Optimal. Leaf size=180 \[ -\frac {3 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^5 d^2}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}} \]
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Rubi [A] time = 0.23, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {4704, 4716, 4658, 4183, 2279, 2391, 261, 266, 43} \[ \frac {3 i b \text {PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \text {PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}-\frac {3 \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^5 d^2}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4183
Rule 4658
Rule 4704
Rule 4716
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cos ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^3}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac {3 \int \frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{2 c^2 d}\\ &=\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {(3 b) \int \frac {x}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d^2}+\frac {b \operatorname {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{4 c d^2}-\frac {3 \int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 c^4 d}\\ &=-\frac {3 b \sqrt {1-c^2 x^2}}{2 c^5 d^2}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {3 \operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \left (1-c^2 x\right )^{3/2}}-\frac {1}{c^2 \sqrt {1-c^2 x}}\right ) \, dx,x,x^2\right )}{4 c d^2}\\ &=\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c^5 d^2}\\ &=\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}+\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {(3 i b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}\\ &=\frac {b}{2 c^5 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sqrt {1-c^2 x^2}}{c^5 d^2}+\frac {3 x \left (a+b \cos ^{-1}(c x)\right )}{2 c^4 d^2}+\frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {3 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c^5 d^2}+\frac {3 i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}-\frac {3 i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c^5 d^2}\\ \end {align*}
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Mathematica [A] time = 0.37, size = 294, normalized size = 1.63 \[ \frac {3 a \log (1-c x)}{4 c^5 d^2}-\frac {3 a \log (c x+1)}{4 c^5 d^2}+\frac {a x}{c^4 d^2}-\frac {a x}{2 c^4 d^2 \left (c^2 x^2-1\right )}+\frac {b \left (-\frac {3 i \left (4 \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )+\cos ^{-1}(c x) \left (\cos ^{-1}(c x)+4 i \log \left (1-e^{i \cos ^{-1}(c x)}\right )\right )\right )}{8 c^5}-\frac {3 \left (-\frac {2 i \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{c}-\frac {i \cos ^{-1}(c x)^2}{2 c}+\frac {2 \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{c}\right )}{4 c^4}+\frac {c x \cos ^{-1}(c x)-\sqrt {1-c^2 x^2}}{c^5}+\frac {\sqrt {1-c^2 x^2}-\cos ^{-1}(c x)}{4 c^4 \left (c^2 x+c\right )}+\frac {\sqrt {1-c^2 x^2}+\cos ^{-1}(c x)}{4 c^4 \left (c-c^2 x\right )}\right )}{d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{4} \arccos \left (c x\right ) + a x^{4}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.89, size = 296, normalized size = 1.64 \[ \frac {a x}{c^{4} d^{2}}-\frac {a}{4 c^{5} d^{2} \left (c x +1\right )}-\frac {3 a \ln \left (c x +1\right )}{4 c^{5} d^{2}}-\frac {a}{4 c^{5} d^{2} \left (c x -1\right )}+\frac {3 a \ln \left (c x -1\right )}{4 c^{5} d^{2}}-\frac {b \sqrt {-c^{2} x^{2}+1}}{c^{5} d^{2}}+\frac {b \arccos \left (c x \right ) x}{c^{4} d^{2}}-\frac {b \arccos \left (c x \right ) x}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 c^{5} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {3 b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}+\frac {3 i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}+\frac {3 b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}}-\frac {3 i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c^{5} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, a {\left (\frac {2 \, x}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac {4 \, x}{c^{4} d^{2}} + \frac {3 \, \log \left (c x + 1\right )}{c^{5} d^{2}} - \frac {3 \, \log \left (c x - 1\right )}{c^{5} d^{2}}\right )} + \frac {{\left ({\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - {\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )} \int \frac {{\left (4 \, c^{3} x^{3} - 6 \, c x - 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{8} d^{2} x^{4} - 2 \, c^{6} d^{2} x^{2} + c^{4} d^{2}}\,{d x}\right )} b}{4 \, {\left (c^{7} d^{2} x^{2} - c^{5} d^{2}\right )}} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a x^{4}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{4} \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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