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 = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 12
Rule 1588
Rule 4840
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.48, size = 52, normalized size = 1.04 \[ \frac {b d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) + a d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.02, size = 48, normalized size = 0.96 \[ a x +b \left (x \arcsinh \left (d \,x^{2}-i\right )+\frac {2 x \left (-d \,x^{2}+2 i\right )}{\sqrt {d^{2} x^{4}-2 i d \,x^{2}}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.70, size = 44, normalized size = 0.88 \[ {\left (x \operatorname {arsinh}\left (d x^{2} - i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2 i} d}\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.45, size = 39, normalized size = 0.78 \[ a\,x+b\,x\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )-\frac {2\,b\,\sqrt {{\left (d\,x^2-\mathrm {i}\right )}^2+1}}{d\,x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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