Optimal. Leaf size=76 \[ -\frac {4 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2+8 b^2 x \]
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Rubi [A] time = 0.02, 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 = {4814, 8} \[ -\frac {4 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2+8 b^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4814
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 = 0.03, size = 76, normalized size = 1.00 \[ -\frac {4 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2+8 b^2 x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 114, normalized size = 1.50 \[ \frac {b^{2} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x - 4 \, \sqrt {d^{2} x^{2} + 2 i \, d} a b + 2 \, {\left (a b d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )}{d} \]
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 [F] time = 0.20, 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 \[ 2 \, {\left (x \operatorname {arsinh}\left (d x^{2} + i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2 i} d}\right )} a b + {\left (x \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{2} - \int \frac {{\left (4 \, d^{2} x^{4} + 8 i \, d x^{2} + {\left (4 \, d^{\frac {3}{2}} x^{3} + 4 i \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )}{d^{2} x^{4} + 3 i \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + 2 i \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2 i} - 2}\,{d x}\right )} b^{2} + a^{2} x \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^2 \,d x \]
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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