Integrand size = 23, antiderivative size = 127 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{9 d \sqrt {-1+c+d x}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {5996, 5883, 104, 12, 118, 117} \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{9 d \sqrt {c+d x-1}}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \sqrt {e (c+d x)}}{9 d} \]
[In]
[Out]
Rule 12
Rule 104
Rule 117
Rule 118
Rule 5883
Rule 5996
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \sqrt {e x} (a+b \text {arccosh}(x)) \, dx,x,c+d x\right )}{d} \\ & = \frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {(2 b) \text {Subst}\left (\int \frac {(e x)^{3/2}}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {(4 b) \text {Subst}\left (\int \frac {e^2}{2 \sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d e} \\ & = -\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {(2 b e) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {e x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d} \\ & = -\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {\left (2 b e \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 )}{9 d \sqrt {-1+c+d x}} \\ & = -\frac {4 b \sqrt {-1+c+d x} \sqrt {e (c+d x)} \sqrt {1+c+d x}}{9 d}+\frac {2 (e (c+d x))^{3/2} (a+b \text {arccosh}(c+d x))}{3 d e}-\frac {4 b \sqrt {e} \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{9 d \sqrt {-1+c+d x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.32 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\frac {\sqrt {e (c+d x)} \left (\frac {2}{3} (c+d x)^{3/2} (a+b \text {arccosh}(c+d x))-\frac {4 b \left (-1+c^2+2 c d x+d^2 x^2+\sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )\right )}{9 \sqrt {\frac {-1+c+d x}{c+d x}} \sqrt {1+c+d x}}\right )}{d \sqrt {c+d x}} \]
[In]
[Out]
Time = 1.26 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(194\) |
default | \(\frac {\frac {2 \left (d e x +c e \right )^{\frac {3}{2}} a}{3}+2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {-d e x -c e +e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(194\) |
parts | \(\frac {2 a \left (d e x +c e \right )^{\frac {3}{2}}}{3 d e}+\frac {2 b \left (\frac {\left (d e x +c e \right )^{\frac {3}{2}} \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{3}-\frac {2 \left (\sqrt {-\frac {1}{e}}\, \left (d e x +c e \right )^{\frac {5}{2}}+\sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {-\frac {d e x +c e -e}{e}}\, \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) e^{2}-\sqrt {-\frac {1}{e}}\, e^{2} \sqrt {d e x +c e}\right )}{9 e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e +e}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\) | \(199\) |
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.12 \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=-\frac {2 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} \sqrt {d e x + c e} b d^{2} - 3 \, {\left (b d^{3} x + b c d^{2}\right )} \sqrt {d e x + c e} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + 2 \, \sqrt {d^{3} e} b {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right ) - 3 \, {\left (a d^{3} x + a c d^{2}\right )} \sqrt {d e x + c e}\right )}}{9 \, d^{3}} \]
[In]
[Out]
\[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\int \sqrt {e \left (c + d x\right )} \left (a + b \operatorname {acosh}{\left (c + d x \right )}\right )\, dx \]
[In]
[Out]
Exception generated. \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
\[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\int { \sqrt {d e x + c e} {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {c e+d e x} (a+b \text {arccosh}(c+d x)) \, dx=\int \sqrt {c\,e+d\,e\,x}\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \]
[In]
[Out]