Optimal. Leaf size=127 \[ \frac{192 b^3 \sqrt{-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )}{d x}-48 b^2 x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^2-\frac{8 b \sqrt{-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^4+384 b^4 x \]
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Rubi [A] time = 0.0287647, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4815, 8} \[ \frac{192 b^3 \sqrt{-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )}{d x}-48 b^2 x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^2-\frac{8 b \sqrt{-d^2 x^4-2 d x^2} \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^3}{d x}+x \left (a+b \cos ^{-1}\left (d x^2+1\right )\right )^4+384 b^4 x \]
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 )^4 \, dx &=-\frac{8 b \sqrt{-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^3}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^4-\left (48 b^2\right ) \int \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2 \, dx\\ &=\frac{192 b^3 \sqrt{-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2-\frac{8 b \sqrt{-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^3}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^4+\left (384 b^4\right ) \int 1 \, dx\\ &=384 b^4 x+\frac{192 b^3 \sqrt{-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2-\frac{8 b \sqrt{-2 d x^2-d^2 x^4} \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^3}{d x}+x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^4\\ \end{align*}
Mathematica [A] time = 0.232636, size = 249, normalized size = 1.96 \[ \frac{d x^2 \left (-48 a^2 b^2+a^4+384 b^4\right )-8 a b \left (a^2-24 b^2\right ) \sqrt{-d x^2 \left (d x^2+2\right )}+6 b^2 \cos ^{-1}\left (d x^2+1\right )^2 \left (a^2 d x^2-4 a b \sqrt{-d x^2 \left (d x^2+2\right )}-8 b^2 d x^2\right )+4 b \cos ^{-1}\left (d x^2+1\right ) \left (-6 a^2 b \sqrt{-d x^2 \left (d x^2+2\right )}+a^3 d x^2-24 a b^2 d x^2+48 b^3 \sqrt{-d x^2 \left (d x^2+2\right )}\right )+4 b^3 \cos ^{-1}\left (d x^2+1\right )^3 \left (a d x^2-2 b \sqrt{-d x^2 \left (d x^2+2\right )}\right )+b^4 d x^2 \cos ^{-1}\left (d x^2+1\right )^4}{d x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.121, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arccos \left ( d{x}^{2}+1 \right ) \right ) ^{4}\, 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 = 2.60548, size = 474, normalized size = 3.73 \begin{align*} \frac{b^{4} d x^{2} \arccos \left (d x^{2} + 1\right )^{4} + 4 \, a b^{3} d x^{2} \arccos \left (d x^{2} + 1\right )^{3} + 6 \,{\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arccos \left (d x^{2} + 1\right )^{2} + 4 \,{\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arccos \left (d x^{2} + 1\right ) +{\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} - 8 \,{\left (b^{4} \arccos \left (d x^{2} + 1\right )^{3} + 3 \, a b^{3} \arccos \left (d x^{2} + 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \,{\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arccos \left (d x^{2} + 1\right )\right )} \sqrt{-d^{2} x^{4} - 2 \, d x^{2}}}{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 )^{4}\, 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 )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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