Integrand size = 22, antiderivative size = 263 \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}+\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}-\frac {\sqrt {-\frac {i}{b}} b \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )} \]
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Time = 0.02 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {4895} \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=-\frac {\sqrt {\pi } \sqrt {-\frac {i}{b}} b x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )}+\frac {\sqrt {\pi } x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )}+x \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \]
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Rule 4895
Rubi steps \begin{align*} \text {integral}& = x \sqrt {a+i b \arcsin \left (1-i d x^2\right )}+\frac {\sqrt {\pi } x \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{\sqrt {-\frac {i}{b}} \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 {-\frac {i}{b}} b \sqrt {\pi } x \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\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 )} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.98 \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\frac {x \left (\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )-\sqrt {\pi } \operatorname {FresnelC}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right )+\sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {-\frac {i}{b}} \sqrt {a+i b \arcsin \left (1-i d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )\right )}{\sqrt {-\frac {i}{b}} \left (\cos \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )-\sin \left (\frac {1}{2} \arcsin \left (1-i d x^2\right )\right )\right )} \]
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\[\int \sqrt {a +b \,\operatorname {arcsinh}\left (d \,x^{2}+i\right )}d x\]
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Exception generated. \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Exception generated. \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\int { \sqrt {b \operatorname {arsinh}\left (d x^{2} + i\right ) + a} \,d x } \]
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Exception generated. \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \sqrt {a+i b \arcsin \left (1-i d x^2\right )} \, dx=\int \sqrt {a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )} \,d x \]
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