Optimal. Leaf size=130 \[ -\frac {\sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^3 c^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]
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Rubi [A] time = 0.31, antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {4634, 4720, 4636, 4406, 12, 3303, 3299, 3302, 4642} \[ -\frac {\sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 3299
Rule 3302
Rule 3303
Rule 4406
Rule 4634
Rule 4636
Rule 4642
Rule 4720
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \cos ^{-1}(c x)\right )^3} \, dx &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {\int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{2 b c}+\frac {c \int \frac {x^2}{\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{b}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac {2 \int \frac {x}{a+b \cos ^{-1}(c x)} \, dx}{b^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {2 \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 (a+b x)} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {\cos \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b^2 c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {1}{2 b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {x^2}{b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac {\text {Ci}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{b^3 c^2}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^3 c^2}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 107, normalized size = 0.82 \[ \frac {\frac {b^2 c x \sqrt {1-c^2 x^2}}{\left (a+b \cos ^{-1}(c x)\right )^2}+\frac {b \left (2 c^2 x^2-1\right )}{a+b \cos ^{-1}(c x)}-2 \sin \left (\frac {2 a}{b}\right ) \text {Ci}\left (2 \left (\frac {a}{b}+\cos ^{-1}(c x)\right )\right )+2 \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\cos ^{-1}(c x)\right )\right )}{2 b^3 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{3} \arccos \left (c x\right )^{3} + 3 \, a b^{2} \arccos \left (c x\right )^{2} + 3 \, a^{2} b \arccos \left (c x\right ) + a^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.77, size = 860, normalized size = 6.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 157, normalized size = 1.21 \[ \frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{4 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {2 \arccos \left (c x \right ) \Si \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b -2 \arccos \left (c x \right ) \Ci \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +2 \Si \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a -2 \Ci \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +\cos \left (2 \arccos \left (c x \right )\right ) b}{2 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{2}} \]
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 \[ \frac {2 \, a c^{2} x^{2} + \sqrt {c x + 1} \sqrt {-c x + 1} b c x + {\left (2 \, b c^{2} x^{2} - b\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - a - \frac {4 \, {\left (b^{4} c^{2} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a^{2} b^{2} c^{2}\right )} \int \frac {x}{b \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a}\,{d x}}{b^{2}}}{2 \, {\left (b^{4} c^{2} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )^{2} + 2 \, a b^{3} c^{2} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) + a^{2} b^{2} c^{2}\right )}} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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