Optimal. Leaf size=91 \[ -\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c \left (a+b \cos ^{-1}(c x)\right )} \]
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Rubi [A] time = 0.10, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4632, 3303, 3299, 3302} \[ -\frac {\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^2 c^2}+\frac {x \sqrt {1-c^2 x^2}}{b c \left (a+b \cos ^{-1}(c x)\right )} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 4632
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b \cos ^{-1}(c x)\right )^2} \, dx &=\frac {x \sqrt {1-c^2 x^2}}{b c \left (a+b \cos ^{-1}(c x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\cos ^{-1}(c x)\right )}{b c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{b c \left (a+b \cos ^{-1}(c x)\right )}-\frac {\cos \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 c^2}-\frac {\sin \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 c^2}\\ &=\frac {x \sqrt {1-c^2 x^2}}{b c \left (a+b \cos ^{-1}(c x)\right )}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^2 c^2}-\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \cos ^{-1}(c x)\right )}{b^2 c^2}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 80, normalized size = 0.88 \[ \frac {\frac {b c x \sqrt {1-c^2 x^2}}{a+b \cos ^{-1}(c x)}-\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (2 \left (\frac {a}{b}+\cos ^{-1}(c x)\right )\right )-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\cos ^{-1}(c x)\right )\right )}{b^2 c^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.18, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{2} \arccos \left (c x\right )^{2} + 2 \, a b \arccos \left (c x\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.22, size = 323, normalized size = 3.55 \[ -\frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, b \arccos \left (c x\right ) \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} - \frac {2 \, a \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b c x}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {b \arccos \left (c x\right ) \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} + \frac {a \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arccos \left (c x\right )\right )}{b^{3} c^{2} \arccos \left (c x\right ) + a b^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 78, normalized size = 0.86 \[ \frac {\frac {\sin \left (2 \arccos \left (c x \right )\right )}{2 \left (a +b \arccos \left (c x \right )\right ) b}-\frac {\Si \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\Ci \left (2 \arccos \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )}{b^{2}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x}{{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x}{\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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