Optimal. Leaf size=244 \[ -\frac {\sqrt {-2 i d x^2+d^2 x^4}}{2 b d x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}+\frac {x \text {CosIntegral}\left (\frac {i \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{2 b}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )\right )}-\frac {x \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \text {ArcSin}\left (1+i d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )\right )} \]
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Rubi [A]
time = 0.02, antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {4909}
\begin {gather*} \frac {x \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right ) \text {CosIntegral}\left (\frac {i \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )\right )}-\frac {x \left (\sinh \left (\frac {a}{2 b}\right )+i \cosh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \text {ArcSin}\left (i d x^2+1\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )\right )}-\frac {\sqrt {d^2 x^4-2 i d x^2}}{2 b d x \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 4909
Rubi steps
\begin {align*} \int \frac {1}{\left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )^2} \, dx &=-\frac {\sqrt {-2 i d x^2+d^2 x^4}}{2 b d x \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}+\frac {x \text {Ci}\left (\frac {i \left (a-i b \sin ^{-1}\left (1+i d x^2\right )\right )}{2 b}\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}-\frac {x \left (i \cosh \left (\frac {a}{2 b}\right )+\sinh \left (\frac {a}{2 b}\right )\right ) \text {Shi}\left (\frac {a-i b \sin ^{-1}\left (1+i d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+i d x^2\right )\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 1.02, size = 196, normalized size = 0.80 \begin {gather*} \frac {-\frac {2 b \sqrt {d x^2 \left (-2 i+d x^2\right )}}{d \left (a-i b \text {ArcSin}\left (1+i d x^2\right )\right )}+\frac {x^2 \left (\text {CosIntegral}\left (\frac {1}{2} \left (\frac {i a}{b}+\text {ArcSin}\left (1+i d x^2\right )\right )\right ) \left (\cosh \left (\frac {a}{2 b}\right )+i \sinh \left (\frac {a}{2 b}\right )\right )-\left (\cosh \left (\frac {a}{2 b}\right )-i \sinh \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {1}{2} \left (\frac {i a}{b}+\text {ArcSin}\left (1+i d x^2\right )\right )\right )\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+i d x^2\right )\right )}}{4 b^2 x} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.26, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arcsinh \left (d \,x^{2}-i\right )\right )^{2}}\, dx\]
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 [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (d\,x^2-\mathrm {i}\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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