Integrand size = 20, antiderivative size = 76 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \]
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Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4898, 8} \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=-\frac {4 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2+8 b^2 x \]
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Rule 8
Rule 4898
Rubi steps \begin{align*} \text {integral}& = -\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \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 \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=8 b^2 x-\frac {4 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \]
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\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}-i\right )\right )}^{2}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.50 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=\frac {b^{2} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{2} + {\left (a^{2} + 8 \, b^{2}\right )} d x - 4 \, \sqrt {d^{2} x^{2} - 2 i \, d} a b + 2 \, {\left (a b d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b^{2}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )}{d} \]
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Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} - i\right ) + a\right )}^{2} \,d x } \]
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Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^2 \,d x \]
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