Optimal. Leaf size=98 \[ \frac{x \sin \left (\frac{a}{2 b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \cos \left (\frac{a}{2 b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}} \]
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Rubi [A] time = 0.0133217, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {4818} \[ \frac{x \sin \left (\frac{a}{2 b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \cos \left (\frac{a}{2 b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}} \]
Antiderivative was successfully verified.
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Rule 4818
Rubi steps
\begin{align*} \int \frac{1}{a+b \cos ^{-1}\left (-1+d x^2\right )} \, dx &=\frac{x \text{Ci}\left (\frac{a+b \cos ^{-1}\left (-1+d x^2\right )}{2 b}\right ) \sin \left (\frac{a}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}-\frac{x \cos \left (\frac{a}{2 b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}\left (-1+d x^2\right )}{2 b}\right )}{\sqrt{2} b \sqrt{d x^2}}\\ \end{align*}
Mathematica [A] time = 0.0998995, size = 85, normalized size = 0.87 \[ \frac{\cos \left (\frac{1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) \left (\sin \left (\frac{a}{2 b}\right ) \text{CosIntegral}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )-\cos \left (\frac{a}{2 b}\right ) \text{Si}\left (\frac{a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )\right )}{b d x} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}-1 \right ) \right ) ^{-1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \arccos \left (d x^{2} - 1\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{b \arccos \left (d x^{2} - 1\right ) + a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{a + b \operatorname{acos}{\left (d x^{2} - 1 \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{b \arccos \left (d x^{2} - 1\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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