Integrand size = 20, antiderivative size = 153 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=384 b^4 x-\frac {192 b^3 \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \]
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Time = 0.03 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4898, 8} \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=-\frac {192 b^3 \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4+384 b^4 x \]
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Rule 8
Rule 4898
Rubi steps \begin{align*} \text {integral}& = -\frac {8 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2 \, dx \\ & = -\frac {192 b^3 \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}+48 b^2 x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx \\ & = 384 b^4 x-\frac {192 b^3 \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}+48 b^2 x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.97 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=-\frac {8 b \sqrt {d x^2 \left (-2 i+d x^2\right )} \left (a-i b \arcsin \left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4+48 b^2 \left (8 b^2 x-\frac {4 b \sqrt {d x^2 \left (-2 i+d x^2\right )} \left (a-i b \arcsin \left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \arcsin \left (1+i d x^2\right )\right )^2\right ) \]
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\[\int {\left (a +b \,\operatorname {arcsinh}\left (d \,x^{2}-i\right )\right )}^{4}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (129) = 258\).
Time = 0.25 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.76 \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=\frac {b^{4} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{4} + 4 \, {\left (a b^{3} d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{3} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x - 6 \, {\left (4 \, \sqrt {d^{2} x^{2} - 2 i \, d} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x - 6 \, {\left (a^{2} b^{2} + 8 \, b^{4}\right )} \sqrt {d^{2} x^{2} - 2 i \, d}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) - 8 \, {\left (a^{3} b + 24 \, a b^{3}\right )} \sqrt {d^{2} x^{2} - 2 i \, d}}{d} \]
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Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=\int { {\left (b \operatorname {arsinh}\left (d x^{2} - i\right ) + a\right )}^{4} \,d x } \]
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Exception generated. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a-i b \arcsin \left (1+i d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^4 \,d x \]
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