Optimal. Leaf size=68 \[ -\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-1}(a+b x)}\right )}{2 d}-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {\cos ^{-1}(a+b x) \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {4806, 12, 4626, 3719, 2190, 2279, 2391} \[ -\frac {i \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}(a+b x)}\right )}{2 d}-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {\cos ^{-1}(a+b x) \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2190
Rule 2279
Rule 2391
Rule 3719
Rule 4626
Rule 4806
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a+b x)}{\frac {a d}{b}+d x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b \cos ^{-1}(x)}{d x} \, dx,x,a+b x\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\cos ^{-1}(x)}{x} \, dx,x,a+b x\right )}{d}\\ &=-\frac {\operatorname {Subst}\left (\int x \tan (x) \, dx,x,\cos ^{-1}(a+b x)\right )}{d}\\ &=-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\cos ^{-1}(a+b x)\right )}{d}\\ &=-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {\cos ^{-1}(a+b x) \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )}{d}-\frac {\operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(a+b x)\right )}{d}\\ &=-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {\cos ^{-1}(a+b x) \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )}{d}+\frac {i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(a+b x)}\right )}{2 d}\\ &=-\frac {i \cos ^{-1}(a+b x)^2}{2 d}+\frac {\cos ^{-1}(a+b x) \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )}{d}-\frac {i \text {Li}_2\left (-e^{2 i \cos ^{-1}(a+b x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 59, normalized size = 0.87 \[ -\frac {i \left (\text {Li}_2\left (-e^{2 i \cos ^{-1}(a+b x)}\right )+\cos ^{-1}(a+b x) \left (\cos ^{-1}(a+b x)+2 i \log \left (1+e^{2 i \cos ^{-1}(a+b x)}\right )\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arccos \left (b x + a\right )}{b d x + a d}, x\right ) \]
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 \[ \int \frac {\arccos \left (b x + a\right )}{d x + \frac {a d}{b}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 85, normalized size = 1.25 \[ -\frac {i \arccos \left (b x +a \right )^{2}}{2 d}+\frac {\arccos \left (b x +a \right ) \ln \left (1+\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{d}-\frac {i \polylog \left (2, -\left (b x +a +i \sqrt {1-\left (b x +a \right )^{2}}\right )^{2}\right )}{2 d} \]
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 \[ \int \frac {\arccos \left (b x + a\right )}{d x + \frac {a d}{b}}\,{d x} \]
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 \[ \int \frac {\mathrm {acos}\left (a+b\,x\right )}{d\,x+\frac {a\,d}{b}} \,d x \]
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 \[ \frac {b \int \frac {\operatorname {acos}{\left (a + b x \right )}}{a + b x}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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