Optimal. Leaf size=63 \[ -\frac {4 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^2-8 b^2 x \]
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Rubi [A] time = 0.01, antiderivative size = 63, 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 = {4815, 8} \[ -\frac {4 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^2-8 b^2 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 4815
Rubi steps
\begin {align*} \int \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2 \, dx &=-\frac {4 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2-\left (8 b^2\right ) \int 1 \, dx\\ &=-8 b^2 x-\frac {4 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2\\ \end {align*}
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Mathematica [A] time = 0.07, size = 98, normalized size = 1.56 \[ x \left (a^2-8 b^2\right )-\frac {4 a b \sqrt {-d x^2 \left (d x^2+2\right )}}{d x}+\frac {2 b \cos ^{-1}\left (d x^2+1\right ) \left (a d x^2-2 b \sqrt {-d x^2 \left (d x^2+2\right )}\right )}{d x}+b^2 x \cos ^{-1}\left (d x^2+1\right )^2 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 91, normalized size = 1.44 \[ \frac {b^{2} d x^{2} \arccos \left (d x^{2} + 1\right )^{2} + 2 \, a b d x^{2} \arccos \left (d x^{2} + 1\right ) + {\left (a^{2} - 8 \, b^{2}\right )} d x^{2} - 4 \, \sqrt {-d^{2} x^{4} - 2 \, d x^{2}} {\left (b^{2} \arccos \left (d x^{2} + 1\right ) + a b\right )}}{d x} \]
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 \[ \int {\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.28, size = 0, normalized size = 0.00 \[ \int \left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{2}\, dx \]
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 = -0.02 \[ \int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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