Optimal. Leaf size=76 \[ 8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4898, 8}
\begin {gather*} -\frac {4 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2+8 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4898
Rubi steps
\begin {align*} \int \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2 \, dx &=-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+\left (8 b^2\right ) \int 1 \, dx\\ &=8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 76, normalized size = 1.00 \begin {gather*} 8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.91, size = 0, normalized size = 0.00 \[\int \left (a +b \arcsinh \left (d \,x^{2}-i\right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 114, normalized size = 1.50 \begin {gather*} \frac {b^{2} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x - 4 \, \sqrt {d^{2} x^{2} - 2 i \, d} a b + 2 \, {\left (a b d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )}{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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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