Optimal. Leaf size=129 \[ 24 a b^2 x-\frac {48 b^3 \sqrt {-2 i d x^2+d^2 x^4}}{d x}-24 i b^3 x \text {ArcSin}\left (1+i d x^2\right )-\frac {6 b \sqrt {-2 i d x^2+d^2 x^4} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^3 \]
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Rubi [A]
time = 0.04, 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 = {4898, 4924, 12,
1602} \begin {gather*} -\frac {6 b \sqrt {d^2 x^4-2 i d x^2} \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^2}{d x}+x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )^3+24 a b^2 x-24 i b^3 x \text {ArcSin}\left (1+i d x^2\right )-\frac {48 b^3 \sqrt {d^2 x^4-2 i d x^2}}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 1602
Rule 4898
Rule 4924
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 = 0.10, size = 180, normalized size = 1.40 \begin {gather*} \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 ) \text {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 ) \text {ArcSin}\left (1+i d x^2\right )^2+i b^3 d x^2 \text {ArcSin}\left (1+i d x^2\right )^3}{d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.93, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 188, normalized size = 1.46 \begin {gather*} \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} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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