Integrand size = 16, antiderivative size = 145 \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=-\frac {2 \sqrt {\frac {1}{b}} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{d x}-\frac {2 \sqrt {\frac {1}{b}} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{d x} \]
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Time = 0.01 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4904} \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=-\frac {2 \sqrt {\pi } \sqrt {\frac {1}{b}} \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{d x}-\frac {2 \sqrt {\pi } \sqrt {\frac {1}{b}} \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{d x} \]
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Rule 4904
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {\frac {1}{b}} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{d x}-\frac {2 \sqrt {\frac {1}{b}} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{d x} \\ \end{align*}
Time = 0.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74 \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=-\frac {2 \sqrt {\pi } \left (\cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right )+\operatorname {FresnelS}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{\sqrt {b} d x} \]
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\[\int \frac {1}{\sqrt {a +b \arccos \left (d \,x^{2}+1\right )}}d x\]
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Exception generated. \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=\int \frac {1}{\sqrt {a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=\int { \frac {1}{\sqrt {b \arccos \left (d x^{2} + 1\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx=\int \frac {1}{\sqrt {a+b\,\mathrm {acos}\left (d\,x^2+1\right )}} \,d x \]
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