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 = 0.0115554, antiderivative size = 63, normalized size of antiderivative = 1., 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 4815
Rule 8
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*}
Mathematica [A] time = 0.0657708, 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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Maple [F] time = 0.117, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}+1 \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.50039, size = 205, normalized size = 3.25 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acos}{\left (d x^{2} + 1 \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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