Integrand size = 21, antiderivative size = 97 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{9 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d} \]
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Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5996, 12, 5883, 102, 75} \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d}-\frac {b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{9 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{9 d} \]
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Rule 12
Rule 75
Rule 102
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int e^2 x^2 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int x^2 (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d} \\ & = -\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{9 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arccosh}(c+d x))}{3 d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.73 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {e^2 \left (-\frac {1}{9} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (2+c^2+2 c d x+d^2 x^2\right )+\frac {1}{3} (c+d x)^3 (a+b \text {arccosh}(c+d x))\right )}{d} \]
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Time = 0.60 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {a \,e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
default | \(\frac {\frac {a \,e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
parts | \(\frac {a \,e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b \left (\frac {\left (d x +c \right )^{3} \operatorname {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(69\) |
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Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.73 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {3 \, a d^{3} e^{2} x^{3} + 9 \, a c d^{2} e^{2} x^{2} + 9 \, a c^{2} d e^{2} x + 3 \, {\left (b d^{3} e^{2} x^{3} + 3 \, b c d^{2} e^{2} x^{2} + 3 \, b c^{2} d e^{2} x + b c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (b d^{2} e^{2} x^{2} + 2 \, b c d e^{2} x + {\left (b c^{2} + 2 \, b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{9 \, d} \]
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\[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=e^{2} \left (\int a c^{2}\, dx + \int a d^{2} x^{2}\, dx + \int b c^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 a c d x\, dx + \int b d^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )}\, dx + \int 2 b c d x \operatorname {acosh}{\left (c + d x \right )}\, dx\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (83) = 166\).
Time = 0.22 (sec) , antiderivative size = 449, normalized size of antiderivative = 4.63 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} e^{2} x + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c^{2} e^{2}}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (83) = 166\).
Time = 0.69 (sec) , antiderivative size = 419, normalized size of antiderivative = 4.32 \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b d^{2} e^{2} + a c^{2} e^{2} x \]
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Timed out. \[ \int (c e+d e x)^2 (a+b \text {arccosh}(c+d x)) \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
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