Integrand size = 22, antiderivative size = 180 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c d^3} \]
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Time = 0.11 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5901, 5903, 4267, 2317, 2438, 75} \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {3 \text {arctanh}\left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{4 c d^3}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {3 b}{8 c d^3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b}{12 c d^3 (c x-1)^{3/2} (c x+1)^{3/2}} \]
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Rule 75
Rule 2317
Rule 2438
Rule 4267
Rule 5901
Rule 5903
Rubi steps \begin{align*} \text {integral}& = \frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{4 d^3}+\frac {3 \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^2} \, dx}{4 d} \\ & = \frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {(3 b c) \int \frac {x}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{8 d^3}+\frac {3 \int \frac {a+b \text {arccosh}(c x)}{d-c^2 d x^2} \, dx}{8 d^2} \\ & = \frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}-\frac {3 \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arccosh}(c x))}{8 c d^3} \\ & = \frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {(3 b) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 c d^3}-\frac {(3 b) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {(3 b) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {(3 b) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{8 c d^3} \\ & = \frac {b}{12 c d^3 (-1+c x)^{3/2} (1+c x)^{3/2}}-\frac {3 b}{8 c d^3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {x (a+b \text {arccosh}(c x))}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {3 x (a+b \text {arccosh}(c x))}{8 d^3 \left (1-c^2 x^2\right )}+\frac {3 (a+b \text {arccosh}(c x)) \text {arctanh}\left (e^{\text {arccosh}(c x)}\right )}{4 c d^3}+\frac {3 b \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )}{8 c d^3}-\frac {3 b \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )}{8 c d^3} \\ \end{align*}
Time = 0.80 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.76 \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\frac {\frac {4 a x}{\left (-1+c^2 x^2\right )^2}-\frac {6 a x}{-1+c^2 x^2}+\frac {b \left (\sqrt {-1+c x} \sqrt {1+c x} (2+c x)-3 \text {arccosh}(c x)\right )}{3 c (1+c x)^2}+\frac {b \left ((2-c x) \sqrt {-1+c x} \sqrt {1+c x}+3 \text {arccosh}(c x)\right )}{3 c (-1+c x)^2}+\frac {3 b \left (-\frac {1}{\sqrt {\frac {-1+c x}{1+c x}}}+\frac {\text {arccosh}(c x)}{1-c x}\right )}{c}+\frac {3 b \left (\sqrt {\frac {-1+c x}{1+c x}}-\frac {\text {arccosh}(c x)}{1+c x}\right )}{c}-\frac {3 a \log (1-c x)}{c}+\frac {3 a \log (1+c x)}{c}-\frac {3 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1+e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,-e^{\text {arccosh}(c x)}\right )\right )}{2 c}+\frac {3 b \left (\text {arccosh}(c x) \left (\text {arccosh}(c x)-4 \log \left (1-e^{\text {arccosh}(c x)}\right )\right )-4 \operatorname {PolyLog}\left (2,e^{\text {arccosh}(c x)}\right )\right )}{2 c}}{16 d^3} \]
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Time = 0.69 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(256\) |
default | \(\frac {-\frac {a \left (\frac {1}{16 \left (c x +1\right )^{2}}+\frac {3}{16 \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16}-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {3}{16 \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3}}}{c}\) | \(256\) |
parts | \(-\frac {a \left (\frac {1}{16 c \left (c x +1\right )^{2}}+\frac {3}{16 c \left (c x +1\right )}-\frac {3 \ln \left (c x +1\right )}{16 c}-\frac {1}{16 c \left (c x -1\right )^{2}}+\frac {3}{16 c \left (c x -1\right )}+\frac {3 \ln \left (c x -1\right )}{16 c}\right )}{d^{3}}-\frac {b \left (\frac {9 c^{3} x^{3} \operatorname {arccosh}\left (c x \right )+9 \sqrt {c x -1}\, \sqrt {c x +1}\, c^{2} x^{2}-15 c x \,\operatorname {arccosh}\left (c x \right )-11 \sqrt {c x -1}\, \sqrt {c x +1}}{24 c^{4} x^{4}-48 c^{2} x^{2}+24}+\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1-c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}+\frac {3 \operatorname {polylog}\left (2, c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \,\operatorname {arccosh}\left (c x \right ) \ln \left (1+c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}-\frac {3 \operatorname {polylog}\left (2, -c x -\sqrt {c x -1}\, \sqrt {c x +1}\right )}{8}\right )}{d^{3} c}\) | \(273\) |
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=- \frac {\int \frac {a}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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