Optimal. Leaf size=63 \[ -8 b^2 x-\frac {4 b \sqrt {-2 d x^2-d^2 x^4} \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )}{d x}+x \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^2 \]
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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 = {4899, 8}
\begin {gather*} -\frac {4 b \sqrt {-d^2 x^4-2 d x^2} \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )}{d x}+x \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )^2-8 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 4899
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 = 0.04, size = 98, normalized size = 1.56 \begin {gather*} \left (a^2-8 b^2\right ) x-\frac {4 a b \sqrt {-d x^2 \left (2+d x^2\right )}}{d x}+\frac {2 b \left (a d x^2-2 b \sqrt {-d x^2 \left (2+d x^2\right )}\right ) \text {ArcCos}\left (1+d x^2\right )}{d x}+b^2 x \text {ArcCos}\left (1+d x^2\right )^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, 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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.99, size = 91, normalized size = 1.44 \begin {gather*} \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 {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 \left (a + b \operatorname {acos}{\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 = -0.02 \begin {gather*} \int {\left (a+b\,\mathrm {acos}\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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