Optimal. Leaf size=50 \[ a x-\frac {2 b \sqrt {2 i d x^2+d^2 x^4}}{d x}+i b x \text {ArcSin}\left (1-i d x^2\right ) \]
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Rubi [A]
time = 0.03, 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 = {4924, 12, 1602}
\begin {gather*} a x+i b x \text {ArcSin}\left (1-i d x^2\right )-\frac {2 b \sqrt {d^2 x^4+2 i d x^2}}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1602
Rule 4924
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 = 0.02, size = 48, normalized size = 0.96 \begin {gather*} a x-\frac {2 b \sqrt {d x^2 \left (2 i+d x^2\right )}}{d x}+i b x \text {ArcSin}\left (1-i d x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 47, normalized size = 0.94
method | result | size |
default | \(a x +b \left (x \arcsinh \left (d \,x^{2}+i\right )-\frac {2 x \left (d \,x^{2}+2 i\right )}{\sqrt {d^{2} x^{4}+2 i d \,x^{2}}}\right )\) | \(47\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 44, normalized size = 0.88 \begin {gather*} {\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 52, normalized size = 1.04 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.53, size = 39, normalized size = 0.78 \begin {gather*} a\,x+b\,x\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )-\frac {2\,b\,\sqrt {{\left (d\,x^2+1{}\mathrm {i}\right )}^2+1}}{d\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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