Optimal. Leaf size=231 \[ -\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]
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Rubi [A] time = 0.0299369, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {4819} \[ -\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4819
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}} \, dx &=-\frac{\sqrt{\pi } x S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}-\frac{\sqrt{\pi } x C\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )}\\ \end{align*}
Mathematica [A] time = 0.0037899, size = 180, normalized size = 0.78 \[ \frac{\sqrt{\pi } x \left (\left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) \left (-S\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{i b} \sqrt{\pi }}\right )\right )-\left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{a+i b \sin ^{-1}\left (1-i d x^2\right )}}{\sqrt{\pi } \sqrt{i b}}\right )\right )}{\sqrt{i b} \left (\cos \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1-i d x^2\right )\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.063, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a+b{\it Arcsinh} \left ( i+d{x}^{2} \right ) }}}\, 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}{\sqrt{b \operatorname{arsinh}\left (d x^{2} + i\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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}{\sqrt{b \operatorname{arsinh}\left (d x^{2} + i\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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