Optimal. Leaf size=132 \[ \frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d^2}+\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}-\frac {i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c d^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4656, 4658, 4183, 2279, 2391, 261} \[ -\frac {i b \text {PolyLog}\left (2,-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {PolyLog}\left (2,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c d^2}+\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 261
Rule 2279
Rule 2391
Rule 4183
Rule 4656
Rule 4658
Rubi steps
\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 d^2}+\frac {\int \frac {a+b \cos ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {\operatorname {Subst}\left (\int (a+b x) \csc (x) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}+\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}-\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}\\ &=\frac {b}{2 c d^2 \sqrt {1-c^2 x^2}}+\frac {x \left (a+b \cos ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{i \cos ^{-1}(c x)}\right )}{c d^2}-\frac {i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}+\frac {i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{2 c d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.30, size = 220, normalized size = 1.67 \[ \frac {-\frac {2 a x}{c^2 x^2-1}-\frac {a \log (1-c x)}{c}+\frac {a \log (c x+1)}{c}+\frac {b \sqrt {1-c^2 x^2}}{c-c^2 x}+\frac {b \sqrt {1-c^2 x^2}}{c^2 x+c}+\frac {b \cos ^{-1}(c x)}{c-c^2 x}-\frac {b \cos ^{-1}(c x)}{c^2 x+c}-\frac {2 i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )}{c}+\frac {2 i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )}{c}-\frac {2 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )}{c}+\frac {2 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{c}}{4 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arccos \left (c x\right ) + a}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {b \arccos \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.13, size = 250, normalized size = 1.89 \[ -\frac {a}{4 c \,d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{4 c \,d^{2}}-\frac {a}{4 c \,d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{4 c \,d^{2}}-\frac {b \arccos \left (c x \right ) x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \sqrt {-c^{2} x^{2}+1}}{2 c \,d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c \,d^{2}}-\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c \,d^{2}}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{2 c \,d^{2}}+\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{2 c \,d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, a {\left (\frac {2 \, x}{c^{2} d^{2} x^{2} - d^{2}} - \frac {\log \left (c x + 1\right )}{c d^{2}} + \frac {\log \left (c x - 1\right )}{c d^{2}}\right )} - \frac {{\left ({\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - {\left (c^{3} d^{2} x^{2} - c d^{2}\right )} \int \frac {{\left (2 \, c x - {\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x}\right )} b}{4 \, {\left (c^{3} d^{2} x^{2} - c d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________