Integrand size = 14, antiderivative size = 151 \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\frac {\sqrt {-2 d x^2-d^2 x^4}}{2 b d x \left (a+b \arccos \left (1+d x^2\right )\right )}+\frac {x \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}} \]
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Time = 0.02 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4910} \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\frac {x \sin \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{2 b d x \left (a+b \arccos \left (d x^2+1\right )\right )} \]
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Rule 4910
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-2 d x^2-d^2 x^4}}{2 b d x \left (a+b \arccos \left (1+d x^2\right )\right )}+\frac {x \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}} \\ \end{align*}
Time = 0.74 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\frac {\sqrt {-d x^2 \left (2+d x^2\right )} \left (\frac {b}{a+b \arccos \left (1+d x^2\right )}-\frac {\cos \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right ) \left (\operatorname {CosIntegral}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )-\cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (1+d x^2\right )}{2 b}\right )\right )}{2+d x^2}\right )}{2 b^2 d x} \]
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\[\int \frac {1}{{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{2}}d x\]
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\[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^2} \,d x \]
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