Optimal. Leaf size=291 \[ -\frac{\sqrt{d^2 x^4-2 i d x^2}}{b d x \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}-\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )}+\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (i d x^2+1\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )} \]
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Rubi [A] time = 0.0486732, antiderivative size = 291, 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 = {4822} \[ -\frac{\sqrt{d^2 x^4-2 i d x^2}}{b d x \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}-\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )}+\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (i d x^2+1\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4822
Rubi steps
\begin{align*} \int \frac{1}{\left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^{3/2}} \, dx &=-\frac{\sqrt{-2 i d x^2+d^2 x^4}}{b d x \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}+\frac{\left (\frac{i}{b}\right )^{3/2} \sqrt{\pi } x S\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}{\sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right )}{\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 )}-\frac{\left (\frac{i}{b}\right )^{3/2} \sqrt{\pi } x C\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}{\sqrt{\pi }}\right ) \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right )}{\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 )}\\ \end{align*}
Mathematica [A] time = 0.356835, size = 291, normalized size = 1. \[ -\frac{\sqrt{d^2 x^4-2 i d x^2}}{b d x \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}-\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )+i \sinh \left (\frac{a}{2 b}\right )\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (1+i d x^2\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )}+\frac{\sqrt{\pi } \left (\frac{i}{b}\right )^{3/2} x \left (\cosh \left (\frac{a}{2 b}\right )-i \sinh \left (\frac{a}{2 b}\right )\right ) S\left (\frac{\sqrt{\frac{i}{b}} \sqrt{a-i b \sin ^{-1}\left (i d x^2+1\right )}}{\sqrt{\pi }}\right )}{\cos \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac{1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\it Arcsinh} \left ( -i+d{x}^{2} \right ) \right ) ^{-{\frac{3}{2}}}\, 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}{{\left (b \operatorname{arsinh}\left (d x^{2} - i\right ) + a\right )}^{\frac{3}{2}}}\,{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}{{\left (b \operatorname{arsinh}\left (d x^{2} - i\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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