Optimal. Leaf size=50 \[ a x-\frac{2 b \sqrt{d^2 x^4+2 i d x^2}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]
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Rubi [A] time = 0.04019, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4840, 12, 1588} \[ a x-\frac{2 b \sqrt{d^2 x^4+2 i d x^2}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 4840
Rule 12
Rule 1588
Rubi steps
\begin{align*} \int \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right ) \, dx &=a x+(i b) \int \sin ^{-1}\left (1-i d x^2\right ) \, dx\\ &=a x+i b x \sin ^{-1}\left (1-i d x^2\right )-(i b) \int -\frac{2 i d x^2}{\sqrt{2 i d x^2+d^2 x^4}} \, dx\\ &=a x+i b x \sin ^{-1}\left (1-i d x^2\right )-(2 b d) \int \frac{x^2}{\sqrt{2 i d x^2+d^2 x^4}} \, dx\\ &=a x-\frac{2 b \sqrt{2 i d x^2+d^2 x^4}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0239312, size = 48, normalized size = 0.96 \[ a x-\frac{2 b \sqrt{d x^2 \left (d x^2+2 i\right )}}{d x}+i b x \sin ^{-1}\left (1-i d x^2\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.014, size = 47, normalized size = 0.9 \begin{align*} ax+b \left ( x{\it Arcsinh} \left ( i+d{x}^{2} \right ) -2\,{\frac{x \left ( d{x}^{2}+2\,i \right ) }{\sqrt{2\,id{x}^{2}+{d}^{2}{x}^{4}}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21383, size = 59, normalized size = 1.18 \begin{align*}{\left (x \operatorname{arsinh}\left (d x^{2} + i\right ) - \frac{2 \,{\left (d^{\frac{3}{2}} x^{2} + 2 i \, \sqrt{d}\right )}}{\sqrt{d x^{2} + 2 i} d}\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.67691, size = 138, normalized size = 2.76 \begin{align*} \frac{b d x^{2} \log \left (d x^{2} + \sqrt{d^{2} x^{4} + 2 i \, d x^{2}} + i\right ) + a d x^{2} - 2 \, \sqrt{d^{2} x^{4} + 2 i \, d x^{2}} b}{d x} \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 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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