Integrand size = 20, antiderivative size = 129 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=24 a b^2 x-\frac {48 b^3 \sqrt {2 i d x^2+d^2 x^4}}{d x}+24 i b^3 x \arcsin \left (1-i d x^2\right )-\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \]
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Time = 0.05 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4898, 4924, 12, 1602} \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=-\frac {6 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3+24 a b^2 x+24 i b^3 x \arcsin \left (1-i d x^2\right )-\frac {48 b^3 \sqrt {d^2 x^4+2 i d x^2}}{d x} \]
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Rule 12
Rule 1602
Rule 4898
Rule 4924
Rubi steps \begin{align*} \text {integral}& = -\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+i b \arcsin \left (1-i d x^2\right )\right ) \, dx \\ & = 24 a b^2 x-\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3+\left (24 i b^3\right ) \int \arcsin \left (1-i d x^2\right ) \, dx \\ & = 24 a b^2 x+24 i b^3 x \arcsin \left (1-i d x^2\right )-\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3-\left (24 i b^3\right ) \int -\frac {2 i d x^2}{\sqrt {2 i d x^2+d^2 x^4}} \, dx \\ & = 24 a b^2 x+24 i b^3 x \arcsin \left (1-i d x^2\right )-\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3-\left (48 b^3 d\right ) \int \frac {x^2}{\sqrt {2 i d x^2+d^2 x^4}} \, dx \\ & = 24 a b^2 x-\frac {48 b^3 \sqrt {2 i d x^2+d^2 x^4}}{d x}+24 i b^3 x \arcsin \left (1-i d x^2\right )-\frac {6 b \sqrt {2 i d x^2+d^2 x^4} \left (a+i b \arcsin \left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.40 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\frac {a \left (a^2+24 b^2\right ) d x^2-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2 \left (2 i+d x^2\right )}+3 i b \left (a^2 d x^2+8 b^2 d x^2-4 a b \sqrt {d x^2 \left (2 i+d x^2\right )}\right ) \arcsin \left (1-i d x^2\right )+3 b^2 \left (-a d x^2+2 b \sqrt {d x^2 \left (2 i+d x^2\right )}\right ) \arcsin \left (1-i d x^2\right )^2-i b^3 d x^2 \arcsin \left (1-i d x^2\right )^3}{d x} \]
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\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}+i\right )\right )}^{3}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.46 \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\frac {b^{3} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{2} - 3 \, {\left (4 \, \sqrt {d^{2} x^{2} + 2 i \, d} a b^{2} - {\left (a^{2} b + 8 \, b^{3}\right )} d x\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right ) - 6 \, \sqrt {d^{2} x^{2} + 2 i \, d} {\left (a^{2} b + 8 \, b^{3}\right )}}{d} \]
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Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} + i\right ) + a\right )}^{3} \,d x } \]
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Exception generated. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+i b \arcsin \left (1-i d x^2\right )\right )^3 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^3 \,d x \]
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