Integrand size = 14, antiderivative size = 149 \[ \int \frac {1}{\left (a+b \arccos \left (-1+d x^2\right )\right )^2} \, dx=\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a+b \arccos \left (-1+d x^2\right )\right )}-\frac {x \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 149, 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 = {4911} \[ \int \frac {1}{\left (a+b \arccos \left (-1+d x^2\right )\right )^2} \, dx=-\frac {x \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (d x^2-1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}+\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a+b \arccos \left (d x^2-1\right )\right )} \]
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Rule 4911
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 \cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {d x^2}}-\frac {x \sin \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 = 0.64 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b \arccos \left (-1+d x^2\right )\right )^2} \, dx=\frac {\sqrt {-d x^2 \left (-2+d x^2\right )} \left (\frac {b}{a+b \arccos \left (-1+d x^2\right )}+\frac {\sin \left (\frac {1}{2} \arccos \left (-1+d x^2\right )\right ) \left (\cos \left (\frac {a}{2 b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )+\sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \arccos \left (-1+d x^2\right )}{2 b}\right )\right )}{-2+d x^2}\right )}{2 b^2 d x} \]
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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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\[ \int \frac {1}{\left (a+b \arccos \left (-1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \arccos \left (-1+d x^2\right )\right )^2} \, dx=\int { \frac {1}{{\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{2}} \,d x } \]
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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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