Optimal. Leaf size=129 \[ 24 a b^2 x-\frac {6 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3-\frac {48 b^3 \sqrt {d^2 x^4+2 i d x^2}}{d x}+24 i b^3 x \sin ^{-1}\left (1-i d x^2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4814, 4840, 12, 1588} \[ 24 a b^2 x-\frac {6 b \sqrt {d^2 x^4+2 i d x^2} \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3-\frac {48 b^3 \sqrt {d^2 x^4+2 i d x^2}}{d x}+24 i b^3 x \sin ^{-1}\left (1-i d x^2\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 1588
Rule 4814
Rule 4840
Rubi steps
\begin {align*} \int \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3 \, dx &=-\frac {6 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 )^2}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3+\left (24 b^2\right ) \int \left (a+i b \sin ^{-1}\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 \sin ^{-1}\left (1-i d x^2\right )\right )^2}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3+\left (24 i b^3\right ) \int \sin ^{-1}\left (1-i d x^2\right ) \, dx\\ &=24 a b^2 x+24 i b^3 x \sin ^{-1}\left (1-i d x^2\right )-\frac {6 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 )^2}{d x}+x \left (a+i b \sin ^{-1}\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 \sin ^{-1}\left (1-i d x^2\right )-\frac {6 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 )^2}{d x}+x \left (a+i b \sin ^{-1}\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 \sin ^{-1}\left (1-i d x^2\right )-\frac {6 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 )^2}{d x}+x \left (a+i b \sin ^{-1}\left (1-i d x^2\right )\right )^3\\ \end {align*}
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Mathematica [A] time = 0.15, size = 180, normalized size = 1.40 \[ \frac {a d x^2 \left (a^2+24 b^2\right )-6 b \left (a^2+8 b^2\right ) \sqrt {d x^2 \left (d x^2+2 i\right )}+3 i b \sin ^{-1}\left (1-i d x^2\right ) \left (a^2 d x^2-4 a b \sqrt {d x^2 \left (d x^2+2 i\right )}+8 b^2 d x^2\right )+3 b^2 \sin ^{-1}\left (1-i d x^2\right )^2 \left (-a d x^2+2 b \sqrt {d x^2 \left (d x^2+2 i\right )}\right )-i b^3 d x^2 \sin ^{-1}\left (1-i d x^2\right )^3}{d x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 188, normalized size = 1.46 \[ \frac {b^{3} d x \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{3} + {\left (a^{3} + 24 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x - 2 \, \sqrt {d^{2} x^{2} + 2 i \, d} b^{3}\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right )^{2} - 3 \, {\left (4 \, \sqrt {d^{2} x^{2} + 2 i \, d} a b^{2} - {\left (a^{2} b + 8 \, b^{3}\right )} d x\right )} \log \left (d x^{2} + \sqrt {d^{2} x^{2} + 2 i \, d} x + i\right ) - 6 \, \sqrt {d^{2} x^{2} + 2 i \, d} {\left (a^{2} b + 8 \, b^{3}\right )}}{d} \]
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 = 0.21, size = 0, normalized size = 0.00 \[ \int \left (a +b \arcsinh \left (d \,x^{2}+i\right )\right )^{3}\, dx \]
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 \[ b^{3} x \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{3} + 3 \, {\left (x \operatorname {arsinh}\left (d x^{2} + i\right ) - \frac {2 \, {\left (d^{\frac {3}{2}} x^{2} + 2 i \, \sqrt {d}\right )}}{\sqrt {d x^{2} + 2 i} d}\right )} a^{2} b + a^{3} x + \int \frac {{\left (3 \, {\left (a b^{2} d^{2} - 2 \, b^{3} d^{2}\right )} x^{4} - 6 \, a b^{2} + {\left (9 i \, a b^{2} d - 12 i \, b^{3} d\right )} x^{2} + {\left (3 \, {\left (a b^{2} d^{\frac {3}{2}} - 2 \, b^{3} d^{\frac {3}{2}}\right )} x^{3} + {\left (6 i \, a b^{2} \sqrt {d} - 6 i \, b^{3} \sqrt {d}\right )} x\right )} \sqrt {d x^{2} + 2 i}\right )} \log \left (d x^{2} + \sqrt {d x^{2} + 2 i} \sqrt {d} x + i\right )^{2}}{d^{2} x^{4} + 3 i \, d x^{2} + {\left (d^{\frac {3}{2}} x^{3} + 2 i \, \sqrt {d} x\right )} \sqrt {d x^{2} + 2 i} - 2}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2+1{}\mathrm {i}\right )\right )}^3 \,d x \]
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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