Optimal. Leaf size=200 \[ -c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} b c^3 d^2 x \sqrt {c x-1} \sqrt {c x+1}+b c^2 d^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac {b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \]
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Rubi [A] time = 0.22, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5729, 97, 12, 38, 52, 5727, 5660, 3718, 2190, 2279, 2391} \[ -b c^2 d^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} b c^3 d^2 x \sqrt {c x-1} \sqrt {c x+1}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac {b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 38
Rule 52
Rule 97
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5727
Rule 5729
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{2} \left (b c d^2\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x^2} \, dx\\ &=-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} \left (b c d^2\right ) \int 3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \, dx-\left (2 c^2 d^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx-\left (b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=-\frac {1}{2} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d^2\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{2} \left (3 b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}+\frac {1}{2} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-\left (4 c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{4} \left (3 b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (2 b c^2 d^2\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (b c^2 d^2\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-b c^2 d^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.27, size = 182, normalized size = 0.91 \[ \frac {d^2 \left (2 a c^4 x^4-8 a c^2 x^2 \log (x)-2 a-b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}+4 b c^2 x^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-4 b c^2 x^2 \cosh ^{-1}(c x)^2-2 b c^2 x^2 \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )+2 b \cosh ^{-1}(c x) \left (c^4 x^4-4 c^2 x^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )-1\right )+2 b c x \sqrt {c x-1} \sqrt {c x+1}\right )}{4 x^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arcosh}\left (c x\right )}{x^{3}}, x\right ) \]
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 = 0.56, size = 220, normalized size = 1.10 \[ \frac {c^{4} d^{2} a \,x^{2}}{2}-2 c^{2} d^{2} a \ln \left (c x \right )-\frac {d^{2} a}{2 x^{2}}+c^{2} d^{2} b \mathrm {arccosh}\left (c x \right )^{2}-\frac {b \,c^{3} d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}+\frac {c^{4} d^{2} b \,\mathrm {arccosh}\left (c x \right ) x^{2}}{2}-\frac {b \,c^{2} d^{2} \mathrm {arccosh}\left (c x \right )}{4}-\frac {d^{2} b \,c^{2}}{2}+\frac {b c \,d^{2} \sqrt {c x -1}\, \sqrt {c x +1}}{2 x}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right )}{2 x^{2}}-2 c^{2} d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-c^{2} d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) \]
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 \[ \frac {1}{2} \, a c^{4} d^{2} x^{2} - 2 \, a c^{2} d^{2} \log \relax (x) + \frac {1}{2} \, b d^{2} {\left (\frac {\sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\operatorname {arcosh}\left (c x\right )}{x^{2}}\right )} - \frac {a d^{2}}{2 \, x^{2}} + \int b c^{4} d^{2} x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - \frac {2 \, b c^{2} d^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \]
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 \[ d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \left (- \frac {2 a c^{2}}{x}\right )\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{4} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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