Optimal. Leaf size=544 \[ \frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}} \]
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Rubi [A]
time = 0.63, antiderivative size = 544, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 9, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5959, 5879,
75, 5909, 5962, 5681, 2221, 2317, 2438} \begin {gather*} \frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {c^2 (-d)-e}}+1\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {c^2 (-d)-e}+c \sqrt {-d}}+1\right )}{2 e^{3/2}}+\frac {a x}{e}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e} \end {gather*}
Antiderivative was successfully verified.
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Rule 75
Rule 2221
Rule 2317
Rule 2438
Rule 5681
Rule 5879
Rule 5909
Rule 5959
Rule 5962
Rubi steps
\begin {align*} \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d+e x^2} \, dx &=\int \left (\frac {a+b \cosh ^{-1}(c x)}{e}-\frac {d \left (a+b \cosh ^{-1}(c x)\right )}{e \left (d+e x^2\right )}\right ) \, dx\\ &=\frac {\int \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{e}-\frac {d \int \frac {a+b \cosh ^{-1}(c x)}{d+e x^2} \, dx}{e}\\ &=\frac {a x}{e}+\frac {b \int \cosh ^{-1}(c x) \, dx}{e}-\frac {d \int \left (\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx}{e}\\ &=\frac {a x}{e}+\frac {b x \cosh ^{-1}(c x)}{e}-\frac {(b c) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{e}-\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 e}-\frac {\sqrt {-d} \int \frac {a+b \cosh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 e}\\ &=\frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}-\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {(a+b x) \sinh (x)}{c \sqrt {-d}+\sqrt {e} \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}\\ &=\frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}-\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}-\frac {\sqrt {-d} \text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d-e}+\sqrt {e} e^x} \, dx,x,\cosh ^{-1}(c x)\right )}{2 e}\\ &=\frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 e^{3/2}}\\ &=\frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^{3/2}}-\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^{3/2}}+\frac {\left (b \sqrt {-d}\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {e} x}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 e^{3/2}}\\ &=\frac {a x}{e}-\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c e}+\frac {b x \cosh ^{-1}(c x)}{e}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {\sqrt {-d} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}-\frac {b \sqrt {-d} \text {Li}_2\left (-\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}+\frac {b \sqrt {-d} \text {Li}_2\left (\frac {\sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{2 e^{3/2}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.56, size = 457, normalized size = 0.84 \begin {gather*} \frac {4 a c \sqrt {e} x-4 a c \sqrt {d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+i b \left (4 i \sqrt {e} \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)-c x \cosh ^{-1}(c x)\right )-c \sqrt {d} \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-2 \left (\log \left (1+\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )-2 \text {PolyLog}\left (2,\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )-2 \text {PolyLog}\left (2,-\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+c \sqrt {d} \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-2 \left (\log \left (1+\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )-2 \text {PolyLog}\left (2,\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )-2 \text {PolyLog}\left (2,\frac {i \sqrt {e} e^{\cosh ^{-1}(c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )}{4 c e^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 15.79, size = 301, normalized size = 0.55
method | result | size |
derivativedivides | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \,c^{3} \mathrm {arccosh}\left (c x \right ) x}{e}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{e}-\frac {b \,c^{4} d \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e}+\frac {b \,c^{4} d \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e}}{c^{3}}\) | \(301\) |
default | \(\frac {\frac {a \,c^{3} x}{e}-\frac {a \,c^{3} d \arctan \left (\frac {x e}{\sqrt {d e}}\right )}{e \sqrt {d e}}+\frac {b \,c^{3} \mathrm {arccosh}\left (c x \right ) x}{e}-\frac {b \,c^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{e}-\frac {b \,c^{4} d \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 e}+\frac {b \,c^{4} d \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d +2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\mathrm {arccosh}\left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c x -1}\, \sqrt {c x +1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right )}{2 e}}{c^{3}}\) | \(301\) |
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 {x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{d + e 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 {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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