Optimal. Leaf size=124 \[ \frac {b c \sqrt {1-c^2 x^2}}{2 d x}-\frac {a+b \text {ArcCos}(c x)}{2 d x^2}+\frac {2 c^2 (a+b \text {ArcCos}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcCos}(c x)}\right )}{d}-\frac {i b c^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )}{2 d}+\frac {i b c^2 \text {PolyLog}\left (2,e^{2 i \text {ArcCos}(c x)}\right )}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, 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 = {4790, 4770,
4504, 4268, 2317, 2438, 270} \begin {gather*} \frac {2 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcCos}(c x)}\right ) (a+b \text {ArcCos}(c x))}{d}-\frac {a+b \text {ArcCos}(c x)}{2 d x^2}-\frac {i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcCos}(c x)}\right )}{2 d}+\frac {i b c^2 \text {Li}_2\left (e^{2 i \text {ArcCos}(c x)}\right )}{2 d}+\frac {b c \sqrt {1-c^2 x^2}}{2 d x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2317
Rule 2438
Rule 4268
Rule 4504
Rule 4770
Rule 4790
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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 150, normalized size = 1.21 \begin {gather*} -\frac {\frac {a}{x^2}-2 a c^2 \log (x)+a c^2 \log \left (1-c^2 x^2\right )+b c^2 \left (-\frac {\sqrt {1-c^2 x^2}}{c x}+\frac {\text {ArcCos}(c x)}{c^2 x^2}+2 \text {ArcCos}(c x) \log \left (1-e^{2 i \text {ArcCos}(c x)}\right )-2 \text {ArcCos}(c x) \log \left (1+e^{2 i \text {ArcCos}(c x)}\right )+i \text {PolyLog}\left (2,-e^{2 i \text {ArcCos}(c x)}\right )-i \text {PolyLog}\left (2,e^{2 i \text {ArcCos}(c x)}\right )\right )}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.96, size = 283, normalized size = 2.28
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a \ln \left (c x +1\right )}{2 d}-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \ln \left (c x \right )}{d}+\frac {i b}{2 d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d c x}-\frac {b \arccos \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}\right )\) | \(283\) |
default | \(c^{2} \left (-\frac {a \ln \left (c x +1\right )}{2 d}-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a}{2 d \,c^{2} x^{2}}+\frac {a \ln \left (c x \right )}{d}+\frac {i b}{2 d}+\frac {b \sqrt {-c^{2} x^{2}+1}}{2 d c x}-\frac {b \arccos \left (c x \right )}{2 d \,c^{2} x^{2}}-\frac {b \arccos \left (c x \right ) \ln \left (1-c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \polylog \left (2, c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}-\frac {b \arccos \left (c x \right ) \ln \left (1+c x +i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {i b \polylog \left (2, -c x -i \sqrt {-c^{2} x^{2}+1}\right )}{d}+\frac {b \arccos \left (c x \right ) \ln \left (1+\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d}-\frac {i b \polylog \left (2, -\left (c x +i \sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2 d}\right )\) | \(283\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{x^3\,\left (d-c^2\,d\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________