Optimal. Leaf size=197 \[ \frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \text {ArcCos}(c x))^2}-\frac {x}{b^2 c^2 (a+b \text {ArcCos}(c x))}+\frac {3 x^3}{2 b^2 (a+b \text {ArcCos}(c x))}-\frac {\text {CosIntegral}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{8 b^3 c^3}-\frac {9 \text {CosIntegral}\left (\frac {3 (a+b \text {ArcCos}(c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcCos}(c x))}{b}\right )}{8 b^3 c^3} \]
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Rubi [A]
time = 0.38, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {4730, 4808,
4732, 4491, 3384, 3380, 3383, 4720} \begin {gather*} -\frac {\sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right )}{8 b^3 c^3}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcCos}(c x))}{b}\right )}{8 b^3 c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}(c x)}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcCos}(c x))}{b}\right )}{8 b^3 c^3}-\frac {x}{b^2 c^2 (a+b \text {ArcCos}(c x))}+\frac {3 x^3}{2 b^2 (a+b \text {ArcCos}(c x))}+\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c (a+b \text {ArcCos}(c x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4720
Rule 4730
Rule 4732
Rule 4808
Rubi steps
\begin {align*} \int \frac {x^2}{\left (a+b \cos ^{-1}(c x)\right )^3} \, dx &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {\int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{b c}+\frac {(3 c) \int \frac {x^3}{\sqrt {1-c^2 x^2} \left (a+b \cos ^{-1}(c x)\right )^2} \, dx}{2 b}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac {9 \int \frac {x^2}{a+b \cos ^{-1}(c x)} \, dx}{2 b^2}+\frac {\int \frac {1}{a+b \cos ^{-1}(c x)} \, dx}{b^2 c^2}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b^3 c^3}+\frac {9 \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{2 b^2 c^3}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {9 \text {Subst}\left (\int \left (\frac {\sin (x)}{4 (a+b x)}+\frac {\sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 b^2 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b^3 c^3}+\frac {\sin \left (\frac {a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \cos ^{-1}(c x)\right )}{b^3 c^3}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {\text {Ci}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right )}{b^3 c^3}+\frac {9 \text {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}+\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {\text {Ci}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right )}{b^3 c^3}+\frac {\left (9 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}+\frac {\left (9 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}-\frac {\left (9 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}-\frac {\left (9 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{8 b^2 c^3}\\ &=\frac {x^2 \sqrt {1-c^2 x^2}}{2 b c \left (a+b \cos ^{-1}(c x)\right )^2}-\frac {x}{b^2 c^2 \left (a+b \cos ^{-1}(c x)\right )}+\frac {3 x^3}{2 b^2 \left (a+b \cos ^{-1}(c x)\right )}-\frac {9 \text {Ci}\left (\frac {a}{b}+\cos ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{8 b^3 c^3}+\frac {\text {Ci}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b^3 c^3}-\frac {9 \text {Ci}\left (\frac {3 a}{b}+3 \cos ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\cos ^{-1}(c x)\right )}{8 b^3 c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \cos ^{-1}(c x)\right )}{8 b^3 c^3}-\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}(c x)}{b}\right )}{b^3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.33, size = 169, normalized size = 0.86 \begin {gather*} \frac {\frac {4 b^2 x^2 \sqrt {1-c^2 x^2}}{c (a+b \text {ArcCos}(c x))^2}-\frac {8 b x}{c^2 (a+b \text {ArcCos}(c x))}+\frac {12 b x^3}{a+b \text {ArcCos}(c x)}-\frac {\text {CosIntegral}\left (\frac {a}{b}+\text {ArcCos}(c x)\right ) \sin \left (\frac {a}{b}\right )}{c^3}-\frac {9 \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcCos}(c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )}{c^3}+\frac {\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcCos}(c x)\right )}{c^3}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcCos}(c x)\right )\right )}{c^3}}{8 b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 290, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \sinIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \cosineIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \sinIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {9 \cosineIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arccos \left (c x \right ) \cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +x b c}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\) | \(290\) |
default | \(\frac {\frac {\sin \left (3 \arccos \left (c x \right )\right )}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\frac {9 \arccos \left (c x \right ) \sinIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b}{8}-\frac {9 \arccos \left (c x \right ) \cosineIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b}{8}+\frac {9 \sinIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a}{8}-\frac {9 \cosineIntegral \left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a}{8}+\frac {3 \cos \left (3 \arccos \left (c x \right )\right ) b}{8}}{\left (a +b \arccos \left (c x \right )\right ) b^{3}}+\frac {\sqrt {-c^{2} x^{2}+1}}{8 \left (a +b \arccos \left (c x \right )\right )^{2} b}+\frac {\arccos \left (c x \right ) \sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -\arccos \left (c x \right ) \cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +\sinIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -\cosineIntegral \left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +x b c}{8 \left (a +b \arccos \left (c x \right )\right ) b^{3}}}{c^{3}}\) | \(290\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1479 vs.
\(2 (183) = 366\).
time = 0.51, size = 1479, normalized size = 7.51 \begin {gather*} \text {Too large to display} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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