Integrand size = 23, antiderivative size = 104 \[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))}{d e}+\frac {4 b \sqrt {1-c-d x} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {1-c-d x}}{\sqrt {2}}\right )\right |2\right )}{d e \sqrt {-1+c+d x} \sqrt {c+d x}} \] Output:
2*(e*(d*x+c))^(1/2)*(a+b*arccosh(d*x+c))/d/e+4*b*(-d*x-c+1)^(1/2)*(e*(d*x+ c))^(1/2)*EllipticE(1/2*(-d*x-c+1)^(1/2)*2^(1/2),2^(1/2))/d/e/(d*x+c-1)^(1 /2)/(d*x+c)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {e (c+d x)} \left (3 (a+b \text {arccosh}(c+d x))-\frac {2 b (c+d x) \sqrt {1-(c+d x)^2} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},(c+d x)^2\right )}{\sqrt {-1+c+d x} \sqrt {1+c+d x}}\right )}{3 d e} \] Input:
Integrate[(a + b*ArcCosh[c + d*x])/Sqrt[c*e + d*e*x],x]
Output:
(2*Sqrt[e*(c + d*x)]*(3*(a + b*ArcCosh[c + d*x]) - (2*b*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Hypergeometric2F1[1/2, 3/4, 7/4, (c + d*x)^2])/(Sqrt[-1 + c + d*x]*Sqrt[1 + c + d*x])))/(3*d*e)
Time = 0.31 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {6411, 6298, 124, 27, 123}
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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx\) |
\(\Big \downarrow \) 6411 |
\(\displaystyle \frac {\int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {e (c+d x)}}d(c+d x)}{d}\) |
\(\Big \downarrow \) 6298 |
\(\displaystyle \frac {\frac {2 \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))}{e}-\frac {2 b \int \frac {\sqrt {e (c+d x)}}{\sqrt {c+d x-1} \sqrt {c+d x+1}}d(c+d x)}{e}}{d}\) |
\(\Big \downarrow \) 124 |
\(\displaystyle \frac {\frac {2 \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))}{e}-\frac {\sqrt {2} b \sqrt {-c-d x+1} \sqrt {e (c+d x)} \int \frac {\sqrt {2} \sqrt {-c-d x}}{\sqrt {-c-d x+1} \sqrt {c+d x+1}}d(c+d x)}{e \sqrt {-c-d x} \sqrt {c+d x-1}}}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {2 \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))}{e}-\frac {2 b \sqrt {-c-d x+1} \sqrt {e (c+d x)} \int \frac {\sqrt {-c-d x}}{\sqrt {-c-d x+1} \sqrt {c+d x+1}}d(c+d x)}{e \sqrt {-c-d x} \sqrt {c+d x-1}}}{d}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {\frac {2 \sqrt {e (c+d x)} (a+b \text {arccosh}(c+d x))}{e}-\frac {4 b \sqrt {-c-d x+1} \sqrt {e (c+d x)} E\left (\left .\arcsin \left (\frac {\sqrt {c+d x+1}}{\sqrt {2}}\right )\right |2\right )}{e \sqrt {-c-d x} \sqrt {c+d x-1}}}{d}\) |
Input:
Int[(a + b*ArcCosh[c + d*x])/Sqrt[c*e + d*e*x],x]
Output:
((2*Sqrt[e*(c + d*x)]*(a + b*ArcCosh[c + d*x]))/e - (4*b*Sqrt[1 - c - d*x] *Sqrt[e*(c + d*x)]*EllipticE[ArcSin[Sqrt[1 + c + d*x]/Sqrt[2]], 2])/(e*Sqr t[-c - d*x]*Sqrt[-1 + c + d*x]))/d
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d *x]*Sqrt[b*((e + f*x)/(b*e - a*f))])) Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x /(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !(GtQ[b/(b*c - a*d), 0] && Gt Q[b/(b*e - a*f), 0]) && !LtQ[-(b*c - a*d)/d, 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]
Result contains complex when optimal does not.
Time = 2.06 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(138\) |
default | \(\frac {2 \sqrt {d e x +c e}\, a +2 b \left (\sqrt {d e x +c e}\, \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {\frac {-d e x -c e +e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {-\frac {-d e x -c e +e}{e}}}\right )}{d e}\) | \(138\) |
parts | \(\frac {2 a \sqrt {d e x +c e}}{d e}+\frac {2 b \left (\sqrt {d e x +c e}\, \operatorname {arccosh}\left (\frac {d e x +c e}{e}\right )-\frac {2 \left (\operatorname {EllipticF}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )-\operatorname {EllipticE}\left (\sqrt {d e x +c e}\, \sqrt {-\frac {1}{e}}, i\right )\right ) \sqrt {-\frac {d e x +c e -e}{e}}}{\sqrt {-\frac {1}{e}}\, \sqrt {\frac {d e x +c e -e}{e}}}\right )}{d e}\) | \(144\) |
Input:
int((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x,method=_RETURNVERBOSE)
Output:
2/d/e*((d*e*x+c*e)^(1/2)*a+b*((d*e*x+c*e)^(1/2)*arccosh((d*e*x+c*e)/e)-2*( EllipticF((d*e*x+c*e)^(1/2)*(-1/e)^(1/2),I)-EllipticE((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 = 0.12 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \, {\left (\sqrt {d e x + c e} b d \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + \sqrt {d e x + c e} a d + 2 \, \sqrt {d^{3} e} b {\rm weierstrassZeta}\left (\frac {4}{d^{2}}, 0, {\rm weierstrassPInverse}\left (\frac {4}{d^{2}}, 0, \frac {d x + c}{d}\right )\right )\right )}}{d^{2} e} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x, algorithm="fricas")
Output:
2*(sqrt(d*e*x + c*e)*b*d*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*sqrt(d^3*e)*b*weierstrassZeta(4/d^2, 0, weiers trassPInverse(4/d^2, 0, (d*x + c)/d)))/(d^2*e)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c + d x \right )}}{\sqrt {e \left (c + d x\right )}}\, dx \] Input:
integrate((a+b*acosh(d*x+c))/(d*e*x+c*e)**(1/2),x)
Output:
Integral((a + b*acosh(c + d*x))/sqrt(e*(c + d*x)), x)
Exception generated. \[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),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
\[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\int { \frac {b \operatorname {arcosh}\left (d x + c\right ) + a}{\sqrt {d e x + c e}} \,d x } \] Input:
integrate((a+b*arccosh(d*x+c))/(d*e*x+c*e)^(1/2),x, algorithm="giac")
Output:
integrate((b*arccosh(d*x + c) + a)/sqrt(d*e*x + c*e), x)
Timed out. \[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{\sqrt {c\,e+d\,e\,x}} \,d x \] Input:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(1/2),x)
Output:
int((a + b*acosh(c + d*x))/(c*e + d*e*x)^(1/2), x)
\[ \int \frac {a+b \text {arccosh}(c+d x)}{\sqrt {c e+d e x}} \, dx=\frac {2 \sqrt {d x +c}\, a +\left (\int \frac {\mathit {acosh} \left (d x +c \right )}{\sqrt {d x +c}}d x \right ) b d}{\sqrt {e}\, d} \] Input:
int((a+b*acosh(d*x+c))/(d*e*x+c*e)^(1/2),x)
Output:
(2*sqrt(c + d*x)*a + int(acosh(c + d*x)/sqrt(c + d*x),x)*b*d)/(sqrt(e)*d)