Integrand size = 23, antiderivative size = 84 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=-\frac {2 (a+b \text {arccosh}(c+d x))}{d e \sqrt {e (c+d x)}}+\frac {4 b \sqrt {1-c-d x} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{d e^{3/2} \sqrt {-1+c+d x}} \] Output:
(-2*a-2*b*arccosh(d*x+c))/d/e/(e*(d*x+c))^(1/2)+4*b*(-d*x-c+1)^(1/2)*Ellip ticF((e*(d*x+c))^(1/2)/e^(1/2),I)/d/e^(3/2)/(d*x+c-1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.11 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\frac {2 \left (-a-b \text {arccosh}(c+d x)+\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{d e \sqrt {e (c+d x)}} \] Input:
Integrate[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(3/2),x]
Output:
(2*(-a - b*ArcCosh[c + d*x] + (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hyperge ometric2F1[1/4, 1/2, 5/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d *x])))/(d*e*Sqrt[e*(c + d*x)])
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {6411, 6298, 127, 126}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6411 |
| \(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{(e (c+d x))^{3/2}}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle \frac {\frac {2 b \int \frac {1}{\sqrt {c+d x-1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)}{e}-\frac {2 (a+b \text {arccosh}(c+d x))}{e \sqrt {e (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 127 |
| \(\displaystyle \frac {\frac {2 b \sqrt {-c-d x+1} \int \frac {1}{\sqrt {-c-d x+1} \sqrt {e (c+d x)} \sqrt {c+d x+1}}d(c+d x)}{e \sqrt {c+d x-1}}-\frac {2 (a+b \text {arccosh}(c+d x))}{e \sqrt {e (c+d x)}}}{d}\) |
| \(\Big \downarrow \) 126 |
| \(\displaystyle \frac {\frac {4 b \sqrt {-c-d x+1} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e (c+d x)}}{\sqrt {e}}\right ),-1\right )}{e^{3/2} \sqrt {c+d x-1}}-\frac {2 (a+b \text {arccosh}(c+d x))}{e \sqrt {e (c+d x)}}}{d}\) |
Input:
Int[(a + b*ArcCosh[c + d*x])/(c*e + d*e*x)^(3/2),x]
Output:
((-2*(a + b*ArcCosh[c + d*x]))/(e*Sqrt[e*(c + d*x)]) + (4*b*Sqrt[1 - c - d *x]*EllipticF[ArcSin[Sqrt[e*(c + d*x)]/Sqrt[e]], -1])/(e^(3/2)*Sqrt[-1 + c + d*x]))/d
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[(2/(b*Sqrt[e]))*Rt[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]* Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && GtQ[c, 0] & & GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])
Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x _] :> Simp[Sqrt[1 + d*(x/c)]*(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x ])) Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x], x] /; Free Q[{b, c, d, e, f}, x] && !(GtQ[c, 0] && GtQ[e, 0])
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[((a_.) + ArcCosh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcCosh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 2.09 (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 | \(-\frac {2 a}{\sqrt {d e x +c e}\, d e}+\frac {2 b \left (-\frac {\operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )}{\sqrt {d e x +c e}}+\frac {2 \operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right ) \sqrt {-\frac {d e x +c e -e}{e}}}{e \sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\) | \(124\) |
Input:
int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x,method=_RETURNVERBOSE)
Output:
2/d/e*(-a/(d*e*x+c*e)^(1/2)+b*(-1/(d*e*x+c*e)^(1/2)*arccosh((d*e*x+c*e)/e) +2/e*EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)*((-d*e*x-c*e+e)/e)^(1/2)/ (-1/e)^(1/2)/(-(-d*e*x-c*e+e)/e)^(1/2)))
Time = 0.11 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.31 \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=-\frac {2 \, {\left (\sqrt {d e x + c e} b d^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {d e x + c e} a d^{2} - 2 \, \sqrt {d^{3} e} {\left (b d x + b c\right )} {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )}}{d^{4} e^{2} x + c d^{3} e^{2}} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="fricas")
Output:
-2*(sqrt(d*e*x + c*e)*b*d^2*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 - 1 )) + sqrt(d*e*x + c*e)*a*d^2 - 2*sqrt(d^3*e)*(b*d*x + b*c)*weierstrassPInv erse(4/d^2, 0, (d*x + c)/d))/(d^4*e^2*x + c*d^3*e^2)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c + d x \right )}}{\left (e \left (c + d x\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**(3/2),x)
Output:
Integral((a + b*acosh(c + d*x))/(e*(c + d*x))**(3/2), x)
Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{{\left (d e x + c e\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(3/2),x, algorithm="giac")
Output:
integrate((b*arccosh(d*x + c) + a)/(d*e*x + c*e)^(3/2), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(3/2),x)
Output:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(3/2), x)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{(c e+d e x)^{3/2}} \, dx=\frac {\sqrt {d x +c}\, \left (\int \frac {\mathit {acosh} \left (d x +c \right )}{\sqrt {d x +c}\, c +\sqrt {d x +c}\, d x}d x \right ) b d -2 a}{\sqrt {e}\, \sqrt {d x +c}\, d e} \] Input:
int((a+b*acosh(d*x+c))/(d*e*x+c*e)^(3/2),x)
Output:
(sqrt(c + d*x)*int(acosh(c + d*x)/(sqrt(c + d*x)*c + sqrt(c + d*x)*d*x),x) *b*d - 2*a)/(sqrt(e)*sqrt(c + d*x)*d*e)