Integrand size = 19, antiderivative size = 75 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=-\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{4 d}-\frac {b e \text {arccosh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d} \]
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Time = 0.03 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5996, 12, 5883, 92, 54} \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {b e \text {arccosh}(c+d x)}{4 d}-\frac {b e \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{4 d} \]
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Rule 12
Rule 54
Rule 92
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int e x (a+b \text {arccosh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e \text {Subst}(\int x (a+b \text {arccosh}(x)) \, dx,x,c+d x)}{d} \\ & = \frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{2 d} \\ & = -\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d}-\frac {(b e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d} \\ & = -\frac {b e \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{4 d}-\frac {b e \text {arccosh}(c+d x)}{4 d}+\frac {e (c+d x)^2 (a+b \text {arccosh}(c+d x))}{2 d} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.08 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arccosh}(c+d x))-\frac {1}{4} b \left (\sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}+2 \text {arctanh}\left (\sqrt {\frac {-1+c+d x}{1+c+d x}}\right )\right )\right )}{d} \]
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Time = 0.02 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(100\) |
default | \(\frac {\frac {e a \left (d x +c \right )^{2}}{2}+e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(100\) |
parts | \(e a \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e b \left (\frac {\left (d x +c \right )^{2} \operatorname {arccosh}\left (d x +c \right )}{2}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right ) \sqrt {\left (d x +c \right )^{2}-1}+\ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )\right )}{4 \sqrt {\left (d x +c \right )^{2}-1}}\right )}{d}\) | \(101\) |
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none
Time = 0.28 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.47 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 \, a d^{2} e x^{2} + 4 \, a c d e x + {\left (2 \, b d^{2} e x^{2} + 4 \, b c d e x + {\left (2 \, b c^{2} - b\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b d e x + b c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{4 \, d} \]
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\[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=e \left (\int a c\, dx + \int a d x\, dx + \int b c \operatorname {acosh}{\left (c + d x \right )}\, dx + \int b d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (65) = 130\).
Time = 0.22 (sec) , antiderivative size = 203, normalized size of antiderivative = 2.71 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b d e + a c e x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c e}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (65) = 130\).
Time = 0.54 (sec) , antiderivative size = 245, normalized size of antiderivative = 3.27 \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{2} \, a d e x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c e + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b d e + a c e x \]
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Timed out. \[ \int (c e+d e x) (a+b \text {arccosh}(c+d x)) \, dx=\int \left (c\,e+d\,e\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
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