Optimal. Leaf size=124 \[ \frac {2 c^2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}-\frac {i b c^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {i b c^2 \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {b c \sqrt {1-c^2 x^2}}{2 d x} \]
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Rubi [A] time = 0.19, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {4702, 4680, 4419, 4183, 2279, 2391, 264} \[ -\frac {i b c^2 \text {PolyLog}\left (2,-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {i b c^2 \text {PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {2 c^2 \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{d}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}+\frac {b c \sqrt {1-c^2 x^2}}{2 d x} \]
Antiderivative was successfully verified.
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Rule 264
Rule 2279
Rule 2391
Rule 4183
Rule 4419
Rule 4680
Rule 4702
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )} \, dx &=-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}+c^2 \int \frac {a+b \cos ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx-\frac {(b c) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx}{2 d}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}-\frac {c^2 \operatorname {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}-\frac {\left (2 c^2\right ) \operatorname {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}+\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}-\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{d}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}-\frac {\left (i b c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {\left (i b c^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 d}\\ &=\frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \cos ^{-1}(c x)}{2 d x^2}+\frac {2 c^2 \left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \cos ^{-1}(c x)}\right )}{d}-\frac {i b c^2 \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )}{2 d}+\frac {i b c^2 \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 150, normalized size = 1.21 \[ -\frac {a c^2 \log \left (1-c^2 x^2\right )-2 a c^2 \log (x)+\frac {a}{x^2}+b c^2 \left (-\frac {\sqrt {1-c^2 x^2}}{c x}+\frac {\cos ^{-1}(c x)}{c^2 x^2}+i \text {Li}_2\left (-e^{2 i \cos ^{-1}(c x)}\right )-i \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )+2 \cos ^{-1}(c x) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )-2 \cos ^{-1}(c x) \log \left (1+e^{2 i \cos ^{-1}(c x)}\right )\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \arccos \left (c x\right ) + a}{c^{2} d x^{5} - d x^{3}}, x\right ) \]
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 \[ \int -\frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.71, size = 301, normalized size = 2.43 \[ -\frac {c^{2} a \ln \left (c x +1\right )}{2 d}-\frac {a}{2 d \,x^{2}}+\frac {c^{2} a \ln \left (c x \right )}{d}-\frac {c^{2} a \ln \left (c x -1\right )}{2 d}+\frac {i c^{2} b}{2 d}+\frac {b c \sqrt {-c^{2} x^{2}+1}}{2 d x}-\frac {b \arccos \left (c x \right )}{2 d \,x^{2}}-\frac {c^{2} b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i c^{2} b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {c^{2} b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {i b \,c^{2} \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}-\frac {c^{2} b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i c^{2} b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d} \]
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 \[ -\frac {1}{2} \, {\left (\frac {c^{2} \log \left (c x + 1\right )}{d} + \frac {c^{2} \log \left (c x - 1\right )}{d} - \frac {2 \, c^{2} \log \relax (x)}{d} + \frac {1}{d x^{2}}\right )} a - b \int \frac {\arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right )}{c^{2} d x^{5} - d x^{3}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{2} x^{5} - x^{3}}\, dx + \int \frac {b \operatorname {acos}{\left (c x \right )}}{c^{2} x^{5} - x^{3}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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