Integrand size = 21, antiderivative size = 99 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4} \]
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Time = 0.05 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883, 105, 94, 209} \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \arctan \left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{6 d e^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^4 (c+d x)^2} \]
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Rule 12
Rule 94
Rule 105
Rule 209
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{e^4 x^4} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{x^4} \, dx,x,c+d x\right )}{d e^4} \\ & = -\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {1}{\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}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{6 d e^4} \\ & = \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4} \\ & = \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^4 (c+d x)^2}-\frac {a+b \text {arccosh}(c+d x)}{3 d e^4 (c+d x)^3}+\frac {b \arctan \left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{6 d e^4} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {-\frac {2 a}{(c+d x)^3}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{(c+d x)^2}-\frac {2 b \text {arccosh}(c+d x)}{(c+d x)^3}+\frac {b \sqrt {-1+(c+d x)^2} \arctan \left (\sqrt {-1+(c+d x)^2}\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}}{6 d e^4} \]
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Time = 0.64 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(110\) |
default | \(\frac {-\frac {a}{3 e^{4} \left (d x +c \right )^{3}}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4}}}{d}\) | \(110\) |
parts | \(-\frac {a}{3 e^{4} \left (d x +c \right )^{3} d}+\frac {b \left (-\frac {\operatorname {arccosh}\left (d x +c \right )}{3 \left (d x +c \right )^{3}}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right ) \left (d x +c \right )^{2}-\sqrt {\left (d x +c \right )^{2}-1}\right )}{6 \left (d x +c \right )^{2} \sqrt {\left (d x +c \right )^{2}-1}}\right )}{e^{4} d}\) | \(112\) |
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Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (85) = 170\).
Time = 0.30 (sec) , antiderivative size = 276, normalized size of antiderivative = 2.79 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=-\frac {2 \, a c^{3} - 2 \, {\left (b c^{3} d^{3} x^{3} + 3 \, b c^{4} d^{2} x^{2} + 3 \, b c^{5} d x + b c^{6}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b c^{3} d x + b c^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{6 \, {\left (c^{3} d^{4} e^{4} x^{3} + 3 \, c^{4} d^{3} e^{4} x^{2} + 3 \, c^{5} d^{2} e^{4} x + c^{6} d e^{4}\right )}} \]
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\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\frac {\int \frac {a}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{e^{4}} \]
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\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^4} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^4} \,d x \]
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