Optimal. Leaf size=158 \[ \frac {2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^5 d}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d}+\frac {11 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^5 d}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 d} \]
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Rubi [A] time = 0.23, antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5766, 100, 12, 74, 5694, 4182, 2279, 2391} \[ \frac {b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{c^5 d}-\frac {b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{c^5 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}+\frac {2 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{c^5 d}+\frac {b x^2 \sqrt {c x-1} \sqrt {c x+1}}{9 c^3 d}+\frac {11 b \sqrt {c x-1} \sqrt {c x+1}}{9 c^5 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 74
Rule 100
Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rule 5766
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx &=-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {\int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{d-c^2 d x^2} \, dx}{c^2}+\frac {b \int \frac {x^3}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 c d}\\ &=\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{c^4}+\frac {b \int \frac {2 x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^3 d}+\frac {b \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{c^3 d}\\ &=\frac {b \sqrt {-1+c x} \sqrt {1+c x}}{c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}-\frac {\operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d}+\frac {(2 b) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 c^3 d}\\ &=\frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d}+\frac {b \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d}-\frac {b \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^5 d}\\ &=\frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d}+\frac {b \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^5 d}-\frac {b \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{c^5 d}\\ &=\frac {11 b \sqrt {-1+c x} \sqrt {1+c x}}{9 c^5 d}+\frac {b x^2 \sqrt {-1+c x} \sqrt {1+c x}}{9 c^3 d}-\frac {x \left (a+b \cosh ^{-1}(c x)\right )}{c^4 d}-\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{3 c^2 d}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d}+\frac {b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{c^5 d}-\frac {b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{c^5 d}\\ \end {align*}
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Mathematica [A] time = 0.34, size = 227, normalized size = 1.44 \[ -\frac {6 a c^3 x^3+18 a c x+9 a \log (1-c x)-9 a \log (c x+1)+6 b c^3 x^3 \cosh ^{-1}(c x)-2 b c^2 x^2 \sqrt {c x-1} \sqrt {c x+1}+18 b \text {Li}_2\left (-e^{-\cosh ^{-1}(c x)}\right )+18 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )-18 b c x \sqrt {\frac {c x-1}{c x+1}}-18 b \sqrt {\frac {c x-1}{c x+1}}-4 b \sqrt {c x-1} \sqrt {c x+1}+18 b c x \cosh ^{-1}(c x)-9 b \cosh ^{-1}(c x)^2-18 b \cosh ^{-1}(c x) \log \left (e^{-\cosh ^{-1}(c x)}+1\right )+18 b \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )}{18 c^5 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}}{c^{2} d x^{2} - d}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 263, normalized size = 1.66 \[ -\frac {a \,x^{3}}{3 c^{2} d}-\frac {a x}{c^{4} d}-\frac {a \ln \left (c x -1\right )}{2 c^{5} d}+\frac {a \ln \left (c x +1\right )}{2 c^{5} d}+\frac {11 b \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{5} d}+\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}+\frac {b \,x^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{9 c^{3} d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) x^{3}}{3 c^{2} d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) x}{c^{4} d}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{c^{5} d} \]
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 \[ \frac {1}{72} \, {\left (4 \, c^{4} {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{8} d} - \frac {3 \, \log \left (c x + 1\right )}{c^{9} d} + \frac {3 \, \log \left (c x - 1\right )}{c^{9} d}\right )} + 36 \, c^{2} {\left (\frac {2 \, x}{c^{6} d} - \frac {\log \left (c x + 1\right )}{c^{7} d} + \frac {\log \left (c x - 1\right )}{c^{7} d}\right )} + 648 \, c \int \frac {x \log \left (c x - 1\right )}{12 \, {\left (c^{6} d x^{2} - c^{4} d\right )}}\,{d x} - \frac {3 \, {\left (4 \, {\left (2 \, c^{3} x^{3} + 6 \, c x - 3 \, \log \left (c x + 1\right ) + 3 \, \log \left (c x - 1\right )\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 3 \, \log \left (c x + 1\right )^{2} + 6 \, \log \left (c x + 1\right ) \log \left (c x - 1\right )\right )}}{c^{5} d} + 72 \, \int -\frac {2 \, c^{3} x^{3} + 6 \, c x - 3 \, \log \left (c x + 1\right ) + 3 \, \log \left (c x - 1\right )}{6 \, {\left (c^{7} d x^{3} - c^{5} d x + {\left (c^{6} d x^{2} - c^{4} d\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x} - 216 \, \int \frac {\log \left (c x - 1\right )}{12 \, {\left (c^{6} d x^{2} - c^{4} d\right )}}\,{d x}\right )} b - \frac {1}{6} \, a {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4} d} - \frac {3 \, \log \left (c x + 1\right )}{c^{5} d} + \frac {3 \, \log \left (c x - 1\right )}{c^{5} d}\right )} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{d-c^2\,d\,x^2} \,d x \]
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 \[ - \frac {\int \frac {a x^{4}}{c^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{2} x^{2} - 1}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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