Optimal. Leaf size=216 \[ -\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \text {ArcSin}\left (1-d x^2\right )\right )}-\frac {x \text {CosIntegral}\left (-\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )} \]
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Rubi [A]
time = 0.02, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4909}
\begin {gather*} -\frac {x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) \text {CosIntegral}\left (-\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right )}-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \text {ArcSin}\left (1-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-b \sin ^{-1}\left (1-d x^2\right )\right )^2} \, dx &=-\frac {\sqrt {2 d x^2-d^2 x^4}}{2 b d x \left (a-b \sin ^{-1}\left (1-d x^2\right )\right )}-\frac {x \text {Ci}\left (-\frac {a-b \sin ^{-1}\left (1-d x^2\right )}{2 b}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}-\frac {x \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a}{2 b}-\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )}{4 b^2 \left (\cos \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1-d x^2\right )\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 183, normalized size = 0.85 \begin {gather*} \frac {2 b \sqrt {d x^2 \left (2-d x^2\right )}+\left (a-b \text {ArcSin}\left (1-d x^2\right )\right ) \left (\cos \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1-d x^2\right )\right )\right ) \left (\text {CosIntegral}\left (\frac {1}{2} \left (-\frac {a}{b}+\text {ArcSin}\left (1-d x^2\right )\right )\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )+\left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right ) \text {Si}\left (\frac {a-b \text {ArcSin}\left (1-d x^2\right )}{2 b}\right )\right )}{4 b^2 d x \left (-a+b \text {ArcSin}\left (1-d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arcsin \left (d \,x^{2}-1\right )\right )^{2}}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (d x^{2} - 1 \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (d\,x^2-1\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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