Integrand size = 23, antiderivative size = 84 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=-\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2} \sqrt {-1+c+d x}} \]
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Time = 0.07 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {5996, 5883, 118, 117} \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\frac {4 b \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2} \sqrt {c+d x-1}}-\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}} \]
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Rule 117
Rule 118
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a+b \text {arccosh}(x)}{(e x)^{3/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e} \\ & = -\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}}+\frac {\left (2 b \sqrt {1-c-d x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d e \sqrt {-1+c+d x}} \\ & = -\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2} \sqrt {-1+c+d x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\frac {2 \left (-a-b \text {arccosh}(c+d x)+\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{d e \sqrt {e (c+d x)}} \]
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Time = 1.23 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.42
method | result | size |
derivativedivides | \(\frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(119\) |
default | \(\frac {-\frac {2 a}{\sqrt {d e x +c e}}+2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(119\) |
parts | \(-\frac {2 a}{\sqrt {d e x +c e}\, d e}+\frac {2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {-\frac {d e x +c e -e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\) | \(124\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {d e x + c e} b d^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {d e x + c e} a d^{2} - 2 \, \sqrt {d^{3} e} {\left (b d x + b c\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{d^{4} e^{2} x + c d^{3} e^{2}} \]
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\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \]
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