Optimal. Leaf size=171 \[ \frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac {b \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac {b \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.19, antiderivative size = 171, 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 = {5936, 5916,
5569, 4267, 2317, 2438, 39, 40} \begin {gather*} \frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^3}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac {2 b c x}{3 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c x}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5916
Rule 5936
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^3} \, dx &=\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {1}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{6 d^3}+\frac {(b c) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^3}+\frac {\int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac {b \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}-\frac {b \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^3}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac {b \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac {b \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac {b c x}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {a+b \cosh ^{-1}(c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \cosh ^{-1}(c x)}{2 d^3 \left (1-c^2 x^2\right )}+\frac {2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^3}+\frac {b \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}-\frac {b \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.88, size = 210, normalized size = 1.23 \begin {gather*} -\frac {-\frac {3 a}{\left (-1+c^2 x^2\right )^2}+\frac {6 a}{-1+c^2 x^2}-12 a \log (x)+6 a \log \left (1-c^2 x^2\right )+b \left (\frac {8 c x \sqrt {\frac {-1+c x}{1+c x}}}{-1+c x}-\frac {c x \left (\frac {-1+c x}{1+c x}\right )^{3/2}}{(-1+c x)^3}-\frac {3 \cosh ^{-1}(c x)}{\left (-1+c^2 x^2\right )^2}+\frac {6 \cosh ^{-1}(c x)}{-1+c^2 x^2}+12 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-12 \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+6 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )-6 \text {PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )\right )}{12 d^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(507\) vs.
\(2(190)=380\).
time = 7.32, size = 508, normalized size = 2.97
method | result | size |
derivativedivides | \(\frac {a \ln \left (c x \right )}{d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {5 a}{16 d^{3} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{2 d^{3}}+\frac {a}{16 d^{3} \left (c x +1\right )^{2}}+\frac {5 a}{16 d^{3} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{2 d^{3}}-\frac {2 b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b \,c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {4 b \,c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \,\mathrm {arccosh}\left (c x \right )}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}-\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{3}}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 d^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}\) | \(508\) |
default | \(\frac {a \ln \left (c x \right )}{d^{3}}+\frac {a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {5 a}{16 d^{3} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{2 d^{3}}+\frac {a}{16 d^{3} \left (c x +1\right )^{2}}+\frac {5 a}{16 d^{3} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{2 d^{3}}-\frac {2 b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{3}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b \,c^{4} x^{4}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \sqrt {c x +1}\, \sqrt {c x -1}\, c x}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {4 b \,c^{2} x^{2}}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {3 b \,\mathrm {arccosh}\left (c x \right )}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 b}{3 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}-\frac {b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}+\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{3}}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2 d^{3}}-\frac {b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}-\frac {b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}\) | \(508\) |
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^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{7} - 3 c^{4} x^{5} + 3 c^{2} x^{3} - x}\, dx}{d^{3}} \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\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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