Integrand size = 16, antiderivative size = 221 \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arccos \left (1+d x^2\right )}}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{3 d x}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )}{3 d x} \]
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Time = 0.04 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4913, 4904} \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {x}{3 b^2 \sqrt {a+b \arccos \left (d x^2+1\right )}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{3 b d x \left (a+b \arccos \left (d x^2+1\right )\right )^{3/2}}+\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{5/2} \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{3 d x}+\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{5/2} \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \arccos \left (d x^2+1\right )\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \arccos \left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{3 d x} \]
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Rule 4904
Rule 4913
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arccos \left (1+d x^2\right )}}-\frac {\int \frac {1}{\sqrt {a+b \arccos \left (1+d x^2\right )}} \, dx}{3 b^2} \\ & = \frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \arccos \left (1+d x^2\right )}}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \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 )}{3 d x}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \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 )}{3 d x} \\ \end{align*}
Time = 1.21 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\frac {2 \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right ) \left (\sqrt {\pi } \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \cos \left (\frac {a}{2 b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\pi } \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2} \operatorname {FresnelS}\left (\frac {\sqrt {a+b \arccos \left (1+d x^2\right )}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )+\sqrt {b} \left (b \cos \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )-\left (a+b \arccos \left (1+d x^2\right )\right ) \sin \left (\frac {1}{2} \arccos \left (1+d x^2\right )\right )\right )\right )}{3 b^{5/2} d x \left (a+b \arccos \left (1+d x^2\right )\right )^{3/2}} \]
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\[\int \frac {1}{{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )}^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \arccos \left (1+d x^2\right )\right )^{5/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^{5/2}} \,d x \]
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