Optimal. Leaf size=249 \[ \frac {3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^5 d^3}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {3 b}{8 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b (c x-1)^{3/2}}{12 c^5 d^3 (c x+1)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {c x-1} (c x+1)^{3/2}}+\frac {b x^3}{12 c^2 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rubi [A] time = 0.24, antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5750, 94, 89, 21, 37, 74, 5694, 4182, 2279, 2391} \[ \frac {3 b \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 c^5 d^3}+\frac {b x^3}{12 c^2 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b (c x-1)^{3/2}}{12 c^5 d^3 (c x+1)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {c x-1} (c x+1)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 37
Rule 74
Rule 89
Rule 94
Rule 2279
Rule 2391
Rule 4182
Rule 5694
Rule 5750
Rubi steps
\begin {align*} \int \frac {x^4 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {x^3}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 c d^3}-\frac {3 \int \frac {x^2 \left (a+b \cosh ^{-1}(c x)\right )}{\left (d-c^2 d x^2\right )^2} \, dx}{4 c^2 d}\\ &=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {(3 b) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 c^3 d^3}-\frac {b \int \frac {x^2}{(-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{4 c^2 d^3}+\frac {3 \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 c^4 d^2}\\ &=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {3 \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5 d^3}-\frac {b \int \frac {-c+c^2 x}{\sqrt {-1+c x} (1+c x)^{5/2}} \, dx}{4 c^5 d^3}\\ &=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5 d^3}-\frac {(3 b) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 c^5 d^3}-\frac {b \int \frac {\sqrt {-1+c x}}{(1+c x)^{5/2}} \, dx}{4 c^4 d^3}\\ &=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {(3 b) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}\\ &=\frac {b x^3}{12 c^2 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {b}{4 c^5 d^3 \sqrt {-1+c x} (1+c x)^{3/2}}-\frac {b (-1+c x)^{3/2}}{12 c^5 d^3 (1+c x)^{3/2}}+\frac {3 b}{8 c^5 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x^3 \left (a+b \cosh ^{-1}(c x)\right )}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {3 x \left (a+b \cosh ^{-1}(c x)\right )}{8 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {3 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 c^5 d^3}+\frac {3 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}-\frac {3 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 c^5 d^3}\\ \end {align*}
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Mathematica [A] time = 1.79, size = 287, normalized size = 1.15 \[ \frac {\frac {30 a c x}{c^2 x^2-1}+\frac {12 a c x}{\left (c^2 x^2-1\right )^2}-9 a \log (1-c x)+9 a \log (c x+1)+18 b \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )-18 b \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )+\frac {b \sqrt {c x-1} (c x+2)}{(c x+1)^{3/2}}-\frac {b (c x-2) \sqrt {c x+1}}{(c x-1)^{3/2}}+\frac {3 b \cosh ^{-1}(c x)}{(c x-1)^2}-\frac {3 b \cosh ^{-1}(c x)}{(c x+1)^2}-15 b \left (\frac {\cosh ^{-1}(c x)}{1-c x}-\frac {1}{\sqrt {\frac {c x-1}{c x+1}}}\right )-15 b \left (\sqrt {\frac {c x-1}{c x+1}}-\frac {\cosh ^{-1}(c x)}{c x+1}\right )+\frac {9}{2} b \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )-\frac {9}{2} b \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )}{48 c^5 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b x^{4} \operatorname {arcosh}\left (c x\right ) + a x^{4}}{c^{6} d^{3} x^{6} - 3 \, c^{4} d^{3} x^{4} + 3 \, c^{2} d^{3} x^{2} - d^{3}}, x\right ) \]
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 \[ \int -\frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (c^{2} d x^{2} - d\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.84, size = 383, normalized size = 1.54 \[ \frac {a}{16 c^{5} d^{3} \left (c x -1\right )^{2}}+\frac {5 a}{16 c^{5} d^{3} \left (c x -1\right )}-\frac {3 a \ln \left (c x -1\right )}{16 c^{5} d^{3}}-\frac {a}{16 c^{5} d^{3} \left (c x +1\right )^{2}}+\frac {5 a}{16 c^{5} d^{3} \left (c x +1\right )}+\frac {3 a \ln \left (c x +1\right )}{16 c^{5} d^{3}}+\frac {5 b \,\mathrm {arccosh}\left (c x \right ) x^{3}}{8 c^{2} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {5 b \sqrt {c x +1}\, \sqrt {c x -1}\, x^{2}}{8 c^{3} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) x}{8 c^{4} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {13 b \sqrt {c x +1}\, \sqrt {c x -1}}{24 c^{5} d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{5} d^{3}}+\frac {3 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{5} d^{3}}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{5} d^{3}}-\frac {3 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 c^{5} d^{3}} \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a x^{4}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b x^{4} \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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