Integrand size = 14, antiderivative size = 127 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=384 b^4 x+\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (-1+d x^2\right )\right )^2-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )^3}{d x}+x \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \]
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Time = 0.02 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4899, 8} \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\frac {192 b^3 \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (d x^2-1\right )\right )^2-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (d x^2-1\right )\right )^3}{d x}+x \left (a+b \arccos \left (d x^2-1\right )\right )^4+384 b^4 x \]
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Rule 8
Rule 4899
Rubi steps \begin{align*} \text {integral}& = -\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )^3}{d x}+x \left (a+b \arccos \left (-1+d x^2\right )\right )^4-\left (48 b^2\right ) \int \left (a+b \arccos \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 \arccos \left (-1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (-1+d x^2\right )\right )^2-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )^3}{d x}+x \left (a+b \arccos \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 \arccos \left (-1+d x^2\right )\right )}{d x}-48 b^2 x \left (a+b \arccos \left (-1+d x^2\right )\right )^2-\frac {8 b \sqrt {2 d x^2-d^2 x^4} \left (a+b \arccos \left (-1+d x^2\right )\right )^3}{d x}+x \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.96 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\frac {\left (a^4-48 a^2 b^2+384 b^4\right ) d x^2-8 a b \left (a^2-24 b^2\right ) \sqrt {d x^2 \left (2-d x^2\right )}+4 b \left (a^3 d x^2-24 a b^2 d x^2-6 a^2 b \sqrt {-d x^2 \left (-2+d x^2\right )}+48 b^3 \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )+6 b^2 \left (a^2 d x^2-8 b^2 d x^2-4 a b \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )^2+4 b^3 \left (a d x^2-2 b \sqrt {-d x^2 \left (-2+d x^2\right )}\right ) \arccos \left (-1+d x^2\right )^3+b^4 d x^2 \arccos \left (-1+d x^2\right )^4}{d x} \]
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\[\int {\left (a +b \arccos \left (d \,x^{2}-1\right )\right )}^{4}d x\]
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none
Time = 0.26 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.63 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\frac {b^{4} d x^{2} \arccos \left (d x^{2} - 1\right )^{4} + 4 \, a b^{3} d x^{2} \arccos \left (d x^{2} - 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arccos \left (d x^{2} - 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arccos \left (d x^{2} - 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} - 8 \, {\left (b^{4} \arccos \left (d x^{2} - 1\right )^{3} + 3 \, a b^{3} \arccos \left (d x^{2} - 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arccos \left (d x^{2} - 1\right )\right )} \sqrt {-d^{2} x^{4} + 2 \, d x^{2}}}{d x} \]
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\[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int \left (a + b \operatorname {acos}{\left (d x^{2} - 1 \right )}\right )^{4}\, dx \]
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\[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int { {\left (b \arccos \left (d x^{2} - 1\right ) + a\right )}^{4} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (123) = 246\).
Time = 1.04 (sec) , antiderivative size = 586, normalized size of antiderivative = 4.61 \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=4 \, {\left (x \arccos \left (d x^{2} - 1\right ) + \frac {2 \, \sqrt {2} \mathrm {sgn}\left (x\right )}{\sqrt {d}} - \frac {2 \, \sqrt {-d^{2} x^{2} + 2 \, d}}{d \mathrm {sgn}\left (x\right )}\right )} a^{3} b + 6 \, {\left (x \arccos \left (d x^{2} - 1\right )^{2} + \frac {4 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 2 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} - \frac {4 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arccos \left (d x^{2} - 1\right ) - \frac {2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} a^{2} b^{2} + 4 \, {\left (x \arccos \left (d x^{2} - 1\right )^{3} + \frac {6 \, {\left (\sqrt {2} \pi ^{2} \sqrt {d} - 8 \, \sqrt {2} \sqrt {d}\right )} \mathrm {sgn}\left (x\right )}{d} - \frac {6 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arccos \left (d x^{2} - 1\right )^{2} + \frac {4 \, {\left (\sqrt {d^{2} x^{2}} \arccos \left (\frac {d^{2} x^{2} - d}{d}\right ) + \frac {2 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {-d^{2} x^{2} + 2 \, d}\right )} d}{{\left | d \right |}} - \frac {2 \, \sqrt {2} d^{\frac {3}{2}}}{{\left | d \right |}}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} a b^{3} + {\left (x \arccos \left (d x^{2} - 1\right )^{4} + \frac {8 \, {\left (\sqrt {2} \pi ^{3} \sqrt {d} {\left | d \right |} - 24 \, \sqrt {2} \pi \sqrt {d} {\left | d \right |} + 48 \, \sqrt {2} d^{\frac {3}{2}}\right )} \mathrm {sgn}\left (x\right )}{d {\left | d \right |}} - \frac {8 \, {\left (\sqrt {-d^{2} x^{2} + 2 \, d} \arccos \left (d x^{2} - 1\right )^{3} + \frac {6 \, {\left (\sqrt {d^{2} x^{2}} \arccos \left (\frac {d^{2} x^{2} - d}{d}\right )^{2} - \frac {2 \, {\left (\pi \sqrt {-d^{2} x^{2} + 2 \, d} + 2 \, \sqrt {-d^{2} x^{2} + 2 \, d} \arcsin \left (-\frac {d^{2} x^{2} - d}{d}\right ) - \frac {4 \, {\left (\sqrt {2} \sqrt {d} - \sqrt {d^{2} x^{2}}\right )} d}{{\left | d \right |}} - \frac {2 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 2 \, \sqrt {2} d^{\frac {3}{2}}\right )}}{{\left | d \right |}}\right )} d}{{\left | d \right |}} - \frac {4 \, {\left (\sqrt {2} \pi \sqrt {d} {\left | d \right |} - 2 \, \sqrt {2} d^{\frac {3}{2}}\right )}}{d}\right )} d}{{\left | d \right |}}\right )}}{d \mathrm {sgn}\left (x\right )}\right )} b^{4} + a^{4} x \]
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Timed out. \[ \int \left (a+b \arccos \left (-1+d x^2\right )\right )^4 \, dx=\int {\left (a+b\,\mathrm {acos}\left (d\,x^2-1\right )\right )}^4 \,d x \]
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