Optimal. Leaf size=521 \[ \frac {\text {ArcCos}(a x) \log \left (1-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {ArcCos}(a x) \log \left (1+\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {ArcCos}(a x) \log \left (1-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {ArcCos}(a x) \log \left (1+\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {PolyLog}\left (2,-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {PolyLog}\left (2,\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \]
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Rubi [A]
time = 0.56, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {4758, 4826,
4618, 2221, 2317, 2438} \begin {gather*} \frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{\sqrt {-c} a+i \sqrt {c a^2+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {ArcCos}(a x) \log \left (1-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {ArcCos}(a x) \log \left (1+\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {ArcCos}(a x) \log \left (1-\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {ArcCos}(a x) \log \left (1+\frac {\sqrt {d} e^{i \text {ArcCos}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 4618
Rule 4758
Rule 4826
Rubi steps
\begin {align*} \int \frac {\cos ^{-1}(a x)}{c+d x^2} \, dx &=\int \left (\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c \left (\sqrt {-c}-\sqrt {d} x\right )}+\frac {\sqrt {-c} \cos ^{-1}(a x)}{2 c \left (\sqrt {-c}+\sqrt {d} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}-\sqrt {d} x} \, dx}{2 \sqrt {-c}}-\frac {\int \frac {\cos ^{-1}(a x)}{\sqrt {-c}+\sqrt {d} x} \, dx}{2 \sqrt {-c}}\\ &=\frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}-\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {x \sin (x)}{a \sqrt {-c}+\sqrt {d} \cos (x)} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}\\ &=\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}-i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}+\frac {\text {Subst}\left (\int \frac {e^{i x} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}+i \sqrt {d} e^{i x}} \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\text {Subst}\left (\int \log \left (1-\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\text {Subst}\left (\int \log \left (1+\frac {i \sqrt {d} e^{i x}}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right ) \, dx,x,\cos ^{-1}(a x)\right )}{2 \sqrt {-c} \sqrt {d}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}-\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Subst}\left (\int \frac {\log \left (1-\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Subst}\left (\int \frac {\log \left (1+\frac {i \sqrt {d} x}{i a \sqrt {-c}+\sqrt {a^2 c+d}}\right )}{x} \, dx,x,e^{i \cos ^{-1}(a x)}\right )}{2 \sqrt {-c} \sqrt {d}}\\ &=\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {\cos ^{-1}(a x) \log \left (1-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {\cos ^{-1}(a x) \log \left (1+\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}-i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}+\frac {i \text {Li}_2\left (-\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}-\frac {i \text {Li}_2\left (\frac {\sqrt {d} e^{i \cos ^{-1}(a x)}}{a \sqrt {-c}+i \sqrt {a^2 c+d}}\right )}{2 \sqrt {-c} \sqrt {d}}\\ \end {align*}
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Mathematica [A]
time = 0.77, size = 811, normalized size = 1.56 \begin {gather*} \frac {4 \text {ArcSin}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (a \sqrt {c}-i \sqrt {d}\right ) \tan \left (\frac {1}{2} \text {ArcCos}(a x)\right )}{\sqrt {a^2 c+d}}\right )-4 \text {ArcSin}\left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \text {ArcTan}\left (\frac {\left (a \sqrt {c}+i \sqrt {d}\right ) \tan \left (\frac {1}{2} \text {ArcCos}(a x)\right )}{\sqrt {a^2 c+d}}\right )+i \text {ArcCos}(a x) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )+2 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-i \text {ArcCos}(a x) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-2 i \text {ArcSin}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-i \text {ArcCos}(a x) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )+2 i \text {ArcSin}\left (\frac {\sqrt {1-\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )+i \text {ArcCos}(a x) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-2 i \text {ArcSin}\left (\frac {\sqrt {1+\frac {i a \sqrt {c}}{\sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1+\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-\text {PolyLog}\left (2,-\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )+\text {PolyLog}\left (2,\frac {i \left (-a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )+\text {PolyLog}\left (2,-\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )-\text {PolyLog}\left (2,\frac {i \left (a \sqrt {c}+\sqrt {a^2 c+d}\right ) e^{i \text {ArcCos}(a x)}}{\sqrt {d}}\right )}{2 \sqrt {c} \sqrt {d}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 11.92, size = 224, normalized size = 0.43
method | result | size |
derivativedivides | \(\frac {-\frac {i a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 a^{2} c +d}\right )}{2}+\frac {i a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 a^{2} c +d \right )}\right )}{2}}{a}\) | \(224\) |
default | \(\frac {-\frac {i a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {\textit {\_R1} \left (i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} d +2 a^{2} c +d}\right )}{2}+\frac {i a^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (d \,\textit {\_Z}^{4}+\left (4 a^{2} c +2 d \right ) \textit {\_Z}^{2}+d \right )}{\sum }\frac {i \arccos \left (a x \right ) \ln \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -a x -i \sqrt {-a^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} d +2 a^{2} c +d \right )}\right )}{2}}{a}\) | \(224\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acos}{\left (a x \right )}}{c + d x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {acos}\left (a\,x\right )}{d\,x^2+c} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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