Optimal. Leaf size=170 \[ \frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}+\frac {3 b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {b c}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2} \]
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Rubi [A] time = 0.18, antiderivative size = 170, 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 = {5746, 104, 21, 92, 205, 5689, 74, 5694, 4182, 2279, 2391} \[ \frac {3 b c \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 b c \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{d^2}-\frac {b c}{2 d^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 74
Rule 92
Rule 104
Rule 205
Rule 2279
Rule 2391
Rule 4182
Rule 5689
Rule 5694
Rule 5746
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c x)}{x^2 \left (d-c^2 d x^2\right )^2} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\left (3 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx-\frac {(b c) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2}\\ &=\frac {b c}{d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b \int \frac {c+c^2 x}{x \sqrt {-1+c x} (1+c x)^{3/2}} \, dx}{d^2}+\frac {\left (3 b c^3\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{2 d^2}+\frac {\left (3 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b c}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(3 c) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {(b c) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {3 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {(3 b c) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}-\frac {(3 b c) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{2 d^2}+\frac {\left (b c^2\right ) \operatorname {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {3 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {(3 b c) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {(3 b c) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{2 d^2 \left (1-c^2 x^2\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^2}+\frac {3 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{d^2}+\frac {3 b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.72, size = 283, normalized size = 1.66 \[ \frac {-\frac {2 a c^2 x}{c^2 x^2-1}-3 a c \log (1-c x)+3 a c \log (c x+1)-\frac {4 a}{x}+\frac {4 b c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b c^2 x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+6 b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )-6 b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )+b c \sqrt {\frac {c x-1}{c x+1}}+\frac {b c \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+\frac {b c \cosh ^{-1}(c x)}{1-c x}-\frac {b c \cosh ^{-1}(c x)}{c x+1}-\frac {4 b \cosh ^{-1}(c x)}{x}-6 b c \cosh ^{-1}(c x) \log \left (1-e^{\cosh ^{-1}(c x)}\right )+6 b c \cosh ^{-1}(c x) \log \left (e^{\cosh ^{-1}(c x)}+1\right )}{4 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{4} d^{2} x^{6} - 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 259, normalized size = 1.52 \[ -\frac {a}{d^{2} x}-\frac {c a}{4 d^{2} \left (c x -1\right )}-\frac {3 c a \ln \left (c x -1\right )}{4 d^{2}}-\frac {c a}{4 d^{2} \left (c x +1\right )}+\frac {3 c a \ln \left (c x +1\right )}{4 d^{2}}-\frac {3 b \,\mathrm {arccosh}\left (c x \right ) c^{2} x}{2 d^{2} \left (c^{2} x^{2}-1\right )}-\frac {c b \sqrt {c x +1}\, \sqrt {c x -1}}{2 d^{2} \left (c^{2} x^{2}-1\right )}+\frac {b \,\mathrm {arccosh}\left (c x \right )}{d^{2} x \left (c^{2} x^{2}-1\right )}+\frac {2 c b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{2}}+\frac {3 c b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 c b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}}+\frac {3 c b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{2 d^{2}} \]
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}{64} \, {\left (576 \, c^{5} \int \frac {x^{3} \log \left (c x - 1\right )}{8 \, {\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )}}\,{d x} - 24 \, c^{4} {\left (\frac {2 \, x}{c^{4} d^{2} x^{2} - c^{2} d^{2}} + \frac {\log \left (c x + 1\right )}{c^{3} d^{2}} - \frac {\log \left (c x - 1\right )}{c^{3} d^{2}}\right )} - 192 \, c^{4} \int \frac {x^{2} \log \left (c x - 1\right )}{8 \, {\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )}}\,{d x} + 9 \, {\left (c {\left (\frac {2}{c^{4} d^{2} x - c^{3} d^{2}} - \frac {\log \left (c x + 1\right )}{c^{3} d^{2}} + \frac {\log \left (c x - 1\right )}{c^{3} d^{2}}\right )} + \frac {4 \, \log \left (c x - 1\right )}{c^{4} d^{2} x^{2} - c^{2} d^{2}}\right )} c^{3} + 16 \, c^{2} {\left (\frac {2 \, x}{c^{2} d^{2} x^{2} - d^{2}} - \frac {\log \left (c x + 1\right )}{c d^{2}} + \frac {\log \left (c x - 1\right )}{c d^{2}}\right )} + 192 \, c^{2} \int \frac {\log \left (c x - 1\right )}{8 \, {\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )}}\,{d x} - \frac {4 \, {\left (3 \, {\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right )^{2} + 6 \, {\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right ) \log \left (c x - 1\right ) + 4 \, {\left (6 \, c^{2} x^{2} - 3 \, {\left (c^{3} x^{3} - c x\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{3} x^{3} - c x\right )} \log \left (c x - 1\right ) - 4\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )\right )}}{c^{2} d^{2} x^{3} - d^{2} x} + 64 \, \int -\frac {6 \, c^{3} x^{2} - 3 \, {\left (c^{4} x^{3} - c^{2} x\right )} \log \left (c x + 1\right ) + 3 \, {\left (c^{4} x^{3} - c^{2} x\right )} \log \left (c x - 1\right ) - 4 \, c}{4 \, {\left (c^{5} d^{2} x^{6} - 2 \, c^{3} d^{2} x^{4} + c d^{2} x^{2} + {\left (c^{4} d^{2} x^{5} - 2 \, c^{2} d^{2} x^{3} + d^{2} x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )}}\,{d x}\right )} b - \frac {1}{4} \, a {\left (\frac {2 \, {\left (3 \, c^{2} x^{2} - 2\right )}}{c^{2} d^{2} x^{3} - d^{2} x} - \frac {3 \, c \log \left (c x + 1\right )}{d^{2}} + \frac {3 \, c \log \left (c x - 1\right )}{d^{2}}\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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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