Optimal. Leaf size=230 \[ \frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {15 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}+\frac {15 b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {7 b c}{8 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^3} \]
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Rubi [A] time = 0.24, antiderivative size = 230, normalized size of antiderivative = 1.00, number of steps used = 17, 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} \[ \frac {15 b c \text {PolyLog}\left (2,-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 b c \text {PolyLog}\left (2,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {15 c \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3}-\frac {7 b c}{8 d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b c}{12 d^3 (c x-1)^{3/2} (c x+1)^{3/2}}+\frac {b c \tan ^{-1}\left (\sqrt {c x-1} \sqrt {c x+1}\right )}{d^3} \]
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 )^3} \, dx &=-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\left (5 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^3}\\ &=-\frac {b c}{3 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {3 c+3 c^2 x}{x (-1+c x)^{3/2} (1+c x)^{5/2}} \, dx}{3 d^3}-\frac {\left (5 b c^3\right ) \int \frac {x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {\left (15 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d}\\ &=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}-\frac {(b c) \int \frac {1}{x (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^3}+\frac {\left (15 b c^3\right ) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac {\left (15 c^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{d-c^2 d x^2} \, dx}{8 d^2}\\ &=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \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^3}-\frac {(15 c) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {(b c) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{d^3}+\frac {(15 b c) \operatorname {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}-\frac {(15 b c) \operatorname {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{8 d^3}\\ &=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {(15 b c) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {(15 b c) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{8 d^3}+\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^3}\\ &=\frac {b c}{12 d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {a+b \cosh ^{-1}(c x)}{d^3 x \left (1-c^2 x^2\right )^2}+\frac {5 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {15 c^2 x \left (a+b \cosh ^{-1}(c x)\right )}{8 d^3 \left (1-c^2 x^2\right )}+\frac {b c \tan ^{-1}\left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{d^3}+\frac {15 c \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{4 d^3}+\frac {15 b c \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 b c \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )}{8 d^3}\\ \end {align*}
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Mathematica [A] time = 1.78, size = 362, normalized size = 1.57 \[ \frac {-\frac {84 a c^2 x}{c^2 x^2-1}+\frac {24 a c^2 x}{\left (c^2 x^2-1\right )^2}-90 a c \log (1-c x)+90 a c \log (c x+1)-\frac {96 a}{x}+\frac {96 b c \sqrt {c^2 x^2-1} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-45 b c \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (e^{\cosh ^{-1}(c x)}+1\right )\right )-4 \text {Li}_2\left (-e^{\cosh ^{-1}(c x)}\right )\right )+45 b c \left (\cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)-4 \log \left (1-e^{\cosh ^{-1}(c x)}\right )\right )-4 \text {Li}_2\left (e^{\cosh ^{-1}(c x)}\right )\right )-\frac {2 b c \left ((c x-2) \sqrt {c x-1} \sqrt {c x+1}-3 \cosh ^{-1}(c x)\right )}{(c x-1)^2}+\frac {2 b c \left (\sqrt {c x-1} \sqrt {c x+1} (c x+2)-3 \cosh ^{-1}(c x)\right )}{(c x+1)^2}+42 b c \left (\frac {\cosh ^{-1}(c x)}{1-c x}-\frac {1}{\sqrt {\frac {c x-1}{c x+1}}}\right )+42 b c \left (\sqrt {\frac {c x-1}{c x+1}}-\frac {\cosh ^{-1}(c x)}{c x+1}\right )-\frac {96 b \cosh ^{-1}(c x)}{x}}{96 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \operatorname {arcosh}\left (c x\right ) + a}{c^{6} d^{3} x^{8} - 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} - d^{3} x^{2}}, x\right ) \]
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 \[ \int -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 392, normalized size = 1.70 \[ -\frac {a}{d^{3} x}+\frac {c a}{16 d^{3} \left (c x -1\right )^{2}}-\frac {7 c a}{16 d^{3} \left (c x -1\right )}-\frac {15 c a \ln \left (c x -1\right )}{16 d^{3}}-\frac {c a}{16 d^{3} \left (c x +1\right )^{2}}-\frac {7 c a}{16 d^{3} \left (c x +1\right )}+\frac {15 c a \ln \left (c x +1\right )}{16 d^{3}}-\frac {15 b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{3}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {7 b \sqrt {c x +1}\, \sqrt {c x -1}\, c^{3} x^{2}}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {25 b \,\mathrm {arccosh}\left (c x \right ) c^{2} x}{8 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {23 c b \sqrt {c x +1}\, \sqrt {c x -1}}{24 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}-\frac {b \,\mathrm {arccosh}\left (c x \right )}{d^{3} x \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )}+\frac {2 c b \arctan \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{d^{3}}+\frac {15 c b \dilog \left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}}+\frac {15 c b \dilog \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}}+\frac {15 c b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8 d^{3}} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{x^2\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{8} - 3 c^{4} x^{6} + 3 c^{2} x^{4} - x^{2}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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