Optimal. Leaf size=151 \[ \frac {x \sin \left (\frac {a}{2 b}\right ) \text {Ci}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{2 b d x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4826} \[ \frac {x \sin \left (\frac {a}{2 b}\right ) \text {CosIntegral}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{2 b d x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )} \]
Antiderivative was successfully verified.
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Rule 4826
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-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 \cos ^{-1}\left (1+d x^2\right )\right )}+\frac {x \text {Ci}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 133, normalized size = 0.88 \[ \frac {\sqrt {-d x^2 \left (d x^2+2\right )} \left (\frac {b}{a+b \cos ^{-1}\left (d x^2+1\right )}-\frac {\cos \left (\frac {1}{2} \cos ^{-1}\left (d x^2+1\right )\right ) \left (\sin \left (\frac {a}{2 b}\right ) \text {Ci}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )-\cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (d x^2+1\right )}{2 b}\right )\right )}{d x^2+2}\right )}{2 b^2 d x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{b^{2} \arccos \left (d x^{2} + 1\right )^{2} + 2 \, a b \arccos \left (d x^{2} + 1\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
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 \[ \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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