Optimal. Leaf size=155 \[ -\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d^2}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i b \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {4704, 4676, 3717, 2190, 2279, 2391, 288, 216} \[ -\frac {i b \text {PolyLog}\left (2,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}+\frac {\log \left (1-e^{2 i \cos ^{-1}(c x)}\right ) \left (a+b \cos ^{-1}(c x)\right )}{c^4 d^2}+\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 288
Rule 2190
Rule 2279
Rule 2391
Rule 3717
Rule 4676
Rule 4704
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \cos ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx &=\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {x^2}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^2}-\frac {\int \frac {x \left (a+b \cos ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2 d}\\ &=\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}+\frac {\operatorname {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}-\frac {b \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d^2}\\ &=\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2}-\frac {(2 i) \operatorname {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}\\ &=\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}-\frac {b \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\cos ^{-1}(c x)\right )}{c^4 d^2}\\ &=\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}\\ &=\frac {b x}{2 c^3 d^2 \sqrt {1-c^2 x^2}}+\frac {x^2 \left (a+b \cos ^{-1}(c x)\right )}{2 c^2 d^2 \left (1-c^2 x^2\right )}-\frac {i \left (a+b \cos ^{-1}(c x)\right )^2}{2 b c^4 d^2}-\frac {b \sin ^{-1}(c x)}{2 c^4 d^2}+\frac {\left (a+b \cos ^{-1}(c x)\right ) \log \left (1-e^{2 i \cos ^{-1}(c x)}\right )}{c^4 d^2}-\frac {i b \text {Li}_2\left (e^{2 i \cos ^{-1}(c x)}\right )}{2 c^4 d^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.48, size = 203, normalized size = 1.31 \[ \frac {-\frac {2 a}{c^2 x^2-1}+2 a \log \left (1-c^2 x^2\right )+\frac {b \sqrt {1-c^2 x^2}}{1-c x}-\frac {b \sqrt {1-c^2 x^2}}{c x+1}-4 i b \text {Li}_2\left (-e^{i \cos ^{-1}(c x)}\right )-4 i b \text {Li}_2\left (e^{i \cos ^{-1}(c x)}\right )-2 i b \cos ^{-1}(c x)^2+\frac {b \cos ^{-1}(c x)}{1-c x}+\frac {b \cos ^{-1}(c x)}{c x+1}+4 b \cos ^{-1}(c x) \log \left (1-e^{i \cos ^{-1}(c x)}\right )+4 b \cos ^{-1}(c x) \log \left (1+e^{i \cos ^{-1}(c x)}\right )}{4 c^4 d^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x^{3} \arccos \left (c x\right ) + a x^{3}}{c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \arccos \left (c x\right ) + a\right )} x^{3}}{{\left (c^{2} d x^{2} - d\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.83, size = 312, normalized size = 2.01 \[ \frac {a}{4 c^{4} d^{2} \left (c x +1\right )}+\frac {a \ln \left (c x +1\right )}{2 c^{4} d^{2}}-\frac {a}{4 c^{4} d^{2} \left (c x -1\right )}+\frac {a \ln \left (c x -1\right )}{2 c^{4} d^{2}}-\frac {i b \arccos \left (c x \right )^{2}}{2 c^{4} d^{2}}-\frac {i b \,x^{2}}{2 c^{2} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b x \sqrt {-c^{2} x^{2}+1}}{2 c^{3} d^{2} \left (c^{2} x^{2}-1\right )}-\frac {b \arccos \left (c x \right )}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {i b}{2 c^{4} d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{c^{4} d^{2}}+\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{c^{4} d^{2}}-\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{c^{4} d^{2}}-\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{c^{4} d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{2} \, a {\left (\frac {1}{c^{6} d^{2} x^{2} - c^{4} d^{2}} - \frac {\log \left (c^{2} x^{2} - 1\right )}{c^{4} d^{2}}\right )} + \frac {{\left ({\left ({\left (c^{2} x^{2} - 1\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} \log \left (-c x + 1\right ) - 1\right )} \arctan \left (\sqrt {c x + 1} \sqrt {-c x + 1}, c x\right ) - {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )} \int \frac {{\left (c^{2} x^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (c x + 1\right ) + {\left (c^{2} x^{2} - 1\right )} e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )} \log \left (-c x + 1\right ) - e^{\left (\frac {1}{2} \, \log \left (c x + 1\right ) + \frac {1}{2} \, \log \left (-c x + 1\right )\right )}}{c^{9} d^{2} x^{6} - 2 \, c^{7} d^{2} x^{4} + c^{5} d^{2} x^{2} - {\left (c^{7} d^{2} x^{4} - 2 \, c^{5} d^{2} x^{2} + c^{3} d^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )}}\,{d x}\right )} b}{2 \, {\left (c^{6} d^{2} x^{2} - c^{4} d^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^3\,\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {a x^{3}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx + \int \frac {b x^{3} \operatorname {acos}{\left (c x \right )}}{c^{4} x^{4} - 2 c^{2} x^{2} + 1}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________