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 = 0.03, antiderivative size = 127, normalized size of antiderivative = 1.00, 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 8
Rule 4815
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*}
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Mathematica [A] time = 0.25, size = 249, normalized size = 1.96 \[ \frac {-8 a b \left (a^2-24 b^2\right ) \sqrt {-d x^2 \left (d x^2+2\right )}+6 b^2 \cos ^{-1}\left (d x^2+1\right )^2 \left (a^2 d x^2-4 a b \sqrt {-d x^2 \left (d x^2+2\right )}-8 b^2 d x^2\right )+d x^2 \left (a^4-48 a^2 b^2+384 b^4\right )+4 b \cos ^{-1}\left (d x^2+1\right ) \left (a^3 d x^2-6 a^2 b \sqrt {-d x^2 \left (d x^2+2\right )}-24 a b^2 d x^2+48 b^3 \sqrt {-d x^2 \left (d x^2+2\right )}\right )+4 b^3 \cos ^{-1}\left (d x^2+1\right )^3 \left (a d x^2-2 b \sqrt {-d x^2 \left (d x^2+2\right )}\right )+b^4 d x^2 \cos ^{-1}\left (d x^2+1\right )^4}{d x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 207, normalized size = 1.63 \[ \frac {b^{4} d x^{2} \arccos \left (d x^{2} + 1\right )^{4} + 4 \, a b^{3} d x^{2} \arccos \left (d x^{2} + 1\right )^{3} + 6 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} d x^{2} \arccos \left (d x^{2} + 1\right )^{2} + 4 \, {\left (a^{3} b - 24 \, a b^{3}\right )} d x^{2} \arccos \left (d x^{2} + 1\right ) + {\left (a^{4} - 48 \, a^{2} b^{2} + 384 \, b^{4}\right )} d x^{2} - 8 \, {\left (b^{4} \arccos \left (d x^{2} + 1\right )^{3} + 3 \, a b^{3} \arccos \left (d x^{2} + 1\right )^{2} + a^{3} b - 24 \, a b^{3} + 3 \, {\left (a^{2} b^{2} - 8 \, b^{4}\right )} \arccos \left (d x^{2} + 1\right )\right )} \sqrt {-d^{2} x^{4} - 2 \, d x^{2}}}{d x} \]
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 \[ \int {\left (b \arccos \left (d x^{2} + 1\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{4}\, dx \]
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 = -0.01 \[ \int {\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^4 \,d x \]
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 \[ \int \left (a + b \operatorname {acos}{\left (d x^{2} + 1 \right )}\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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