Integrand size = 23, antiderivative size = 186 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4} \]
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Time = 0.25 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5996, 12, 5883, 5933, 5947, 4265, 2317, 2438, 30} \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {2 b \arctan \left (e^{\text {arccosh}(c+d x)}\right ) (a+b \text {arccosh}(c+d x))}{3 d e^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {b^2}{3 d e^4 (c+d x)} \]
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Rule 12
Rule 30
Rule 2317
Rule 2438
Rule 4265
Rule 5883
Rule 5933
Rule 5947
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^2}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {(a+b \text {arccosh}(x))^2}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {(2 b) \text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4} \\ & = \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e^4}-\frac {b^2 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,c+d x\right )}{3 d e^4} \\ & = \frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}(\int (a+b x) \text {sech}(x) \, dx,x,\text {arccosh}(c+d x))}{3 d e^4} \\ & = \frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c+d x)\right )}{3 d e^4}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c+d x)\right )}{3 d e^4} \\ & = \frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(c+d x)}\right )}{3 d e^4} \\ & = \frac {b^2}{3 d e^4 (c+d x)}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x} (a+b \text {arccosh}(c+d x))}{3 d e^4 (c+d x)^2}-\frac {(a+b \text {arccosh}(c+d x))^2}{3 d e^4 (c+d x)^3}+\frac {2 b (a+b \text {arccosh}(c+d x)) \arctan \left (e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}-\frac {i b^2 \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4}+\frac {i b^2 \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c+d x)}\right )}{3 d e^4} \\ \end{align*}
Time = 0.95 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {-\frac {a^2}{(c+d x)^3}+a b \left (\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x)}{(c+d x)^2}-\frac {2 \text {arccosh}(c+d x)}{(c+d x)^3}+2 \arctan \left (\tanh \left (\frac {1}{2} \text {arccosh}(c+d x)\right )\right )\right )+b^2 \left (\frac {1}{c+d x}+\frac {\sqrt {\frac {-1+c+d x}{1+c+d x}} (1+c+d x) \text {arccosh}(c+d x)}{(c+d x)^2}-\frac {\text {arccosh}(c+d x)^2}{(c+d x)^3}-i \text {arccosh}(c+d x) \log \left (1-i e^{-\text {arccosh}(c+d x)}\right )+i \text {arccosh}(c+d x) \log \left (1+i e^{-\text {arccosh}(c+d x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c+d x)}\right )+i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c+d x)}\right )\right )}{3 d e^4} \]
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Time = 0.94 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.69
method | result | size |
derivativedivides | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(314\) |
default | \(\frac {-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4}}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(314\) |
parts | \(-\frac {a^{2}}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b^{2} \left (-\frac {-\left (d x +c \right ) \operatorname {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+\operatorname {arccosh}\left (d x +c \right )^{2}-\left (d x +c \right )^{2}}{3 \left (d x +c \right )^{3}}-\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {arccosh}\left (d x +c \right ) \ln \left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}-\frac {i \operatorname {dilog}\left (1+i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}+\frac {i \operatorname {dilog}\left (1-i \left (d x +c +\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )\right )}{3}\right )}{e^{4} d}+\frac {2 a b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4} d}\) | \(319\) |
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\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\frac {\int \frac {a^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 a b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
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\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \text {arccosh}(c+d x))^2}{(c e+d e x)^4} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^2}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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