Integrand size = 10, antiderivative size = 111 \[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {x}{2 b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{2 b^3 c}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b^3 c} \]
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Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4718, 4808, 4720, 3384, 3380, 3383} \[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b^3 c}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b^3 c}+\frac {x}{2 b^2 (a+b \arccos (c x))}+\frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 4718
Rule 4720
Rule 4808
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {c \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arccos (c x))^2} \, dx}{2 b} \\ & = \frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {x}{2 b^2 (a+b \arccos (c x))}-\frac {\int \frac {1}{a+b \arccos (c x)} \, dx}{2 b^2} \\ & = \frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {x}{2 b^2 (a+b \arccos (c x))}-\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{2 b^3 c} \\ & = \frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {x}{2 b^2 (a+b \arccos (c x))}+\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{2 b^3 c}-\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arccos (c x)\right )}{2 b^3 c} \\ & = \frac {\sqrt {1-c^2 x^2}}{2 b c (a+b \arccos (c x))^2}+\frac {x}{2 b^2 (a+b \arccos (c x))}-\frac {\operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{2 b^3 c}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{2 b^3 c} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.80 \[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\frac {\frac {b \left (a c x+b \sqrt {1-c^2 x^2}+b c x \arccos (c x)\right )}{(a+b \arccos (c x))^2}-\operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )+\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )}{2 b^3 c} \]
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Time = 0.55 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{2 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b -\arccos \left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a +x b c}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c}\) | \(139\) |
default | \(\frac {\frac {\sqrt {-c^{2} x^{2}+1}}{2 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b -\arccos \left (c x \right ) \sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) b +\cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a -\sin \left (\frac {a}{b}\right ) \operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) a +x b c}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c}\) | \(139\) |
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\[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\int \frac {1}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \]
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\[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\int { \frac {1}{{\left (b \arccos \left (c x\right ) + a\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 481 vs. \(2 (101) = 202\).
Time = 0.30 (sec) , antiderivative size = 481, normalized size of antiderivative = 4.33 \[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=-\frac {b^{2} \arccos \left (c x\right )^{2} \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {b^{2} \arccos \left (c x\right )^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {b^{2} c x \arccos \left (c x\right )}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a b \arccos \left (c x\right ) \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c} + \frac {a b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c} + \frac {a b c x}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} - \frac {a^{2} \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {a^{2} \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} + \frac {\sqrt {-c^{2} x^{2} + 1} b^{2}}{2 \, {\left (b^{5} c \arccos \left (c x\right )^{2} + 2 \, a b^{4} c \arccos \left (c x\right ) + a^{2} b^{3} c\right )}} \]
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Timed out. \[ \int \frac {1}{(a+b \arccos (c x))^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \]
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