Integrand size = 20, antiderivative size = 245 \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=-\frac {\sqrt {2 i d x^2+d^2 x^4}}{2 b d x \left (a+i b \arcsin \left (1-i d x^2\right )\right )}+\frac {x \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}+\frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]
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Time = 0.04 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {4909} \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\frac {x \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}+\frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {\sqrt {d^2 x^4+2 i d x^2}}{2 b d x \left (a+i b \arcsin \left (1-i d x^2\right )\right )} \]
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Rule 4909
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {2 i d x^2+d^2 x^4}}{2 b d x \left (a+i b \arcsin \left (1-i d x^2\right )\right )}+\frac {x \operatorname {CosIntegral}\left (-\frac {i \left (a+i b \arcsin \left (1-i d x^2\right )\right )}{2 b}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}+\frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \\ \end{align*}
Time = 1.10 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\frac {-\frac {2 b \sqrt {d x^2 \left (2 i+d x^2\right )}}{d \left (a+i b \arcsin \left (1-i d x^2\right )\right )}+\frac {x^2 \left (\operatorname {CosIntegral}\left (\frac {1}{2} \left (-\frac {i a}{b}+\arcsin \left (1-i d x^2\right )\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )+\left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {i a}{2 b}-\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}}{4 b^2 x} \]
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\[\int \frac {1}{{\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}+i\right )\right )}^{2}}d x\]
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\[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\left (a+i b \arcsin \left (1-i d x^2\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^2} \,d x \]
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