Optimal. Leaf size=157 \[ \frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac {7 b c^3 \text {ArcTan}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d} \]
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Rubi [A]
time = 0.17, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5932, 5903,
4267, 2317, 2438, 94, 211, 105, 12} \begin {gather*} \frac {2 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {7 b c^3 \text {ArcTan}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}+\frac {b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 105
Rule 211
Rule 2317
Rule 2438
Rule 4267
Rule 5903
Rule 5932
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^4 \left (d-c^2 d x^2\right )} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+c^2 \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {1}{x^3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 d}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+c^4 \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx+\frac {(b c) \int \frac {c^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}-\frac {c^3 \text {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{6 d}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac {b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d}+\frac {2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\left (b c^4\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac {7 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {\left (b c^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{6 d x^2}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac {7 b c^3 \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{6 d}+\frac {2 c^3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 223, normalized size = 1.42 \begin {gather*} \frac {-\frac {2 a}{x^3}-\frac {6 a c^2}{x}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{x^2}-\frac {2 b \cosh ^{-1}(c x)}{x^3}-\frac {6 b c^2 \cosh ^{-1}(c x)}{x}+\frac {7 b c^3 \sqrt {-1+c^2 x^2} \text {ArcTan}\left (\sqrt {-1+c^2 x^2}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-6 a c^3 \log \left (1-e^{\cosh ^{-1}(c x)}\right )-6 b c^3 \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 a c^3 \log \left (1+e^{\cosh ^{-1}(c x)}\right )+6 b c^3 \cosh ^{-1}(c x) \log \left (1+e^{\cosh ^{-1}(c x)}\right )+6 b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )-6 b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{6 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 5.82, size = 219, normalized size = 1.39
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a}{3 d \,c^{3} x^{3}}-\frac {a}{d c x}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{6 d \,c^{2} x^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d c x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d \,c^{3} x^{3}}+\frac {7 b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d}+\frac {b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}\right )\) | \(219\) |
default | \(c^{3} \left (-\frac {a \ln \left (c x -1\right )}{2 d}-\frac {a}{3 d \,c^{3} x^{3}}-\frac {a}{d c x}+\frac {a \ln \left (c x +1\right )}{2 d}+\frac {b \sqrt {c x -1}\, \sqrt {c x +1}}{6 d \,c^{2} x^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d c x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d \,c^{3} x^{3}}+\frac {7 b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d}+\frac {b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}\right )\) | \(219\) |
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*} - \frac {\int \frac {a}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx}{d} \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 = -0.01 \begin {gather*} \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\left (d-c^2\,d\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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