Optimal. Leaf size=157 \[ \frac {2 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{d}+\frac {7 b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]
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Rubi [A] time = 0.23, 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 = {5746, 103, 12, 92, 205, 5694, 4182, 2279, 2391} \[ \frac {b c^3 \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {b c^3 \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d x}+\frac {2 c^3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d}-\frac {a+b \cosh ^{-1}(c x)}{3 d x^3}+\frac {7 b c^3 \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{6 d}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{6 d x^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 92
Rule 103
Rule 205
Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rule 5746
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 \operatorname {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 ) \operatorname {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 ) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d}+\frac {\left (b c^4\right ) \operatorname {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 ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{d}-\frac {\left (b c^3\right ) \operatorname {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 = 0.36, size = 223, normalized size = 1.42 \[ \frac {-6 a c^3 \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 a c^3 \log \left (e^{\cosh ^{-1}(c x)}+1\right )-\frac {6 a c^2}{x}-\frac {2 a}{x^3}+6 b c^3 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )-6 b c^3 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )-6 b c^3 \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 b c^3 \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )-\frac {6 b c^2 \cosh ^{-1}(c x)}{x}+\frac {7 b c^3 \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 b \cosh ^{-1}(c x)}{x^3}+\frac {b c \sqrt {c x-1} \sqrt {c x+1}}{x^2}}{6 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{2} d x^{6} - d x^{4}}, x\right ) \]
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 \[ \int -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 225, normalized size = 1.43 \[ -\frac {a}{3 d \,x^{3}}-\frac {c^{2} a}{d x}-\frac {c^{3} a \ln \left (c x -1\right )}{2 d}+\frac {c^{3} a \ln \left (c x +1\right )}{2 d}-\frac {c^{2} b \,\mathrm {arccosh}\left (c x \right )}{d x}+\frac {b c \sqrt {c x -1}\, \sqrt {c x +1}}{6 d \,x^{2}}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{3 d \,x^{3}}+\frac {7 c^{3} b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{3 d}+\frac {c^{3} b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {c^{3} b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d}+\frac {c^{3} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d} \]
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 \[ \frac {1}{6} \, {\left (\frac {3 \, c^{3} \log \left (c x + 1\right )}{d} - \frac {3 \, c^{3} \log \left (c x - 1\right )}{d} - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{d x^{3}}\right )} a + \frac {1}{24} \, {\left (216 \, c^{5} \int \frac {x^{3} \log \left (c x - 1\right )}{12 \, {\left (c^{2} d x^{4} - d x^{2}\right )}}\,{d x} - 12 \, c^{4} {\left (\frac {\log \left (c x + 1\right )}{c d} - \frac {\log \left (c x - 1\right )}{c d}\right )} - 72 \, c^{4} \int \frac {x^{2} \log \left (c x - 1\right )}{12 \, {\left (c^{2} d x^{4} - d x^{2}\right )}}\,{d x} - 4 \, c^{2} {\left (\frac {c \log \left (c x + 1\right )}{d} - \frac {c \log \left (c x - 1\right )}{d} - \frac {2}{d x}\right )} - \frac {3 \, c^{3} x^{3} \log \left (c x + 1\right )^{2} + 6 \, c^{3} x^{3} \log \left (c x + 1\right ) \log \left (c x - 1\right ) - 4 \, {\left (3 \, c^{3} x^{3} \log \left (c x + 1\right ) - 3 \, c^{3} x^{3} \log \left (c x - 1\right ) - 6 \, c^{2} x^{2} - 2\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{d x^{3}} + 24 \, \int \frac {3 \, c^{4} x^{3} \log \left (c x + 1\right ) - 3 \, c^{4} x^{3} \log \left (c x - 1\right ) - 6 \, c^{3} x^{2} - 2 \, c}{6 \, {\left (c^{3} d x^{6} - c d x^{4} + {\left (c^{2} d x^{5} - d x^{3}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x}\right )} b \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^4\,\left (d-c^2\,d\,x^2\right )} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{2} x^{6} - x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{6} - x^{4}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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