Optimal. Leaf size=98 \[ \frac {x \sin \left (\frac {a}{2 b}\right ) \text {Ci}\left (\frac {a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )}{\sqrt {2} b \sqrt {d x^2}} \]
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Rubi [A] time = 0.01, antiderivative size = 98, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.11, size = 85, normalized size = 0.87 \[ \frac {\cos \left (\frac {1}{2} \cos ^{-1}\left (d x^2-1\right )\right ) \left (\sin \left (\frac {a}{2 b}\right ) \text {Ci}\left (\frac {a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )-\cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (d x^2-1\right )}{2 b}\right )\right )}{b d x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b \arccos \left (d x^{2} - 1\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \arccos \left (d x^{2} - 1\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{a +b \arccos \left (d \,x^{2}-1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{b \arccos \left (d x^{2} - 1\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{a+b\,\mathrm {acos}\left (d\,x^2-1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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