Optimal. Leaf size=173 \[ \frac {\sqrt {-2 d x^2-d^2 x^4}}{4 b d x \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^2}+\frac {x}{8 b^2 \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}}-\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4913, 4901}
\begin {gather*} -\frac {x \cos \left (\frac {a}{2 b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (d x^2+1\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}}-\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (d x^2+1\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}}+\frac {x}{8 b^2 \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{4 b d x \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 4901
Rule 4913
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^3} \, dx &=\frac {\sqrt {-2 d x^2-d^2 x^4}}{4 b d x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2}+\frac {x}{8 b^2 \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}-\frac {\int \frac {1}{a+b \cos ^{-1}\left (1+d x^2\right )} \, dx}{8 b^2}\\ &=\frac {\sqrt {-2 d x^2-d^2 x^4}}{4 b d x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2}+\frac {x}{8 b^2 \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Ci}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}}-\frac {x \sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right )}{8 \sqrt {2} b^3 \sqrt {-d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 147, normalized size = 0.85 \begin {gather*} \frac {\frac {2 b^2 \sqrt {-d x^2 \left (2+d x^2\right )}}{d \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^2}+\frac {b x^2}{a+b \text {ArcCos}\left (1+d x^2\right )}+\frac {\sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right ) \left (\cos \left (\frac {a}{2 b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )+\sin \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )\right )}{d}}{8 b^3 x} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{3}}\, dx\]
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 \begin {gather*} \text {Exception raised: RuntimeError} \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 {acos}{\left (d x^{2} + 1 \right )}\right )^{3}}\, 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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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