Optimal. Leaf size=152 \[ -\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b c^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c^2 \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{d^2} \]
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Rubi [A]
time = 0.20, antiderivative size = 152, 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 = {5932, 5936,
5916, 5569, 4267, 2317, 2438, 39, 105, 12} \begin {gather*} \frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac {b c^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c^2 \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c}{2 d^2 x \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 39
Rule 105
Rule 2317
Rule 2438
Rule 4267
Rule 5569
Rule 5916
Rule 5932
Rule 5936
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\left (2 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {1}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {2 c^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2}+\frac {\left (2 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (d-c^2 d x^2\right )} \, dx}{d}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 x}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {\left (2 c^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac {\left (b c^3\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}-\frac {\left (4 c^2\right ) \text {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}-\frac {\left (2 b c^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {\left (b c^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2}\\ &=-\frac {b c}{2 d^2 x \sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^2 \left (a+b \cosh ^{-1}(c x)\right )}{d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{2 d^2 x^2 \left (1-c^2 x^2\right )}+\frac {4 c^2 \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}+\frac {b c^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{d^2}-\frac {b c^2 \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{d^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(319\) vs. \(2(152)=304\).
time = 0.33, size = 319, normalized size = 2.10 \begin {gather*} \frac {a-2 a c^2 x^2-b c x \sqrt {\frac {-1+c x}{1+c x}}-b c^2 x^2 \sqrt {\frac {-1+c x}{1+c x}}+b \cosh ^{-1}(c x)-2 b c^2 x^2 \cosh ^{-1}(c x)+4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-4 b c^2 x^2 \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+4 b c^4 x^4 \cosh ^{-1}(c x) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )-4 a c^2 x^2 \log (x)+4 a c^4 x^4 \log (x)+2 a c^2 x^2 \log \left (1-c^2 x^2\right )-2 a c^4 x^4 \log \left (1-c^2 x^2\right )-2 b c^2 x^2 \left (-1+c^2 x^2\right ) \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )+2 b c^2 x^2 \left (-1+c^2 x^2\right ) \text {PolyLog}\left (2,e^{-2 \cosh ^{-1}(c x)}\right )}{2 d^2 x^2 \left (-1+c^2 x^2\right )} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 5.91, size = 347, normalized size = 2.28
method | result | size |
derivativedivides | \(c^{2} \left (-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{d^{2}}+\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{d^{2}}-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 a \ln \left (c x \right )}{d^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}+\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}\right )\) | \(347\) |
default | \(c^{2} \left (-\frac {a}{4 d^{2} \left (c x -1\right )}-\frac {a \ln \left (c x -1\right )}{d^{2}}+\frac {a}{4 d^{2} \left (c x +1\right )}-\frac {a \ln \left (c x +1\right )}{d^{2}}-\frac {a}{2 d^{2} c^{2} x^{2}}+\frac {2 a \ln \left (c x \right )}{d^{2}}-\frac {b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d^{2} \left (c^{2} x^{2}-1\right ) c x}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{2 d^{2} \left (c^{2} x^{2}-1\right ) c^{2} x^{2}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 b \polylog \left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}+\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}+\frac {b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{d^{2}}-\frac {2 b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}-\frac {2 b \polylog \left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}\right )\) | \(347\) |
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^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{7} - 2 c^{2} x^{5} + x^{3}}\, dx}{d^{2}} \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^3\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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