2}-\frac {b \arcsin \left (c+d x^2\right )}{4 x^4}+\frac {b c d^2 \arctan \left (\frac {\sqrt {-d^2} x^2-\sqrt {1-c^2-2 c d x^2-d^2 x^4}}{\sqrt {-1+c^2}}\right )}{2 (-1+c) (1+c) \sqrt {-1+c^2}} \]
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Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.96
method | result | size |
default | \(-\frac {a}{4 x^{4}}-\frac {b \arcsin \left (d \,x^{2}+c \right )}{4 x^{4}}-\frac {b d \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right ) x^{2}}-\frac {b \,d^{2} c \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {3}{2}}}\) | \(132\) |
parts | \(-\frac {a}{4 x^{4}}-\frac {b \arcsin \left (d \,x^{2}+c \right )}{4 x^{4}}-\frac {b d \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{4 \left (-c^{2}+1\right ) x^{2}}-\frac {b \,d^{2} c \ln \left (\frac {-2 c^{2}+2-2 c d \,x^{2}+2 \sqrt {-c^{2}+1}\, \sqrt {-d^{2} x^{4}-2 c d \,x^{2}-c^{2}+1}}{x^{2}}\right )}{4 \left (-c^{2}+1\right )^{\frac {3}{2}}}\) | \(132\) |
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Time = 0.33 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.86 \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=\left [-\frac {\sqrt {-c^{2} + 1} b c d^{2} x^{4} \log \left (\frac {{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \, {\left (c^{3} - c\right )} d x^{2} - 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 2 \, a c^{4} - 2 \, \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (b c^{2} - b\right )} d x^{2} - 4 \, a c^{2} + 2 \, {\left (b c^{4} - 2 \, b c^{2} + b\right )} \arcsin \left (d x^{2} + c\right ) + 2 \, a}{8 \, {\left (c^{4} - 2 \, c^{2} + 1\right )} x^{4}}, -\frac {\sqrt {c^{2} - 1} b c d^{2} x^{4} \arctan \left (\frac {\sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (c d x^{2} + c^{2} - 1\right )} \sqrt {c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \, {\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) + a c^{4} - \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} {\left (b c^{2} - b\right )} d x^{2} - 2 \, a c^{2} + {\left (b c^{4} - 2 \, b c^{2} + b\right )} \arcsin \left (d x^{2} + c\right ) + a}{4 \, {\left (c^{4} - 2 \, c^{2} + 1\right )} x^{4}}\right ] \]
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\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=\int \frac {a + b \operatorname {asin}{\left (c + d x^{2} \right )}}{x^{5}}\, dx \]
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Exception generated. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=\int { \frac {b \arcsin \left (d x^{2} + c\right ) + a}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {a+b \arcsin \left (c+d x^2\right )}{x^5} \, dx=\int \frac {a+b\,\mathrm {asin}\left (d\,x^2+c\right )}{x^5} \,d x \]
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