Optimal. Leaf size=485 \[ \frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}} \]
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Rubi [A]
time = 0.59, antiderivative size = 485, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5793, 5827,
5680, 2221, 2317, 2438} \begin {gather*} \frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {e-c^2 d}+c \sqrt {-d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{\sqrt {-d} c+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 5680
Rule 5793
Rule 5827
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{d+e x^2} \, dx &=\int \left (\frac {\sqrt {-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \left (a+b \sinh ^{-1}(c x)\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx\\ &=-\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}}\\ &=-\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}-\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {(a+b x) \cosh (x)}{c \sqrt {-d}+\sqrt {e} \sinh (x)} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}\\ &=-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}-\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}-\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}-\frac {\text {Subst}\left (\int \frac {e^x (a+b x)}{c \sqrt {-d}+\sqrt {-c^2 d+e}+\sqrt {e} e^x} \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d}}\\ &=\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Subst}\left (\int \log \left (1-\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Subst}\left (\int \log \left (1+\frac {\sqrt {e} e^x}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \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^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \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^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \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^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \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^{\sinh ^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 434, normalized size = 0.89 \begin {gather*} \frac {2 a \sqrt {-d} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )-b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \sinh ^{-1}(c x) \log \left (1-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \sinh ^{-1}(c x) \log \left (1+\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{-c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )-b \sqrt {d} \text {PolyLog}\left (2,-\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )+b \sqrt {d} \text {PolyLog}\left (2,\frac {\sqrt {e} e^{\sinh ^{-1}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d^2} \sqrt {e}} \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 = 11.79, size = 233, normalized size = 0.48
method | result | size |
derivativedivides | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}}{c}\) | \(233\) |
default | \(\frac {\frac {a c \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e +2 c^{2} d -e \right )}\right )}{2}+\frac {b \,c^{2} \left (\munderset {\textit {\_R1} =\RootOf \left (e \,\textit {\_Z}^{4}+\left (4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\textit {\_R1} \left (\arcsinh \left (c x \right ) \ln \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -c x -\sqrt {c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e +2 c^{2} d -e}\right )}{2}}{c}\) | \(233\) |
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 {a + b \operatorname {asinh}{\left (c x \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 {a+b\,\mathrm {asinh}\left (c\,x\right )}{e\,x^2+d} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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