Optimal. Leaf size=168 \[ \frac{x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (-\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}-\frac{x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )} \]
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Rubi [A] time = 0.0228204, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {4816} \[ \frac{x \left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (-\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}-\frac{x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4816
Rubi steps
\begin{align*} \int \frac{1}{a-b \sin ^{-1}\left (1-d x^2\right )} \, dx &=\frac{x \text{Ci}\left (-\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac{a}{2 b}\right )+\sin \left (\frac{a}{2 b}\right )\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}-\frac{x \left (\cos \left (\frac{a}{2 b}\right )-\sin \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a}{2 b}-\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{2 b \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.195105, size = 130, normalized size = 0.77 \[ \frac{\left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right ) \left (\left (\sin \left (\frac{a}{2 b}\right )+\cos \left (\frac{a}{2 b}\right )\right ) \text{CosIntegral}\left (\frac{1}{2} \left (\sin ^{-1}\left (1-d x^2\right )-\frac{a}{b}\right )\right )+\left (\sin \left (\frac{a}{2 b}\right )-\cos \left (\frac{a}{2 b}\right )\right ) \text{Si}\left (\frac{a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right )\right )}{2 b d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arcsin \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 \arcsin \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 \arcsin \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{asin}{\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 \arcsin \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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