Optimal. Leaf size=184 \[ -\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x} \]
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Rubi [A] time = 0.02, antiderivative size = 184, normalized size of antiderivative = 1.00, 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 = {4812} \[ -\frac {2 \sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}+\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{d x} \]
Antiderivative was successfully verified.
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Rule 4812
Rubi steps
\begin {align*} \int \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \, dx &=\frac {2 \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {\pi } C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{\sqrt {\frac {1}{b}} d x}-\frac {2 \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )} \sin ^2\left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{d x}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 157, normalized size = 0.85 \[ -\frac {2 \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \left (\sqrt {\pi } \sin \left (\frac {a}{2 b}\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )-\sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )+\sqrt {\frac {1}{b}} \sin \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \sqrt {a+b \cos ^{-1}\left (d x^2+1\right )}\right )}{\sqrt {\frac {1}{b}} d x} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \sqrt {b \arccos \left (d x^{2} + 1\right ) + a}\,{d x} \]
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 \[ \int \sqrt {a +b \arccos \left (d \,x^{2}+1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \sqrt {a+b\,\mathrm {acos}\left (d\,x^2+1\right )} \,d x \]
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 \[ \int \sqrt {a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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