Optimal. Leaf size=153 \[ -\frac {192 b^3 \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^4+384 b^4 x \]
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Rubi [A] time = 0.03, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4814, 8} \[ -\frac {192 b^3 \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2-\frac {8 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^4+384 b^4 x \]
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 )^4 \, dx &=-\frac {8 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 )^3}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^4+\left (48 b^2\right ) \int \left (a-i b \sin ^{-1}\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 \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \sin ^{-1}\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 \sin ^{-1}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \sin ^{-1}\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 \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+48 b^2 x \left (a-i b \sin ^{-1}\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 \sin ^{-1}\left (1+i d x^2\right )\right )^3}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^4\\ \end {align*}
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Mathematica [A] time = 0.12, size = 149, normalized size = 0.97 \[ 48 b^2 \left (-\frac {4 b \sqrt {d x^2 \left (d x^2-2 i\right )} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{d x}+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2+8 b^2 x\right )+x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^4-\frac {8 b \sqrt {d x^2 \left (d x^2-2 i\right )} \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^3}{d x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 269, normalized size = 1.76 \[ \frac {b^{4} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{4} + 4 \, {\left (a b^{3} d x - 2 \, \sqrt {d^{2} x^{2} - 2 i \, d} b^{4}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{3} + {\left (a^{4} + 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x - 6 \, {\left (4 \, \sqrt {d^{2} x^{2} - 2 i \, d} a b^{3} - {\left (a^{2} b^{2} + 8 \, b^{4}\right )} d x\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right )^{2} + 4 \, {\left ({\left (a^{3} b + 24 \, a b^{3}\right )} d x - 6 \, {\left (a^{2} b^{2} + 8 \, b^{4}\right )} \sqrt {d^{2} x^{2} - 2 i \, d}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} - 2 i \, d} x - i\right ) - 8 \, {\left (a^{3} b + 24 \, a b^{3}\right )} \sqrt {d^{2} x^{2} - 2 i \, d}}{d} \]
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 = 0.19, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d \,x^{2}-i\right )\right )^{4}\, dx \]
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 \[ b^{4} x \log \left (d x^{2} + \sqrt {d x^{2} - 2 i} \sqrt {d} x - i\right )^{4} + 4 \, {\left (x \operatorname {arsinh}\left (d x^{2} - i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} - 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} - 2 i} d}\right )} a^{3} b + a^{4} x + \int \frac {{\left (4 \, {\left (a b^{3} d^{2} - 2 \, b^{4} d^{2}\right )} x^{4} - 8 \, a b^{3} + {\left (-12 i \, a b^{3} d + 16 i \, b^{4} d\right )} x^{2} + {\left (4 \, {\left (a b^{3} d^{\frac {3}{2}} - 2 \, b^{4} d^{\frac {3}{2}}\right )} x^{3} + {\left (-8 i \, a b^{3} \sqrt {d} + 8 i \, b^{4} \sqrt {d}\right )} x\right )} \sqrt {d x^{2} - 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2 i} \sqrt {d} x - i\right )^{3} + {\left (6 \, a^{2} b^{2} d^{2} x^{4} - 18 i \, a^{2} b^{2} d x^{2} - 12 \, a^{2} b^{2} + {\left (6 \, a^{2} b^{2} d^{\frac {3}{2}} x^{3} - 12 i \, a^{2} b^{2} \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} - 2 i} \sqrt {d} x - i\right )^{2}}{d^{2} x^{4} - 3 i \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} - 2 i \, \sqrt {d} x\right )} \sqrt {d x^{2} - 2 i} - 2}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^4 \,d x \]
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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